This is a preprint of: “Limit cycles of continuous piecewise differential systems formed by linear and quadratic isochronous centers II”, Bilal Ghermoul, Jaume Llibre, Tayeb Salhi, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 32(6), 2250091 (33 pages), 2022. DOI: [10.1142/S0218127422500912] LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS FORMED BY LINEAR AND QUADRATIC ISOCHRONOUS CENTERS II BILAL GHERMOUL1 , JAUME LLIBRE2 AND TAYEB SALHI1 Abstract. We study the crossing periodic orbits and limit cycles of the planar continuous piecewise differ- ential systems separated by the straight-line x = 0 having in x > 0 the general quadratic isochronous center ẋ = −y + x2 , ẏ = x(1 + y) after an affine transformation, and in x < 0 an arbitrary quadratic isochronous center except for the quadratic isochronous center ẋ = −y + x2 − y 2 , ẏ = x(1 + 2y) which has been studied in [4]. For these continuous, piecewise differential systems the upper bound of crossing limit cycles is 2, and there are realized examples having one crossing limit cycle. 1. Introduction In the qualitative theory of planar differential systems a limit cycle is an isolated periodic solution in the set of all periodic solutions, which remained the most sought solutions when modeling physical systems in the plane. As far as we known the notion of limit cycle appeared in the year 1885 in the work of Poincaré [13]. Most of the early examples in the theory of limit cycles in planar differential systems were commonly related to practical problems with mechanical and electronic systems, but periodic behavior appears in all branches of the sciences. To determine the existence or non–existence of limit cycles is one of the more difficult objects in the qualitative theory of planar differential equations. A large amount of references deals with the subject of limit cycles, many of them motivated for the famous Hilbert’s 16th problem, see for details [5, 7, 8]. Since 1930’s the study of the limit cycles also became important in the continuous and discontinuous piecewise differential systems separated by a straight line, due to their applications to mechanics, electrical circuits, ... see for instance the books [1, 2, 14] and the references therein. As usual a center p of a planar differential system is a singular point for which there is a neighborhood U such that U \ {p} is filled with periodic orbits. When all the periodic orbits surrounding a center have the same period this center is called isochronous. The centers started to be studied by Poincaré [12] and Dulac [3], but the notion of isochronocity goes back to Huygens [6] in 1673. In this paper we consider continuous piecewise differential systems separated by the straight line x = 0 having in x ≤ 0 and in x ≥ 0 quadratic isochronous centers, and we want to study the non–existence, and the existence of crossing periodic orbits and of crossing limit cycles, and in this last case we want also to know the maximum number of crossing limit cycles for these systems. Here a crossing periodic orbit or a crossing limit cycle is a periodic orbit or a limit cycle which intersects exactly in two points the line of separation x = 0. The continuity of a piecewise differential system separated by the straight line x = 0 formed by two centers means that the two vector fields defined by these two quadratic systems with isochronous centbbers coincide on the line of separation x = 0. So a continuous piecewise differential system is a continuous differential system in R2 and is an analytic differential system in R2 \ {x = 0}. 2010 Mathematics Subject Classification. Primary 34C05, 34A34. Key words and phrases. Limit cycles, isochronous quadratic centers, continuous piecewise linear differential systems, first integrals. 1 LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 2 1.1. Quadratic isochronous centers. We consider the quadratic polynomial differential systems having an isochronous center. This kind of centers were classified by Loud in the paper [11]. Those systems after an affine change of coordinates become one of the following four systems: ẋ = − y + x2 − y 2 , ẏ =x(1 + 2y), (1) {nle1} 2 ẋ = − y + x , ẏ =x(1 + y), (2) {nle2} 2   4x 16 ẋ = − y − , ẏ =x 1 − y , (3) {nle3} 3 3   16 4 8 ẋ = − y + x2 − y 2 , ẏ =x 1 + y . (4) {nle4} 3 3 3 We are interested in the general expressions of the quadratic isochronous centers. So we transform their normal forms (1), (2), (3) and (4) through the following general affine change of variables (x, y) → (a1 x + b1 y + c1 , a2 x + b2 y + c2 ), (5) {achv} with a1 b2 − a2 b1 6= 0. (6) {el} Generalized isochronous system (1). Using the change of variables (5) the quadratic system (1) becomes ẋ = ( − b2 c21 + b1 c1 + 2b1 c2 c1 + b2 c22 + b2 c2 + (2a2 b1 c1 + 2a1 b1 c2 − 2a1 b2 c1 + 2a2 b2 c2 + a1 b1 + a2 b2 ) x   + 2b21 c2 + 2b22 c2 + b21 + b22 y + 2a2 b21 + 2a2 b22 xy    + a21 (−b2 ) + 2a2 a1 b1 + a22 b2 x2 + b32 + b21 b2 y 2 /(a2 b1 − a1 b2 ),  (7) {gnle1} ẏ = ( a2 c21 − a1 c1 − 2a1 c2 c1 − a2 c22 − a2 c2 + −2a21 c2 − 2a22 c2 − a21 − a22 x + (2a2 b1 c1 − 2a1 b1 c2 − 2a1 b2 c1 − 2a2 b2 c2 − a1 b1 − a2 b2 ) y   + −2a21 b2 − 2a22 b2 xy + −a32 − a21 a2 x2   + a2 b21 − 2a1 b2 b1 − a2 b22 y 2 /(a2 b1 − a1 b2 ). Since (x2 + y 2 )/(2y + 1) is a first integral of system (1), doing to it the change of variables (5) we get the following first integral of the generalized isochronous quadratic system (7) (a1 x + b1 y + c1 ) 2 + (a2 x + b2 y + c2 ) 2 H2 (x, y) = . 2 (a2 x + b2 y + c2 ) + 1 Generalized isochronous system (2). System (2) is equivalent to the following generalized isochronous system after the linear change of variables (5) ẋ = ( b2 c21 − b1 c1 − b1 c2 c1 − b2 c2 + (−a2 b1 c1 − a1 b1 c2 + 2a1 b2 c1 − a1 b1 − a2 b2 ) x   + b21 (−c2 ) + b2 b1 c1 − b21 − b22 y + a1 b1 b2 − a2 b21 xy   + a21 b2 − a1 a2 b1 x2 /(a1 b2 − a2 b1 ), (8) {gnle2}  ẏ = ( a2 c21 − a1 c1 − a1 c2 c1 − a2 c2 + a21 (−c2 ) + a2 a1 c1 − a21 − a22 x + (2a2 b1 c1 − a1 b1 c2 − a1 b2 c1 − a1 b1 − a2 b2 ) y    + a1 a2 b1 − a21 b2 xy + a2 b21 − a1 b1 b2 y 2 /(a1 b2 − a2 b1 ). The quadratic system (2) has the first integral (x2 + y 2 )/(y + 1)2 . Therefore a first integral of system (8) is 2 2 (a1 x + b1 y + c1 ) + (a2 x + b2 y + c2 ) H3 (x, y) = 2 . (a2 x + b2 y + c2 + 1) LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 3 Generalized isochronous system (3). The quadratic system (3) is equivalent to the following generalized quadratic system after the linear change of variables (5) ẋ = ( 4b2 c21 + 3b1 c1 − 16b1 c2 c1 + 3b2 c2 + (−16a2 b1 c1 − 16a1 b1 c2 + 8a1 b2 c1 + 3a1 b1 + 3a2 b2 ) x   + −16b21 c2 − 8b2 b1 c1 + 3b21 + 3b22 y + −16a2 b21 − 8a1 b2 b1 xy   + 4a21 b2 − 16a1 a2 b1 x2 − 12b21 b2 y 2 /3 (a2 b1 − a1 b2 ) , ẏ = ( − 4a2 c21 − 3a1 c1 + 16a1 c2 c1 − 3a2 c2 (9) {gnle3}  + 16a21 c2 + 8a2 a1 c1 − 3a21 − 3a22 x + (−8a2 b1 c1 + 16a1 b1 c2 + 16a1 b2 c1 − 3a1 b1 − 3a2 b2 ) y  + 16a21 b2 + 8a2 a1 b1 xy + 12a21 a2 x2   + 16a1 b1 b2 − 4a2 b21 y 2 /3 (a2 b1 − a1 b2 ) . Since (32x2 − 24y + 9)2 /(3 − 16y) is a first integral of system (9), then a first integral of system (9) is  32 (a1 x + b1 y + c1 ) 2 − 24 (a2 x + b2 y + c2 ) + 9 2 H4 (x, y) = . 3 − 16 (a2 x + b2 y + c2 ) Generalized isochronous system (4). Doing the affine change of variables (5) the generalized isochronous system for the quadratic system (4) is ẋ = ( − 16b2 c21 + 3b1 c1 + 8b1 c2 c1 + 4b2 c22 + 3b2 c2 + (8a2 b1 c1 + 8a1 b1 c2 − 32a1 b2 c1 + 8a2 b2 c2 + 3a1 b1 + 3a2 b2 ) x  + 8b21 c2 − 24b2 b1 c1 + 8b22 c2 + 3b21 + 3b22 y   + 8a2 b21 − 24a1 b2 b1 + 8a2 b22 xy + −16a21 b2 + 8a2 a1 b1 + 4a22 b2 x2   + 4b32 − 8b21 b2 y 2 /3 (a2 b1 − a1 b2 ) , (10) {gnle4} ẏ = ( 16a2 c21 − 3a1 c1 − 8a1 c2 c1 − 4a2 c22 − 3a2 c2  + −8a21 c2 + 24a2 a1 c1 − 8a22 c2 − 3a21 − 3a22 x + (32a2 b1 c1 − 8a1 b1 c2 − 8a1 b2 c1 − 8a2 b2 c2 − 3a1 b1 − 3a2 b2 ) y   + −8a21 b2 + 24a2 a1 b1 − 8a22 b2 xy + 8a21 a2 − 4a32 x2   + 16a2 b21 − 8a1 b2 b1 − 4a2 b22 y 2 /3 (a2 b1 − a1 b2 ) . The quadratic system (4) has the first integral (−256x2 + 128y 2 + 96y + 9)/(8y + 3)4 , which gives the following first integral for system (10) 2 −256 (a1 x + b1 y + c1 ) 2 + 128 (a2 x + b2 y + c2 ) + 96 (a2 x + b2 y + c2 ) + 9 H5 (x, y) = 4 . (8 (a2 x + b2 y + c2 ) + 3) 1.2. Statement of the main results. In what follows we characterize the existence and non-existence of crossing periodic orbits and crossing limit cycles for continuous piecewise linear differential systems separated by one straight line formed by two quadratic isochronous centers. {thm2} Theorem 1. The following statements hold for the continuous piecewise differential systems formed by two generalized isochronous quadratic centers separated by the straight line x = 0. (a) If the generalized centers are (1) and (1), then the piecewise differential systems can have crossing periodic orbits but they cannot have crossing limit cycles. (b) If the generalized centers are (1) and (2), then the piecewise differential systems can have crossing periodic orbits but they cannot have crossing limit cycles. (c) If the generalized centers are (1) and (3) then the piecewise differential systems can have at most one limit cycle. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 4 (d) If the generalized centers are (1) and (4) then the piecewise differential systems can have at most one limit cycle. (e) If the generalized centers are (2) and (2) then the piecewise differential systems have no crossing periodic orbits, and consequently no crossing limit cycles. (f ) If the generalized centers are (2) and (3) then the piecewise differential systems have no limit cycles. (g) If the generalized centers are (2) and (4) then the piecewise differential systems can have at most two limit cycles, and there are systems in this case with exactly one limit cycle. The first four statement of Theorem 1 were proved in [4], here we prove all the other statements of Theorem 1. 2. Proof of Theorem 1 {s3} In what follows we consider a continuous piecewise differential system formed by two generalized isochronous systems (i) and (j) separated by the straight line x = 0, where i, j ∈ {8, 9, 10}. In the first generalized systems (i) we rename the parameters a2 , b2 and c2 by α1 , β1 and γ1 , respectively; and in the second generalized systems (j) we rename the parameters a1 , b1 , c1 , a2 , b2 and c2 by a2 , b2 , c2 , α2 , β2 and γ2 , respectively. Doing this condition (6) becomes α1 b1 − a1 β1 6= 0 and α2 b2 − a2 β2 6= 0, (11) {ell} for system (i) and (j) respectively. It the piecewise differential system (i)-(j) has a crossing periodic orbit intersecting the line of separation in the points (0, yi ) for i = 1, 2, these two points must satisfy the following algebraic system Hi (0, y1 ) − Hi (0, y2 ) = 0, Hj (0, y1 ) − Hj (0, y2 ) = 0, (12) {eij} where Hi and Hj are first integrals of the generalized systems (i) and (j). Proof of statement (e) of Theorem 1. Consider the continuous piecewise differential system formed by two generalized isochronous quadratic systems (8) separated by the straight line x = 0. Before getting the conditions for which the piecewise differential systems formed by these two systems (8) be continuous we provide an upper bound for the maximum number of limit cycles which these systems can exhibit. Then we must solve equations (12) for finding this upper boound. For our piecewise differential system equations (12) become (y1 − y2 )(2b1 c1 γ12 + 4b1 c1 γ1 − 2b1 β12 c1 y1 y2 + 2β1 γ12 + 2β1 γ1 + 2b1 c1 + 2b21 β1 γ1 y1 y2 +2b21 β1 y1 y2 + b21 γ12 y1 + b21 γ12 y2 + 2b21 γ1 y1 + 2b21 γ1 y2 + b21 y1 + b21 y2 − 2β1 c21 γ1 − 2β1 c21 −β12 c21 y1 − β12 c21 y2 + 2β12 γ1 y1 + 2β12 γ1 y2 + 2β13 y1 y2 + β12 y1 + β12 y2 )/[(γ1 + β1 y1 + 1)2 (γ1 + β1 y2 + 1)2 ] = 0, (13) {e4} (y1 − y2 )(2b2 c2 γ22 + 4b2 c2 γ2 − 2b2 β22 c2 y1 y2 + 2β2 γ22 + 2β2 γ2 + 2b2 c2 + 2b22 β2 γ2 y1 y2 +2b22 β2 y1 y2 + b22 γ22 y1 + b22 γ22 y2 + 2b22 γ2 y1 + 2b22 γ2 y2 + b22 y1 + b22 y2 − 2β2 c22 γ2 − 2β2 c22 −β22 c22 y1 − β22 c22 y2 + 2β22 γ2 y1 + 2β22 γ2 y2 + 2β23 y1 y2 + β22 y1 + β22 y2 )/[(γ2 + β2 y1 + 1)2 (γ2 + β2 y2 + 1)2 ] = 0, solving the first equation of (13) with respect to y1 we obtain −2b1 c1 (γ1 + 1)2 − b21 (γ1 + 1)2 y2 + β1 (c21 (2γ1 + β1 y2 + 2) − 2γ1 (γ1 + 1) − y2 (2β1 γ1 + β1 )) y1 = . −2β12 b1 c1 y2 + b21 (γ1 + 1)(γ1 + 2β1 y2 + 1) + β12 (−c21 + 2γ1 + 2β1 y2 + 1) Replacing this value of y1 into the second equation of (13) and solving it with respect to y2 , we get a unique solution solution excluding the trivial one y1 = y2 . Replacing this value of y2 into the expression of y1 , we LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 5 conclude that we have at most 1 pair (y1 , y2 ) given by y1 = [−b2 c2 γ22 β13 − β2 γ22 β13 − b2 c2 β13 + c22 β2 β13 − 2b2 c2 γ2 β13 + c22 β2 γ2 β13 − β2 γ2 β13 +b1 b2 c1 c2 γ22 β12 + b1 c1 β2 γ22 β12 + b1 b2 c1 c2 β12 − b1 c1 c22 β2 β12 + 2b1 b2 c1 c2 γ2 β12 − b1 c1 c22 β2 γ2 β12 +b1 c1 β2 γ2 β12 − c21 β23 β1 + b2 c21 c2 β22 β1 + β23 γ12 β1 − b2 c2 β22 γ12 β1 + b22 β2 γ12 β1 − b21 b2 c2 γ22 β1 −b21 β2 γ22 β1 − b21 b2 c2 γ1 γ22 β1 − b21 β2 γ1 γ22 β1 − b21 b2 c2 β1 − b22 c21 β2 β1 + b21 c22 β2 β1 − c21 β23 γ1 β1 +β23 γ1 β1 + b2 c21 c2 β22 γ1 β1 − b2 c2 β22 γ1 β1 − b21 b2 c2 γ1 β1 + b22 β2 γ1 β1 − b22 c21 β2 γ1 β1 + b21 c22 β2 γ1 β1 +b22 β2 γ12 γ2 β1 − 2b21 b2 c2 γ2 β1 − b21 β2 γ2 β1 − b22 c21 β2 γ2 β1 + b21 c22 β2 γ2 β1 − 2b21 b2 c2 γ1 γ2 β1 −b21 β2 γ1 γ2 β1 + b22 β2 γ1 γ2 β1 − b22 c21 β2 γ1 γ2 β1 + b21 c22 β2 γ1 γ2 β1 + b1 c1 β23 − b1 b2 c1 c2 β22 + b1 c1 β23 γ12 −b1 b2 c1 c2 β22 γ12 + b1 b22 c1 β2 γ12 + [(β2 (γ1 + 1) − β1 (γ2 + 1))((γ1 + 1)(γ2 + 1)b21 − c1 (γ2 β1 + β1 +β2 + β2 γ1 )b1 + β1 (β2 c21 + β1 − β2 γ1 + β1 γ2 ))((γ1 + 1)(γ2 + 1)b22 − c2 (γ2 β1 + β1 + β2 (14) {e5} +β2 γ1 )b2 + β2 (β1 c22 + β2 + β2 γ1 − β1 γ2 ))((γ1 + 1)(b2 c2 (γ2 + 1) + β2 (γ2 − c22 ))b21 −c1 ((γ1 + 1)(γ2 + 1)b22 + c2 (β1 (γ2 + 1) − β2 (γ1 + 1))b2 + β2 (−c22 β1 + β2 + β2 γ1 + β1 γ2 ))b1 +β1 ((c21 − γ1 )(γ2 + 1)b22 + c2 (−β2 c21 + β1 + β2 γ1 + β1 γ2 )b2 + β2 (β2 c21 − c22 β1 − β2 γ1 +β1 γ2 )))]1/2 + b1 b22 c1 β2 + 2b1 c1 β23 γ1 − 2b1 b2 c1 c2 β22 γ1 + 2b1 b22 c1 β2 γ1 + b1 b22 c1 β2 γ12 γ2 +b1 b22 c1 β2 γ2 + 2b1 b22 c1 β2 γ1 γ2 ]/[b21 (−(γ1 + 1))(b22 (−(γ2 + 1))(β1 (γ2 + 1) − β2 (γ1 + 1)) −b2 β22 c2 (γ1 + 1) + β22 (β2 (γ1 + 1) + β1 (c22 − 2γ2 − 1))) − b1 β12 c1 (b22 (γ2 + 1)2 +β22 (−c22 + 2γ2 + 1)) + β12 (b22 (γ2 + 1)(β1 (γ2 + 1) + β2 (c21 − 2γ1 − 1)) + β22 b2 c2 (−c21 + 2γ1 + 1) +β22 (β2 (c21 − 2γ1 − 1) + β1 (−c22 + 2γ2 + 1)))] and y2 = −[b2 c2 γ22 β13 + β2 γ22 β13 + b2 c2 β13 − c22 β2 β13 + 2b2 c2 γ2 β13 − c22 β2 γ2 β13 + β2 γ2 β13 −b1 b2 c1 c2 γ22 β12 − b1 c1 β2 γ22 β12 − b1 b2 c1 c2 β12 + b1 c1 c22 β2 β12 − 2b1 b2 c1 c2 γ2 β12 + b1 c1 c22 β2 γ2 β12 −b1 c1 β2 γ2 β12 + c21 β23 β1 − b2 c21 c2 β22 β1 − β23 γ12 β1 + b2 c2 β22 γ12 β1 − b22 β2 γ12 β1 + b21 b2 c2 γ22 β1 +b21 β2 γ22 β1 + b21 b2 c2 γ1 γ22 β1 + b21 β2 γ1 γ22 β1 + b21 b2 c2 β1 + b22 c21 β2 β1 − b21 c22 β2 β1 + c21 β23 γ1 β1 −β23 γ1 β1 − b2 c21 c2 β22 γ1 β1 + b2 c2 β22 γ1 β1 + b21 b2 c2 γ1 β1 − b22 β2 γ1 β1 + b22 c21 β2 γ1 β1 − b21 c22 β2 γ1 β1 −b22 β2 γ12 γ2 β1 + 2b21 b2 c2 γ2 β1 + b21 β2 γ2 β1 + b22 c21 β2 γ2 β1 − b21 c22 β2 γ2 β1 + 2b21 b2 c2 γ1 γ2 β1 +b21 β2 γ1 γ2 β1 − b22 β2 γ1 γ2 β1 + b22 c21 β2 γ1 γ2 β1 − b21 c22 β2 γ1 γ2 β1 − b1 c1 β23 + b1 b2 c1 c2 β22 − b1 c1 β23 γ12 +b1 b2 c1 c2 β22 γ12 − b1 b22 c1 β2 γ12 + [(β2 (γ1 + 1) − β1 (γ2 + 1))((γ1 + 1)(γ2 + 1)b21 −c1 (γ2 β1 + β1 + β2 + β2 γ1 )b1 + β1 (β2 c21 + β1 − β2 γ1 + β1 γ2 ))((γ1 + 1)(γ2 + 1)b22 (15) {e6} −c2 (γ2 β1 + β1 + β2 + β2 γ1 )b2 + β2 (β1 c22 + β2 + β2 γ1 − β1 γ2 ))((γ1 + 1)(b2 c2 (γ2 + 1) +β2 (γ2 − c22 ))b21 − c1 ((γ1 + 1)(γ2 + 1)b22 + c2 (β1 (γ2 + 1) − β2 (γ1 + 1))b2 + β2 (−c22 β1 + β2 +β2 γ1 + β1 γ2 ))b1 + β1 ((c21 − γ1 )(γ2 + 1)b22 + c2 (−β2 c21 + β1 + β2 γ1 + β1 γ2 )b2 + β2 (β2 c21 −c22 β1 − β2 γ1 + β1 γ2 )))]1/2 − b1 b22 c1 β2 − 2b1 c1 β23 γ1 + 2b1 b2 c1 c2 β22 γ1 − 2b1 b22 c1 β2 γ1 −b1 b22 c1 β2 γ12 γ2 − b1 b22 c1 β2 γ2 − 2b1 b22 c1 β2 γ1 γ2 ]/[b21 (−(γ1 + 1))(b22 (−(γ2 + 1))(β1 (γ2 + 1) −β2 (γ1 + 1)) − b2 β22 c2 (γ1 + 1) + β22 (β2 (γ1 + 1) + β1 (c22 − 2γ2 − 1))) − b1 β12 c1 (b22 (γ2 + 1)2 +β22 (−c22 + 2γ2 + 1)) + β12 (b22 (γ2 + 1)(β1 (γ2 + 1) + β2 (c21 − 2γ1 − 1)) + β22 b2 c2 (−c21 + 2γ1 + 1) +β22 (β2 (c21 − 2γ1 − 1) + β1 (−c22 + 2γ2 + 1)))]. In summary the piecewise differential systems formed by two systems (8) can have at most one limit cycle. In addition, if we calculate (y1 − y2 )2 /4 from (14) and (15) we get (y1 − y2 )2 /4 = (β2 (γ1 + 1) − β1 (γ2 + 1))(−b1 c1 (β1 γ2 + β2 γ1 + β1 + β2 ) + b21 (γ1 + 1)(γ2 + 1) +β1 (−β2 γ1 + β1 γ2 + β1 + β2 c21 ))(−b2 c2 (β1 γ2 + β2 γ1 + β1 + β2 ) + b22 (γ1 + 1)(γ2 + 1) +β2 (β2 γ1 − β1 γ2 + β2 + β1 c22 ))(b21 (γ1 + 1)(b2 c2 (γ2 + 1) + β2 (γ2 − c22 )) − b1 c1 (b2 c2 (β1 (γ2 + 1) (16) {e7} −β2 (γ1 + 1)) + b22 (γ1 + 1)(γ2 + 1) + β2 (β2 γ1 + β1 γ2 + β2 + β1 (−c22 ))) + β1 (b2 c2 (β2 γ1 + β1 γ2 +β1 − β2 c21 ) + b22 (γ2 + 1)(c21 − γ1 ) + β2 (−β2 γ1 + β1 γ2 + β2 c21 − β1 c22 ))). Of course, if the value of (y1 − y2 )2 /4 = 0, then there are no crossing periodic orbtis. But,we will prove that if the piecewise differential systems formed by two systems (8) is continuous then (y1 − y2 )2 /4 = 0, which proves that there are no crossing periodic orbits and consequently no limit cycles in this case. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 6 Now in order that the piecewise differential system formed by two systems (8) be continuous they must coincide on x = 0; i.e. the coefficients of the system must satisfy the following algebraic system −a2 b1 β2 γ1 c1 + a1 b2 β1 c2 γ2 − a2 b1 β2 c1 + a1 b2 β1 c2 − a2 β1 β2 γ1 + a1 β1 β2 γ2 + a2 β1 β2 c21 −a1 β1 β2 c22 + α2 b2 β1 γ1 − α1 b1 β2 γ2 + α2 (−b2 )β1 c21 + α1 b1 β2 c22 + α2 b1 b2 γ1 c1 − α1 b1 b2 c2 γ2 +α2 b1 b2 c1 − α1 b1 b2 c2 = 0, −a2 β2 b21 γ1 + a1 b22 β1 γ2 − a2 β2 b21 + a1 b22 β1 + a2 β1 β2 b1 c1 − a1 b2 β1 β2 c2 + a1 β1 β22 − a2 β12 β2 −α1 β22 b1 + α2 b2 β12 + α2 b2 b21 γ1 − α1 b22 b1 γ2 + α2 b2 b21 − α1 b1 b22 − α2 b2 β1 b1 c1 + α1 b2 β2 b1 c2 = 0, a2 α1 β2 γ1 − a1 α2 β1 γ2 − a1 α2 b2 γ1 c1 + a2 α1 b1 c2 γ2 − a1 α2 b2 c1 + a2 α1 b1 c2 − a2 α1 β2 c21 +a1 α2 β1 c22 + a1 a2 β2 γ1 c1 − a1 a2 β1 c2 γ2 + a1 a2 β2 c1 − a1 a2 β1 c2 − α1 α2 b2 γ1 + α1 α2 b1 γ2 (17) {cfpc.2.2} +α1 α2 b2 c21 − α1 α2 b1 c22 = 0, a2 α1 β1 β2 − a1 α2 β1 β2 + a2 α1 b1 b2 γ2 − a1 α2 b1 b2 γ1 + a2 α1 b1 b2 − a1 α2 b1 b2 + a1 a2 b1 β2 γ1 −a1 a2 b2 β1 γ2 − a1 a2 b2 β1 + a1 a2 b1 β2 − 2a2 α1 b1 β2 c1 + a2 α1 b1 β2 c2 − a1 α2 b2 β1 c1 +2a1 α2 b2 β1 c2 + a1 a2 β1 β2 c1 − a1 a2 β1 β2 c2 − α2 α1 b2 β1 + α2 α1 b1 β2 + 2α2 α1 b1 b2 c1 −2α2 α1 b1 b2 c2 = 0, b1 − b2 = 0, together with conditions (11). We can easily see from the last equation that b2 = b1 . Solving the second equation of (17) we get a1 = [b21 (β2 (a2 γ1 + a2 − α1 c2 ) + α2 β1 c1 ) − b1 (α2 β12 + β2 (a2 β1 c1 − α1 β2 )) + a2 β12 β2 (18) {B1} +b31 (α1 (γ2 + 1) − α2 (γ1 + 1))]/[β1 (−b1 β2 c2 + b21 (γ2 + 1) + β22 )],  except when denominator β1 −b1 β2 c2 + b21 (γ2 + 1) + β22 = 0. This denominator vanishes if and only if one of the following three conditions holds: d1 = {β1 = 0}, d2 = {γ2 = (β2 b1 c2 − b21 − β22 )/b21 }, (19) {d5} d3 = {b1 = 0, β2 = 0}. Since b2 = b1 then d3 of (19) gives a contradiction with (11). To continue solving (17) we must discuss all cases when β1 (−b1 β2 c2 +b21 (γ2 +1)+β22 ) = 0 and β1 (−b1 β2 c2 + b21 (γ2 + 1) + β22 ) 6= 0.  Case 1: Assume that β1 −b1 β2 c2 + b21 (γ2 + 1) + β22 6= 0. Using a1 given by (18) we solve the fourth equation of the algebraic system (17) and we obtain the solutions u1 = {γ1 = −(−β1 b1 c1 + b21 + β12 )/b21 }, u2 = {γ2 = [b1 (a2 (β1 − β2 (γ1 + 1)) + α2 β1 c1 − 2α2 β1 c2 + α1 β2 c2 ) +β2 (α2 β1 − α1 β2 + a2 β1 (c2 − c1 )) + b21 (α2 γ1 − α1 + α2 )]/[b1 (α1 b1 − a2 β1 )]}, u3 = {b1 = 0, c1 = (α2 β1 − α1 β2 )/(a2 β1 ) + c2 }, u4 = {b1 = 0, β1 = 0}, u5 = {b1 = 0, β2 = 0}, (20) {u4} u6 = {α1 = a2 β1 /b1 , γ1 = −(β1 b1 (c1 − 2c2 ) + b21 + β1 β2 )/b21 }, u7 = {α1 = a2 β1 /b1 , α2 = a2 β2 /b1 }, u8 = {a2 = 0, α1 = 0, γ1 = −(β1 b1 (c1 − 2c2 ) + b21 + β1 β2 )/b21 }, u9 = {a2 = 0, b1 = 0, α1 = α2 β1 /β2 } and u10 = {b1 = 0, α1 = 0, β1 = 0}. The allowed solutions which do not contradict (11) are u2 , u3 , u6 and u8 . Then we must discuss all these cases. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 7 Subcase 1.1: Consider the set u2 of (20). Solving the first equation of (17) we obtain one of the following sets of real solutions v1 = {α1 = [b41 (γ1 + 1)(α2 (β2 γ1 − β1 γ1 + β2 ) − a2 β2 (γ1 + 1)c2 + a2 β2 c1 (γ1 + 1) −2α2 β1 c22 + 2α2 β1 c1 c2 ) + b31 (a2 (β1 − β2 )β2 (γ1 + 1)2 + β1 c1 (−α2 ((β1 + β2 )γ1 + β2 ) − 2a2 β2 (γ1 + 1)c2 + 2α2 β1 c22 ) + 2a2 β1 β2 c22 (γ1 + 1) +α2 β1 c2 (3β1 γ1 − β2 (γ1 + 1) + β1 ) + α2 β12 c31 − 3α2 β12 c2 c21 ) + β1 b21 (α2 β2 (β2 γ1 + β1 + β2 ) + c1 (a2 β2 ((β1 + β2 )γ1 − β1 + β2 ) − 2a2 β1 β2 c22 +α2 β1 (β1 + β2 )c2 ) + a2 β2 c2 (−3β1 γ1 + β2 γ1 − β1 + β2 ) − a2 β1 β2 c31 +3a2 β1 β2 c2 c21 − 2α2 β12 c22 ) − β12 β2 b1 (a2 (β2 + β1 ((c1 − 2c2 )c2 − 1) + β2 c1 c2 ) +α2 (β1 c2 + (β2 − β1 )c1 )) + β13 β22 (a2 (c2 − c1 ) + α2 ) +α2 b51 (−(c1 − c2 ))(γ1 + 1)2 ]/[β2 (b21 + β22 )(−b1 β1 c1 + b21 (γ1 + 1) + β12 )]}, v2 = {b1 = 0, β1 = 0}, v3 = {b1 = 0, β2 = 0}, v4 = {a2 = α2 b1 /β2 , c1 = b1 (γ1 + 1)/β1 + β1 /b1 }, (21) {v5} v5 = {β1 = 0, γ1 = −1}, v6 = {c1 = 2c2 − b1 (γ1 + 1)/β1 , β2 = 0}, v7 = {c1 = (b1 (b1 γ1 + b1 + β1 c2 ) − [b1 (−2β1 b21 c2 (γ1 + 1) + β12 b1 (c22 + 4γ1 ) +b31 (γ1 + 1)2 − 4β13 c2 )]1/2 )/(2b1 β1 ), β2 = 0}, v8 = {c1 = (b1 (b1 γ1 + b1 + β1 c2 ) + [b1 (−2β1 b21 c2 (γ1 + 1) + β12 b1 (c22 + 4γ1 ) +b31 (γ1 + 1)2 − 4β13 c2 )]1/2 )/(2b1 β1 ), β2 = 0}, v9 = {α2 = 0, β2 = 0}, v10 = {c1 = (2b1 c2 + β1 − β2 )/(2b1 ), γ1 = −(β1 (−2b1 c2 + β1 + β2 ))/(2b21 ) − 1}, v11 = {c1 = c2 , β1 = 0, β2 = 0}, v12 = {α2 = 0, β1 = 0, β2 = 0} and v13 = {β1 = 0, β2 = 0, γ1 = −1}. The allowed real solutions which do not contradict (11) are v1 , v7 and v8 of (21). Then we must discuss these three cases. Subcase 1.1.1: Consider the set v1 of (21). We solve the third equation of (17) to obtain one of the following sets of real solutions w1 = {b1 = 0}, w2 = {β2 = α2 b1 /a2 }, w3 = {β2 = −b1 (b1 γ1 + b1 + β1 c1 − 2β1 c2 )/β1 }, w4 = {β2 = −[(β1 b1 (c1 (c2 − c1 ) + γ1 ) + b21 (c1 − c2 )(γ1 + 1) − β12 c2 )(β1 b1 (−2c2 γ1 −c1 ((c1 − c2 )2 + 2)) + b21 (γ1 + 1)((c1 − c2 )2 + γ1 + 1) + β12 (c22 + 1))] /[(β1 c1 − b1 γ1 )(−2b1 β1 c1 (γ1 + 1) + b21 (γ1 + 1)2 + β12 (c21 + 1))]}, u5 = {a2 = 0, b1 = 0}, (22) {w5} w6 = {a2 = 0, α2 = 0}, w7 = {b1 = 0, β1 = 0}, w8 = {c2 = c1 , γ1 = β1 c1 /b1 }, w9 = {β1 = 0, γ1 = −1}, w10 = {b1 = 0, c1 = 0, c2 = 0}, w11 = {c2 = c1 , β1 = 0, γ1 = −1} and w12 = {c2 = c1 , β1 = 0, γ1 = 0}. The allowed real solutions which do not contradict (11) are w4 and w8 of (22). Then we must consider these two cases. Subcase 1.1.1.1: Consider the set w4 of (22). Then we have now a continuous piecewise differential system formed by two systems (8). From (16) we get (y1 − y2 )2 /4 = 0 then we have no periodic orbits formed by (8) and (8) in this subcase. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 8 Subcase 1.1.1.2: Consider the set w8 of (22). Then we have now a continuous piecewise differential system formed by two systems (8). Solving the algebraic system (12) becomes   b21 (y1 − y2 ) b21 + β12 2β1 b1 c1 y1 + 2β1 b1 c1 y2 + 2b1 c1 + 2β1 b21 y1 y2 + b21 y1 + b21 y2 + 2β1 c21 = 0, (β1 b1 y1 + b1 + β1 c1 ) 2 (β1 b1 y2 + b1 + β1 c1 ) 2 and   b21 (y1 − y2 ) b21 + β22 2β2 b1 c1 y1 + 2β2 b1 c1 y2 + 2b1 c1 + 2β2 b21 y1 y2 + b21 y1 + b21 y2 + 2β2 c21 = 0. (β2 b1 y1 + b1 + β2 c1 ) 2 (β2 b1 y2 + b1 + β2 c1 ) 2 From these last two equation we get y1 = y2 or y1 = y2 = −c1 /b1 , then we have no periodic orbits in this subcase. Subcase 1.1.2: Consider the set v7 of (21). In this case (14) and (15) give y1 = −c2 /b1 , and y2 = −c2 /b1 . Then we have no periodic orbits in this subcase. Subcase 1.1.3: Consider the set v8 of (21). Also in this case from (14) and (15) we get y1 = −c2 /b1 , and y2 = −c2 /b1 . Hence we have no periodic orbits in this subcase. Subcase 1.2: Consider the set u3 of (20). The first equation of (17) becomes  β2 (α2 β1 − α1 β2 ) 2 + 2a2 β2 β1 c2 (α2 β1 − α1 β2 ) + a22 β12 −β2 γ1 + β1 γ2 + (β2 − β1 ) c22 = 0, which gives one of the following sets of solutions s1 = {γ2 = (−β2 (α2 β1 − α1 β2 )2 + 2a2 β2 β1 c2 (α1 β2 − α2 β1 ) + a22 β12 (β2 γ1 + (β1 − β2 )c22 )) /(a22 β13 )}, s2 = {a2 = 0, α1 = α2 β1 /β2 }, s3 = {a2 = 0, β2 = 0}, s4 = {α1 = 0, β1 = 0} and s5 = {β1 = 0, β2 = 0}. The allowed solution is s1 which gives  −β2 (α2 β1 − α1 β2 ) 2 + 2a2 β2 β1 c2 (α1 β2 − α2 β1 ) + a22 β12 β2 γ1 + (β1 − β2 ) c22 γ2 = . a22 β13 Then (14) and (15) become  (α2 β1 − α1 β2 ) 2 + 2a2 β1 c2 (α2 β1 − α1 β2 ) + a22 β12 c22 − γ1 y1 = , a22 β13 and  (α2 β1 − α1 β2 ) 2 + 2a2 β1 c2 (α2 β1 − α1 β2 ) + a22 β12 c22 − γ1 y2 = . a22 β13 So y1 = y2 which gives no periodic orbits in this subcase. Subcase 1.3: Consider the set u6 of (20). The first of (17) becomes −β1 (α2 b1 − a2 β2 )(b31 (c1 (β1 γ2 − β2 γ2 + β1 + β2 + 4β2 c22 ) − c2 (β1 γ2 − β2 γ2 + β1 + 2β2 (c22 + 1)) −2β2 c2 c21 ) + β2 b21 (β2 (2c21 − 3c2 c1 + c22 − γ2 + 1) + β1 (c22 − c1 c2 + 2γ2 + 1)) + β22 b1 ((β1 + β2 )c1 −3β1 c2 ) + b41 (2(c1 − c2 )2 + 1)(γ2 + 1) + β1 β23 ) = 0, which gives one of the following sets of real solutions s1 = {β1 = 0}, s2 = {β1 = [b1 (β2 b21 (c1 (−4c22 + γ2 − 1) + c2 (2c22 − γ2 + 2) + 2c2 c21 ) + β22 b1 (−2c21 + 3c2 c1 −c22 + γ2 − 1) + b31 (−(2(c1 − c2 )2 + 1))(γ2 + 1) − β23 c1 )]/[β2 b21 (c22 − c1 c2 + 2γ2 + 1) +β22 b1 (c1 − 3c2 ) + b31 (c1 − c2 )(γ2 + 1) + β23 ]}, s3 = {β2 = α2 b1 /a2 }, s4 = {a2 = 0, b1 = 0}, s5 = {a2 = 0, α2 = 0}, s6 = {b1 = 0, β2 = 0} and s7 = {β2 = 0, γ2 = −1}. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 9 The allowed solution is s2 which gives β1 = [b1 (β2 b21 (c1 (−4c22 + γ2 − 1) + c2 (2c22 − γ2 + 2) + 2c2 c21 ) + β22 b1 (−2c21 + 3c2 c1 −c22 + γ2 − 1) + b31 (−(2(c1 − c2 )2 + 1))(γ2 + 1) − β23 c1 )]/[β2 b21 (c22 − c1 c2 + 2γ2 + 1) +β22 b1 (c1 − 3c2 ) + b31 (c1 − c2 )(γ2 + 1) + β23 ]. Then (14) and (15) become   β2 c22 − γ2 − b1 c2 (γ2 + 1) β2 c22 − γ2 − b1 c2 (γ2 + 1) y1 = , and y2 = . −b1 β2 c2 + b21 (γ2 + 1) + β22 −b1 β2 c2 + b21 (γ2 + 1) + β22 Therefore y1 = y2 which gives no periodic orbits in this subcase. Subcase 1.4: Consider the set u8 of (20). Solving the first and the third equation of (17) simultaneously we get one of the following sets of real solutions s1 = {α2 = 0}, s2 = {β1 = 0}, s3 = {b1 = 0, β2 = 0}, s4 = {β1 = [b21 (β2 b1 (−((c1 − c2 )2 + 2))(c1 − c2 ) − b21 (2(c1 − c2 )4 + 3(c1 − c2 )2 + 1) − β22 )] /[(b1 (c1 − c2 ) + β2 )(2β2 b1 (c1 − c2 ) + b21 ((c1 − c2 )2 + 1) + β22 )], γ2 = β2 c1 /b1 + (c1 − c2 )2 }, s5 = {β1 = 2b1 (c1 − c2 ) + β2 , γ2 = −[−β2 b1 c2 + b21 + β22 ]/b21 }, s6 = {b1 = 0, c1 = 0, β1 = 0}, s7 = {b1 = 0, β1 = 0, β2 = 0} and s8 = {β2 = 0, γ2 = −1}. The allowed solution is s4 which gives    b21 β2 b1 − (c1 − c2 ) 2 + 2 (c1 − c2 ) − b21 2 (c1 − c2 ) 4 + 3 (c1 − c2 ) 2 + 1 − β22 β1 = , (b1 (c1 − c2 ) + β2 ) (2β2 b1 (c1 − c2 ) + b21 ((c1 − c2 ) 2 + 1) + β22 ) and β2 c1 γ2 = + (c1 − c2 ) 2 . b1 Then (14) and (15) become  β2 b1 c1 (c2 − c1 ) − b21 c2 c21 − 2c2 c1 + c22 + 1 − β22 c1 y1 = , b1 (β2 b1 (c1 − c2 ) + b21 (c21 − 2c2 c1 + c22 + 1) + β22 ) and  β2 b1 c1 (c2 − c1 ) − b21 c2 c21 − 2c2 c1 + c22 + 1 − β22 c1 y2 = . b1 (β2 b1 (c1 − c2 ) + b21 (c21 − 2c2 c1 + c22 + 1) + β22 ) Hence y1 = y2 which gives no periodic orbits in this subcase.  Case 2: Assume that β1 −b1 β2 c2 + b21 (γ2 + 1) + β22 = 0. This case is divided in two subcases regarding d1 and d2 of (19). LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 10 Subcase 2.1: Consider the set d1 of (19). Solving the first and the second equation of (17) simultaneously we get one of the following sets of real solutions w1 = {b1 = 0}, w2 = {b1 = 0, β2 = 0}, w3 = {c1 = −[a2 β22 b1 (γ1 + 1) + β2 b21 (a2 c2 (γ1 + 1) − α2 (γ1 + 1) + α1 ) + α1 β23 +α2 b31 (−c2 )(γ1 + 1)]/[b21 (γ1 + 1)(α2 b1 − a2 β2 )], γ2 = [−β2 b1 (a2 γ1 + a2 − α1 c2 ) − α1 β22 + b21 (α2 γ1 − α1 + α2 )]/[α1 b21 ]}, w4 = {α1 = 0, α2 = a2b1β2 }, u5 = {α1 = 0, γ1 = −1}, w6 = {a2 = 0, b1 = 0, α1 = 0}, (23) {w6} w7 = {b1 = 0, c1 = 0, α1 = 0}, w8 = {b1 = 0, α1 = 0, γ1 = −1}, w9 = {α1 = 0, α2 = 0, β2 = 0}, w10 = {α1 = 0, β2 = 0, γ1 = −1}, w11 = {α2 = 0, β2 = 0, γ2 = −1}, w12 = {β2 = 0, γ1 = −1, γ2 = −1} and w13 = {a2 = 0, α2 = 0, β2 = 0, γ2 = −1}. The allowed real solutions which do not contradict (11) are w3 and w12 of (23). Then we must discuss these two subcases. Subcase 2.1.1: Consider the set w3 of (23). Then (14) and (15) become a2 β22 b1 (γ1 + 1) + β2 b21 (a2 c2 (γ1 + 1) − α2 (γ1 + 1) + α1 ) + α1 β23 + α2 b31 (−c2 ) (γ1 + 1) y1 = , b31 (γ1 + 1) (α2 b1 − a2 β2 ) and a2 β22 b1 (γ1 + 1) + β2 b21 (a2 c2 (γ1 + 1) − α2 (γ1 + 1) + α1 ) + α1 β23 + α2 b31 (−c2 ) (γ1 + 1) y2 = . b31 (γ1 + 1) (α2 b1 − a2 β2 ) So y1 = y2 which gives no periodic orbits in this subcase. Subcase 2.1.2: Consider the set w12 of (23). First integrals have a singularity on the y−axis. Then we have no periodic orbits in this subcase. Subcase 2.2: Consider the set d2 of (19). Then we have β2 b1 c2 − b21 − β22 γ2 = . b21 Solving the first and the second equation of (17) simultaneously we get one of the following sets of real solutions w1 = {b1 = 0, β1 = 0}, w2 = {b1 = 0, β2 = 0}, w3 = {α1 = a1b1β1 , α2 = a2b1β2 }, w4 = {α2 = 0, β2 = 0}, β ((a −a )β b2 +a β 3 −a2 β12 β2 +α2 β12 b1 +α2 b31 ) −β b c +b2 +β 2 u5 = {α1 = 1 1 2 2 1 b11β22(b2 +β 2) , γ1 = − 1 1 b12 1 1 }, 1 2 1 (24) {w7} w6 = {a1 = 0, a2 = 0, b1 = 0}, w7 = {a2 = 0, b1 = 0, β1 = 0}, w8 = {b1 = 0, β1 = 0, β2 = 0}, w9 = {β1 = 0, β2 = 0, γ1 = −1} and −β b c +b2 +β 2 w10 = {α2 = 0, β2 = 0, γ1 = − 1 1 b12 1 1 }. 1 The allowed real solutions which do not contradict (11) are w5 and w9 of (24). Then we must discuss these two cases. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 11 Subcase 2.2.1: Consider the set w5 of (24). Solving the third and the fourth equation of (17) simultaneously we get one of the following sets of real solutions s1 = {β1 = 0}, s2 = {β2 = α2 b1 /a2 }, s3 = {a2 = 0, b1 = 0}, s4 = {a2 = 0, α2 = 0}, β b2 (a (3(c −c )2 +1)+3α2 (c2 −c1 ))+3a2 β22 b1 (c1 −c2 )+a2 β23 −3α2 b31 (c1 −c2 )2 s5 = {a1 = 2 1 2 1 2 β2 (b21 +β22 ) , β1 = 2b1 (c1 − c2 ) + β2 }, s6 = {b1 = 0, β1 = 0, β2 = 0}, s7 = {c2 = c1 , β1 = 0, β2 = 0} and s8 = {α2 = 0, β1 = 2b1 (c1 − c2 ), β2 = 0}. The only allowed real solution which does not contradict (11) is s5 . Then we have now a continuous piecewise differential systems formed by two systems (8). Now we will solve the algebraic system (12) which equivalent to  (y1 − y2 ) −2b1 c1 + 4b1 c2 + b21 y1 + b21 y2 − 2β2 = 0, (2b1 c2 − b1 c1 + b21 y1 − β2 ) 2 (2b1 c2 − b1 c1 + b21 y2 − β2 ) 2 and  (y1 − y2 ) 2b1 c2 + b21 y1 + b21 y2 − 2β2 = 0. (b1 c2 + b21 y1 − β2 ) 2 (b1 c2 + b21 y2 − β2 ) 2 From these last two equation we get y1 = y2 , then we have no periodic orbits in this subcase. Subcase 2.2.2: Consider the set w9 of (24). Then the third and the fourth equation of (17) become α1 α2 b1 (c1 − c2 ) (c1 + c2 ) = 0, and 2α1 α2 b21 (c1 − c2 ) = 0. The only solution of these last equations which do not contradict (11) is c2 = c1 . In this case first integrals have a singularity on the y−axis, then no periodic orbits in this subcase.  Proof of statement (f ) of Theorem 1. Consider the generalized isochronous quadratic systems (8) and (9). In order that the piecewise differential systems formed by systems (8) and (9) be continuous, they must coincide on x = 0, which means that the coefficients of the system must satisfy the following algebraic system −3a2 b1 β2 γ1 c1 − 16a1 b2 β1 c2 γ2 − 3a2 b1 β2 c1 + 3a1 b2 β1 c2 − 3a2 β1 β2 γ1 + 3a1 β1 β2 γ2 +3a2 β1 β2 c21 + 4a1 β1 β2 c22 + 3α2 b2 β1 γ1 − 3α1 b1 β2 γ2 − 3α2 b2 β1 c21 − 4α1 b1 β2 c22 +3α2 b1 b2 γ1 c1 + 16α1 b1 b2 c2 γ2 + 3α2 b1 b2 c1 − 3α1 b1 b2 c2 = 0, −3a2 β2 b21 γ1 − 16a1 b22 β1 γ2 − 3a2 β2 b21 + 3a1 b22 β1 + 3a2 β1 β2 b1 c1 − 8a1 b2 β1 β2 c2 + 3a1 β1 β22 −3a2 β12 β2 − 3α1 β22 b1 + 3α2 b2 β12 + 3α2 b2 b21 γ1 + 16α1 b22 b1 γ2 + 3α2 b2 b21 − 3α1 b22 b1 −3α2 b2 β1 b1 c1 + 8α1 b2 β2 b1 c2 = 0, 4b22 β2 = 0, 3a2 α1 β2 γ1 − 3a1 α2 β1 γ2 − 3a1 α2 b2 γ1 c1 − 16a2 α1 b1 c2 γ2 − 3a1 α2 b2 c1 + 3a2 α1 b1 c2 (25) {cfpc.2.3} −3a2 α1 β2 c21 − 4a1 α2 β1 c22 + 3a1 a2 β2 γ1 c1 + 16a1 a2 β1 c2 γ2 + 3a1 a2 β2 c1 − 3a1 a2 β1 c2 −3α1 α2 b2 γ1 + 3α1 α2 b1 γ2 + 3α1 α2 b2 c21 + 4α1 α2 b1 c22 = 0, 3a2 α1 β1 β2 − 3a1 α2 β1 β2 − 16a2 α1 b1 b2 γ2 − 3a1 α2 b1 b2 γ1 + 3a2 α1 b1 b2 − 3a1 α2 b1 b2 +3a1 a2 b1 β2 γ1 + 16a1 a2 b2 β1 γ2 − 3a1 a2 b2 β1 + 3a1 a2 b1 β2 − 6a2 α1 b1 β2 c1 − 16a2 α1 b1 β2 c2 −3a1 α2 b2 β1 c1 − 8a1 α2 b2 β1 c2 + 3a1 a2 β1 β2 c1 + 16a1 a2 β1 β2 c2 − 3α2 α1 b2 β1 + 3α2 α1 b1 β2 +6α2 α1 b1 b2 c1 + 8α2 α1 b1 b2 c2 = 0, −16a2 β2 b2 − 3a2 b1 β2 + 4α2 b22 + 3α2 b1 b2 = 0, together with the conditions (11). LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 12 Solving the third and the sixth equation of the algebraic system (25) we get one of the following sets of solutions s1 = {a2 = 0, b2 = 0}, s2 = {b1 = 0, b2 = 0}, s3 = {b2 = 0, β2 = 0}, (26) {s7} s4 = {b1 = −4b2 /3, β2 = 0} and s5 = {α2 = 0, β2 = 0}. The allowed solutions which does not contradict (11) are s2 and s4 of (26). Then we have two cases. Case 1: We consider s2 of (26), then we have b1 = 0 and b2 = 0. Solving the second and the fifth equation of (25) we obtain one of the following sets of solutions u1 = {β1 = 0}, u2 = {β2 = 0}, u3 = {a1 = 0, a2 = 0}, a2 β1 u4 = {α2 = a2 ( αa11 + c1 + 16c 3 ), β2 = 2 a1 } and u5 = {a1 = 0, α1 = 0, β1 = 0}. The only allowed solution which doe not contradict (11) is u4 . Then we have   α1 16c2 a2 β1 α2 = a2 + c1 + and β2 = . a1 3 a1 Solving the first and the fourth equation of (25) we obtain one of the following sets of real solutions z1 = {a2 = 0}, z2 = {β1 = 0}, a c2 (64c22 +9) c (−(64c22 +9)γ1 +12c2 c31 +(64c22 +9)c21 +12c2 c1 ) (27) {z} z3 = {a2 = 1 9(c 3 +c ) 1 , γ2 = − 2 9(c31 +c1 ) } and 1 z4 = {c1 = 0, c2 = 0, γ2 = aa2 γ1 1 }. The allowed solutions which do not contradict (11) are z3 and s4 of (27). Then we get two subcases. Subcase 1.1: We consider z3 of (27). We have now a piecewise continuous differential systems formed by systems (8) and (9). The algebraic system (12) is equivalent to the following equations β1 (y1 − y2 )(−2γ1 c21 + β1 c21 (−y1 ) − β1 c21 y2 − 2c21 + 2γ12 + 2γ1 + 2β1 γ1 y1 + 2β1 γ1 y2 + 2β12 y1 y2 +β1 y1 + β1 y2 )/[(γ1 + β1 y1 + 1)2 (γ1 + β1 y2 + 1)2 ] = 0, and 64β1 c22 (64c22 + 9)(y1 − y2 )(−6γ1 c31 − 32c2 γ1 c21 − 6γ1 c1 + 16c2 γ12 + 16β1 c2 γ1 y1 + 16β1 c2 γ1 y2 −3β1 c31 y1 − 3β1 c31 y2 − 16β1 c2 c21 y1 − 16β1 c2 c21 y2 − 3β1 c1 y1 − 3β1 c1 y2 + 16β12 c2 y1 y2 + 6c51 +16c2 c41 + 6c31 )/[c1 (c21 + 1)(−16c2 γ1 − 16β1 c2 y1 + 3c31 + 16c2 c21 + 3c1 )(−16c2 γ1 − 16β1 c2 y2 +3c31 + 16c2 c21 + 3c1 )] = 0. Solving the first equation for y1 we get c21 (2γ1 + β1 y2 + 2) − 2γ1 (γ1 + 1) − y2 (2β1 γ1 + β1 ) y1 = , β1 (−c21 + 2γ1 + 2β1 y2 + 1) replacing this value into the second equation which, by excluding the trivial case y1 = y2 , becomes   2 c21 + 1 (3c1 + 8c2 ) c21 − γ1 − β1 y2 2 = 0. c21 − 2γ1 − 2β1 y2 − 1 Finally we obtain y1 = y2 = (c21 − γ1 )/β1 . Then we have no periodic orbits, and then no limit cycles formed by systems (8) and (9) in this subcase. Subcase 1.2: We consider z4 of (27). We have now a piecewise continuous differential systems formed by systems (8) and (9). The algebraic system (12) is equivalent to the following equations  β1 (y1 − y2 ) 2γ12 + 2γ1 + 2β1 γ1 y1 + 2β1 γ1 y2 + 2β12 y1 y2 + β1 y1 + β1 y2 , (γ1 + β1 y1 + 1) 2 (γ1 + β1 y2 + 1) 2 LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 13 and  (y1 − y2 ) −16a2 γ12 + 6a1 γ1 − 16a2 β1 γ1 y1 − 16a2 β1 γ1 y2 − 16a2 β12 y1 y2 + 3a1 β1 y1 + 3a1 β1 y2 . a1 (−16a2 γ1 − 16a2 β1 y1 + 3a1 ) (−16a2 γ1 − 16a2 β1 y2 + 3a1 ) Solving the first equation for y1 we get 2γ1 (γ1 + 1) + y2 (2β1 γ1 + β1 ) y1 = − , β1 (2γ1 + 2β1 y2 + 1) replacing this value into the second equation which, by excluding the trivial case y1 = y2 , becomes 2 (3a1 + 8a2 ) (γ1 + β1 y2 ) 2 = 0. 2γ1 + 2β1 y2 + 1 Finally we obtain y1 = y2 = −γ1 /β1 . Then we have no periodic orbits, and then no limit cycles in this subcase. Case 2: We consider s4 of (26). Solving the second equation of (25) we obtain one of the following sets of solutions s1 = {b2 = 0}, 3β b (a (3−16γ2 )+4α2 c1 )+9α2 β12 +4b22 (α1 (3−16γ2 )+4α2 ) s2 = {γ1 = − 1 2 1 16α2 b22 }, s3 = {b2 = 0, α2 = 0}, s4 = {b2 = 0, β1 = 0}, s5 = {α1 = − 3a4b1 2β1 , α2 = 0} and 3 s6 = {α2 = 0, γ2 = 16 }. The allowed solution is s2 , which means that 3β1 b2 (a1 (3 − 16γ2 ) + 4α2 c1 ) + 9α2 β12 + 4b22 (α1 (3 − 16γ2 ) + 4α2 ) γ1 = − . 16α2 b22 Solving the first and the fifth equation of (25) we obtain one of the following sets of real solutions z1 = {α1 = − 3a4b1 2β1 , α2 = 0}, z2 = {a2 = −[b2 (12b2 β1 (a1 (18c21 + 48c2 c1 + 32c22 + 9) − 3α1 (3c1 + 4c2 )) −27β12 (3a1 c1 + 4a1 c2 + 3α1 ) + 32α1 b22 (3c1 + 4c2 )2 )]/[9(16b22 β1 + 9β13 )], α2 = − b2 (16γ2 −3)(4b2 (3c 1 +4c2 )−9β1 )(3a1 β1 +4α1 b2 ) 9(16b22 β1 +9β13 ) }, z3 = {α1 = 0, β1 = 0}, 3 z4 = {α2 = 0, γ2 = 16 }, (28) {z1} z5 = {a1 = 0, α1 = 0, α2 = 0}, z6 = {a2 = − 3a41 , c2 = − 3c41 , β1 = 0}, z7 = {a2 = a1 (9−48γ64γ 2 )+8α2 (3c1 +4c2 ) 2 −12 , α1 = 0, β1 = 0}, 3c1 3 z8 = {c2 = − 4 , β1 = 0, γ2 = 16 }, 3 z9 = {α2 = 0, β1 = 0, γ2 = 16 } and 2α2 (3c1 +4c2 ) z10 = {a1 = 0, a2 = 16γ2 −3 , α1 = 0, β1 = 0}. The allowed solutions which do not contradict (11) are z2 , z6 and z8 of (28). Then we must discuss these three subcases. Subcase 2.1: We consider z2 of (28). We solve now the fourth equation of (25) to obtain one of the following sets of real solutions v1 = {b2 = 0}, v2 = {β1 = − 4α 1 b2 3a1 }, b (3c +4c )(9c2 +24c c1 +16c22 −12γ2 +9) v3 = {β1 = − 2 1 2 1 27γ22 }, 3 v4 = {γ2 = 16 }, (29) {z2} v5 = {a1 = 0, b2 = 0}, v6 = {a1 = 0, α1 = 0}, v7 = {b2 = 0, γ2 = 0} and v8 = {c2 = − 3c41 , γ2 = 0}. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 14 The allowed solutions which do not contradict (11) are v3 and v8 of (29). Then we must discuss these two subcases. Subcase 2.1.1: We consider v3 of (29). We have now a piecewise continuous differential systems formed by systems (8) and (9). We solve the algebraic system (12) we get two different pairs (y1 , y2 ). The first pair is √ √ 6γ2 3p 3p − 8c2 y1 1 = − R1 − R2 + 3c1 +4c2 , 8 8 8b2 and √ √ 6γ2 3p 3p − 8c2 y2 1 = R1 − R2 + 3c1 +4c2 . 8 8 8b2 Where √ √ 4 3b2 (3c1 + 4c2 ) γ2 R2 + 72γ2 c21 + 192c2 γ2 c1 + 128c22 γ2 − 27c21 − 72c2 c1 − 48c22 − 24γ22 R1 = , b22 (3c1 + 4c2 ) 2 and   4γ2 (3c13γ +4c2 )2 + 2 − 3 2 R2 = . b22 The second pair is √ √ 6γ2 3√ 3p − 8c2 y1 2 = − S+ R2 + 3c1 +4c2 , 8 8 8b2 and √ √ 6γ2 3√ 3p − 8c2 y2 2 = S+ R2 + 3c1 +4c2 . 8 8 8b2 Where √ √ −4 3b2 (3c1 + 4c2 ) γ2 R2 + 72γ2 c21 + 192c2 γ2 c1 + 128c22 γ2 − 27c21 − 72c2 c1 − 48c22 − 24γ22 S= . b22 (3c1 + 4c2 ) 2 Then in this subcase we can have at most two limit cycles formed by systems (8) and (9). We have y2 2 > y1 2 > y2 1 > y1 1 and since the invariant straight line  3 (3c1 + 4c2 ) (3c1 + 8c2 ) γ2 − 36γ22 + 27γ2 − c2 (3c1 + 4c2 ) (3c1 + 4c2 ) 2 + 9 y= b2 (3c1 + 4c2 ) ((3c1 + 4c2 ) 2 − 12γ2 + 9) 27α1 γ2 + x. b2 (3c1 + 4c2 ) ((3c1 + 4c2 ) 2 − 12γ2 + 9) intersects the y−axis at 4c2 + 3c1 c2 − 3γ2 y0 = − 2 , 3b2 c1 + 4b2 c2 and because y0 ∈ [y1 2 , y2 2 ] or y0 ∈ [y1 1 , y2 1 ] (y0 ∈ / (−∞, y1 1 ) ∪ (y2 1 , y1 2 ) ∪ (y2 2 , +∞)) we conclude that we have at most one limit cycle. Then the unique limit cycle, if it exists, must be constructed in the segment [y1 2 , y2 2 ] or on the other segment [y1 1 , y2 1 ]. In what follows we prove that there is no limit cycle under these conditions, for this reason, we consider the generalized system (10), and from all the sets of parameters that give continuity we choose only β2 = 0, then the generalized system (10) becomes 1 ẋ = − (16γ2 + 16α2 x − 3)(a2 x + b2 y + c2 ), 3α2 1  ẏ = a2 (b2 y + c2 )(16γ2 + 8α2 x − 3) + a22 x(16γ2 + 12α2 x − 3) − 4α2 (b2 y + c2 )2 − 3α2 (γ2 + α2 x) , 3α2 b2 which is equivalent to the following ordinary differential equation 3α2 (γ2 + α2 x) + a2 (−16γ2 − 12α2 x + 3) + 4α2 c2 a2 x + b2 y(x) + c2 + 4α2 y(x) dy b2 = . (30) {odeGnle4} dx 16γ2 + 16α2 x − 3 LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 15 Solving the initial value problem (30) and y(0) = y0 , we obtain the following set of solution √ q √ s b2 W1 b2 W1 3/2 W2 + 8a2 x + 8c2 1p 1 p a2 x + c2 2 1 y(x) = − and y(x) = W1 3/2 W2 − , 8 b2 8 2 b2 W 1 b2 where W1 = 16γ2 + 16α2 x − 3, and √ √  2W1 (32(b2 y0 +c2 )2 −24γ2 +9) W2 = 64 (a2 x + c2 ) 2 W1 − 2 W1 32 (a2 x + c2 ) 2 − 24γ2 − 24α2 x + 9 + √ 16γ2 −3 . Replacing by x = 0 in the value of y(x) we get r √ b2 1 3/2 ( 16γ2 −3(b2 y0 +c2 )2 )3/2 b2 2 (16γ2 −3) (b2 y0 +c2 )2 + c2 y1 ( or y2 ) = − , b2 and r √ b2 1 3/2 ( 16γ2 −3(b2 y0 +c2 )2 )3/2 b2 2 (16γ2 −3) (b2 y0 +c2 )2 − c2 y2 ( or y1 ) = . b2 Since the midpoint of this last point y1 and y2 is (y1 + y2 )/2 = −c2 /b2 , (31) {mp1} and because the midpoints of the segments [y1 2 , y2 2 ] and [y1 1 , y2 1 ] are given, respectively, by r   √ 3γ2 4γ2 (3c +4c 2 +2 −3 3b2 1 b22 2) + 3c16γ +4c2 − 8c2 2 (y1 2 + y2 2 )/2 = , (32) {mp2} 8b2 and r   √ 4γ2 3γ2 (3c1 +4c2 )2 +2 −3 6γ2 3b2 b22 − 3c1 +4c2 + 8c2 (y1 1 + y2 1 )/2 = − . (33) {mp3} 8b2 Equation the right-hand sides of (31) and (32) (and (33)) we get γ2 = 3/8. This value of γ2 gives complex values for y1 2 , y2 2 , y1 1 and y2 1 . Finally, the continuous piecewise differential systems formed by (8) and (10) have no limit cycles. Subcase 2.1.2: We consider v8 of (29). We have now a piecewise continuous differential system formed by systems (8) and (9), we solve the algebraic system (12) we get two different pairs (y1 , y2 ). The first pair is p √  − 4b22 − 9β12 + b2 β1 T1 − 2 + 6β1 c1 y1 = , 8b2 β1 and p √  4b22 − 9β12 + b2 β1 T1 + 2 − 6β1 c1 y2 = − . 8b2 β1 Where √  4b2 4b22 −9β12 +2b2 − β12 −9 T1 = . b22 The second pair is p √  4b22 − 9β12 + b2 β1 T2 − 2 + 6β1 c1 y1 = , 8b2 β1 and p √  4b22 − 9β12 − b2 β1 T2 + 2 + 6β1 c1 y2 = . 8b2 β1 LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 16 Where √  4b2 4b22 −9β12 −2b2 β12 −9 T2 = . b22 Since we have T1 < 0 and T2 < 0. Then we have no limit cycles formed by systems (8) and (9). Subcase 2.2: We consider z6 of (28). We solve now the fourth equation of (25) to obtain one of the following sets of real solutions u1 = {b2 = 0}, u2 = {α1 = 0}, u3 = {α2 = α4(4γ 1 (9−48γ2 ) 2 −3) } and 3 u4 = {α1 = 0, γ2 = 4 }. The allowed solution which does not contradict (11) is v3 . Then we have α1 (9 − 48γ2 ) α2 = . 4 (4γ2 − 3) We have now a piecewise continuous differential systems formed by systems (8) and (9), the algebraic system (12) is equivalent to b2 (y1 − y2 ) (−2b2 y1 − 2b2 y2 + 3c1 ) = 0, (4γ2 − 3) 2 and  b2 (y1 − y2 ) (−2b2 y1 − 2b2 y2 + 3c1 ) 24b2 c1 y1 + 24b2 c1 y2 − 16b22 y12 − 16b22 y22 − 18c21 + 24γ2 − 9 = 0. 16γ2 − 3 These two last equations have the solution y1 = y2 or y1 = −y2 + 3c1 /(2b2 ). Then we have a continuum of periodic orbits and then no limit cycles formed by systems (8) and (9). Subcase 2.3: We consider z8 of (28). We solve now the fourth equation of (25) to obtain one of the following sets solutions u1 = {b2 = 0}, u2 = {α1 = 0} and u3 = {α2 = 0}. All allowed solution contradict (11). Then we have we have no piecewise continuous differential systems formed by systems (8) and (9) in this case.  Proof of statement (g) of Theorem 1. In order that the piecewise differential system formed by two systems (8) and (10) be continuous they must coincide on x = 0; i.e. the coefficients of the system must satisfy the LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 17 following algebraic system −3a2 b1 β2 γ1 c1 + 8a1 b2 β1 c2 γ2 − 3a2 b1 β2 c1 + 3a1 b2 β1 c2 + 4a1 β1 β2 γ22 − 3a2 β1 β2 γ1 +3a1 β1 β2 γ2 + 3a2 β1 β2 c21 − 16a1 β1 β2 c22 − 4α1 b1 β2 γ22 + 3α2 b2 β1 γ1 − 3α1 b1 β2 γ2 −3α2 b2 β1 c21 + 16α1 b1 β2 c22 + 3α2 b1 b2 γ1 c1 − 8α1 b1 b2 c2 γ2 + 3α2 b1 b2 c1 − 3α1 b1 b2 c2 = 0, −3a2 β2 b21 γ1 + 8a1 b22 β1 γ2 − 3a2 β2 b21 + 3a1 b22 β1 + 3a2 β1 β2 b1 c1 − 24a1 b2 β1 β2 c2 + 8a1 β1 β22 γ2 +3a1 β1 β22 − 3a2 β12 β2 − 8α1 β22 b1 γ2 − 3α1 β22 b1 + 3α2 b2 β12 + 3α2 b2 b21 γ1 − 8α1 b22 b1 γ2 +3α2 b2 b21 − 3α1 b22 b1 − 3α2 b2 β1 b1 c1 + 24α1 b2 β2 b1 c2 = 0, 4β2 (β22 − 2b22 ) = 0, −4a1 α2 β1 γ22 + 3a2 α1 β2 γ1 − 3a1 α2 β1 γ2 − 3a1 α2 b2 γ1 c1 + 8a2 α1 b1 c2 γ2 − 3a1 α2 b2 c1 (34) {cfpc.2.4} +3a2 α1 b1 c2 − 3a2 α1 β2 c21 + 16a1 α2 β1 c22 + 3a1 a2 β2 γ1 c1 − 8a1 a2 β1 c2 γ2 + 3a1 a2 β2 c1 −3a1 a2 β1 c2 + 4α1 α2 b1 γ22 + 2a1 α2 b2 β1 c2 + a1 a2 β1 β2 c1 − a1 a2 β1 β2 c2 − α2 α1 b2 β1 +α2 α1 b1 β2 + 2α2 α1 b1 b2 c1 − 3α1 α2 b2 γ1 + 3α1 α2 b1 γ2 + 3α1 α2 b2 c21 − 16α1 α2 b1 c22 = 0, −8a1 α2 β1 β2 γ2 + 3a2 α1 β1 β2 − 3a1 α2 β1 β2 + 8a2 α1 b1 b2 γ2 − 3a1 α2 b1 b2 γ1 + 3a2 α1 b1 b2 −3a1 α2 b1 b2 + 3a1 a2 b1 β2 γ1 − 8a1 a2 b2 β1 γ2 − 3a1 a2 b2 β1 + 3a1 a2 b1 β2 − 6a2 α1 b1 β2 c1 +8a2 α1 b1 β2 c2 − 3a1 α2 b2 β1 c1 + 32a1 α2 b2 β1 c2 + 3a1 a2 β1 β2 c1 − 8a1 a2 β1 β2 c2 +8α2 α1 b1 β2 γ2 − 3α2 α1 b2 β1 + 3α2 α1 b1 β2 + 6α2 α1 b1 b2 c1 − 32α2 α1 b1 b2 c2 = 0, 8a2 β2 b2 − 3a2 b1 β2 + 4α2 β22 − 16α2 b22 + 3α2 b1 b2 = 0, together with conditions (11). Solving the second and the third equation of (34), we get the following sets of real solutions s1 = {b2 = 0, β2 = 0}, s2 = {b2 = 0, β2 = 0}, −β (a b (8γ +3)+3α2 β1 )−3α2 b21 +b1 (α1 b2 (8γ2 +3)+3α2 β1 c1 ) s3 = {β2 = 0, γ1 = 1 1 2 2 3α2 b21 }, √ s4 = {β2 = − √ 2b2 , √ √ β1 (a1 b2 (8 2c2 +8γ2 +3)+α2 β1 )+ 2a2 (−β1 b1 c1 +b21 +β12 )+α2 b21 −b1 (α1 b2 (8 2c2 +8γ2 +3)+α2 β1 c1 ) γ1 = − √ b21 ( 2a2 +α2 ) }, √ s5 = {β2 = 2b , √2 √ √ a1 b2 β1 (−8 2c2 +8γ2 +3)− 2a2 (−β1 b1 c1 +b21 +β12 )+α2 β12 +α2 b21 −α2 β1 b1 c1 +α1 b2 b1 (8 2c2 −8γ2 −3) γ1 = 2 √ b1 ( 2a2 −α2 ) }, s6 = {b1 = 0, b2 = 0, β2 = 0}, s7 = {b1 = 0, b2 = 0, β2 = 0}, s8 = {b1 = 0, α2 = − a1 b2 (8γ 3β1 2 +3) , β2 = 0}, s9 = {b1 = 0, β1 = 0, β2 = 0}, √ √ √ s10 = {b1 = 0, c2 = 16 1 (− β1 (2aa21+b2 2α2 ) − 2(8γ2 + 3)), β2 = − 2b2 }, √ s11 = {b1 = 0, β1 = 0, β2 =√− 2b2 }, (35) {s8} √ s12 = {b1 = 0, c2 = β1 (α2 − 2a √2 )+a1 b2 (8γ2 +3) , β2 = 2b2 }, √ 1 2 8 2a b s13 = {b1 = 0, β1 = 0, β2 = 2b2 }, s14 = {α1 = a1b1β1 , α2 = 0, β2 = 0}, 3 s15 = {α2 = 0, β2 = 0, γ2 = √ − 8 }, √ a1 β1 s16 = {α1 = b1 , α2 = − 2a2 , β2 = − 2b2 }, √ √ s17 = {c2 = − 8γ8√2 +3 , α2 = − 2a 2 , β 2 = − 2b2 }, a1 β1 2 √ √ s18 = {α1 = b1 , α2 = 2a2 , β2 = 2b2 }, √ √ s19 = {c2 = 8γ8√ 2 +3 , α2 = 2a2 , β2 = 2b2 }, 2 √ √ s20 = {a1 = 0, b1 = 0, α2 = − √ 2a2 , β2 =√− 2b2 }, s21 = {a1 = 0, b1 = 0, α2 = 2a2 , β2 = 2b√ 2 }, s22 = {a2 = 0, c2 = − 8γ8√ 2 +3 , α 2 = 0, β 2 = − 2b2 } and 2 √ s23 = {a2 = 0, c2 = 8γ8√ 2 +3 2 , α 2 = 0, β2 = 2b 2 }. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 18 The allowed solutions which do not contradict (11) are s3 , s4 , s5 , s8 , s10 and s12 of (35). Then we have six different cases. Case 1: We consider s3 of (35). We solve now the first, the fifth and the sixth equation of (34) to obtain one of the following sets of real solutions d1 = {b2 = 0}, d2 = {α1 = a1b1β1 , α2 = 0}, d3 = {α2 = 0, γ2 = − 38 }, d4 = {a1 = 0, b1 = 0, α2 = 0}, d5 = {b1 = 0, b2 = 0, α2 = 0}, d6 = {b1 = 0, b2 = 0, β1 = 0}, d7 = {b1 = 0, α2 = 0, β1 = 0}, d8 = {b1 = 0, α2 = 0, γ2 = − 83 }, a1 β1 d9 = {b2 = 3b 16 , α1 = b1 , α2 = 0}, 1 −16b1 c2 +3β1 )(α1 b1 −a1 β1 ) d10 = {b2 = 16 , α2 = b1 (8γ2 +3)(3b1 c148β 3b1 2 2 1 (b +β ) , 1 1 b (b β (a (18c2 −192c c +512c2 +9)−9α c +48α c )+3β12 (3a1 c1 −16a1 c2 +3α1 )−2α1 b21 (3c1 −16c2 )2 ) a2 = 1 1 1 1 1 2 1 2 1 1 1 2 48β1 (b21 +β12 ) }, (36) {d6} 3b1 3 d11 = {b2 = 16 , α2 = 0, γ2 = − 8 }, d12 = {a2 = ab11b2 , c2 = b2 (b1 bc21 +β1 ) , α2 = 0}, 1 d13 = {a2 = ab11b2 , α1 = a1b1β1 , α2 = 0}, d14 = {b2 = 3b 16 , α1 = 0, β1 = 0}, 1 d15 = {a1 = 0, b1 = 0, b2 = 0, α2 = 0}, d16 = {a2 = 3a 3b1 3c1 16 , b2 = 16 , c2 = 16 , β1 = 0}, 1 24a1 γ2 +9a1 −96α2 c1 +512α2 c2 d17 = {a2 = 128γ2 +48 , b2 = 3b 16 , α1 = 0, β1 = 0}, 1 3b1 3c1 3 d18 = {b2 = 16 , c2 = 16 , β1 = 0, γ2 = − 8 }, d19 = {b2 = 3b 16 , α2 = 0, β1 = 0, γ2 = − 8 } and 1 3 32α2 c2 −6α2 c1 3b1 d20 = {a1 = 0, a2 = 8γ2 +3 , b2 = 16 , α1 = 0, β1 = 0}. All real solutions contradict (11) except d10 , d16 and d18 of (36). Then we consider these three different subcases. Subcase 1.1: We consider d10 of (36). The fourth equation of (34) becomes   b1 (8γ2 + 3) (α1 b1 − a1 β1 ) 2 b1 (3c1 − 16c2 ) (3c1 − 16c2 ) 2 + (8γ2 + 3) 2 + 48β1 γ2 (4γ2 + 3) = 0, 768β1 (b21 + β12 ) which has one of the following sets of real solutions z1 = {b1 = 0}, z2 = {β1 = αa11b1 }, b (3c −16c )(9c21 −96c2 c1 +256c22 +(8γ2 +3)2 ) z3 = {β1 = − 1 1 2 48γ2 (4γ2 +3) }, z4 = {γ2 = − 38 }, z5 = {a1 = 0, b1 = 0}, (37) {z3} z6 = {a1 = 0, α1 = 0}, z7 = {b1 = 0, γ2 = − 34 }, z8 = {b1 = 0, γ2 = 0}, z9 = {c2 = 3c 1 3 16 , γ2 = − 4 } and 3c1 z10 = {c2 = 16 , γ2 = 0}. All real solutions contradict (11) except z3 , z9 and z10 of (37). Then we have three different subcases. For these cases we have now a piecewise continuous differential systems formed by systems (8) and (10). Subcase 1.1.1: We consider z3 of (37). We solve the algebraic system (12) and we get the pair (y1 , y2 ) such that 16c2 16c2 y1 = − and y2 = − . 3b1 3b1 Then we have no periodic orbits, and then no limit cycles formed by systems (8) and (10) in this case. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 19 Subcase 1.1.2: We consider z9 of (37). We solve the algebraic system (12) and we get the pair (y1 , y2 ) such that c1 c1 y1 = − and y2 = − . b1 b1 Then we have no periodic orbits, and then no limit cycles formed by systems (8) and (10) in this case. Subcase 1.1.3: We consider z10 of (37). We solve the algebraic system (12) and we get the pair (y1 , y2 ) such that c1 c1 y1 = − and y2 = − . b1 b1 Then we have no periodic orbits, and then no limit cycles formed by systems (8) and (10) in this case. Subcase 1.2: We consider d16 of (36). The fourth equation of (34) becomes 1 α1 b1 (8γ2 + 3) (16α2 (8γ2 + 3) − 9α1 ) = 0, 256 which has one of the following sets of real solutions z1 = {b1 = 0}, z2 = {α1 = 0}, (38) {z4} z3 = {α1 = 16 9 α2 (8γ2 + 3)} and z4 = {γ2 = − 83 }. All real solutions contradict (11) except z3 and z4 of (38). Then we have three different subcases. We have now a piecewise continuous differential systems formed by systems (8) and (10). Subcase 1.2.1: We consider z3 of (38). We solve the algebraic system (12) and we get 2c1 y1 = − − y2 . b1 Then we have a continuum of periodic orbits, and then no limit cycles formed by systems (8) and (10) in this case. Subcase 1.2.2: We consider z4 of (38). First integrals have a singularity on the y−axis. Then we have no periodic orbits, and then no limit cycles formed by systems (8) and (10) in this case. Subcase 1.3: We consider d18 of (36). We have immediately a piecewise continuous differential system formed by systems (8) and (10). the first integral of (10) has a singularity on the y−axis. Then we have no periodic orbits, and then no limit cycles formed by systems (8) and (10) in this case. Case 2: We consider s4 of (35). We solve now the sixth equation of (34) to get one of the following sets of real solutions s1 = {b2 = 0}, s2 = {b2 = 3b81√} and s3 = {α2 = − 2a2 }. The only allowed real solution is s2 , we consider then b2 = 3 b1 /8. Solving the first equation of (34), this gives all the following sets of real solutions √ √ u1 = {α2 = 24β1 (b12 +β 2 ) [β1 b21 (8a1 (8γ2 + 3)c2 − 9a1 c1 (8 2c2 + 8γ2 + 3) + 128 2a1 c22 √ 1 1 √ √ 2 √ −8 √2a1 γ2 (4γ2 + 3) − 24 2a2 + 9α1 (8 2c√ 2 + 8γ2 + 3)) − 9a1 β1 b√ 1 (8 2c2 + 8γ2 + 3) −24√ 2a2 β13 + α1 b31 (−8(8γ2 + 3)c2 + 9c1 (8 2c2 + 8γ2 + 3) − 128 2c22 +8 2γ2 (4γ2 + 3))]}, (39) {u5} u2 = {b1 = 0, √ β1 = 0}, 8( 2(8γ +3)c +32c2 −2γ (4γ2 +3)) 2 2 2 u3 = {c1 = √ 2 9(16c2 + 2(8γ2 +3)) , β1 = 0} and u4 = {α1 = 0, β1 = 0}. All real solutions contradict (11) except u1 and u3 of (39). Then we have two different subcases. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 20 Subcase 2.1: We consider u1 of (39). Solving the fifth equation of (34) we obtain 1 √ z1 = {a2 = 72β (b2 +β 2 )(16c + √ 2(8γ +3)) [b1 (β1 b1 (a1 (512 2(8γ2 + 3)c22 − 64(16γ22 + 12γ2 − 9)c2 1 1√ 1 2 2 √ + 54c21 (16c2√ + 2(8γ2 + 3)) − 6c1 (80 2(8γ2 + 3)c2 + 1024c 2 2 √2 + 128γ2 + 96γ2 + 227) 3 3 2 2 + 8192c2 + 2(−512γ2 − 576γ √ 2 + 72γ2 + 81)) +23α1 (2(68 2(8γ√ 2 + 3)c2 + 640c2 + 224γ2 + 168γ2 + √27) − 9c1 (16c2 + 2(8γ2 + 3)))) + 3β1 (9a1 c1 (16c √2 + 2(8γ2 + 3)) − 2a1 (68 2(8γ2 + 3)c√2 + 640c22 + 224γ22 + 168γ √ 2 + 27) + 9 2α1 (8γ2 + 3) + 144α1 c2 ) 2 2 − 2α1 b√1 (27c1 (16c2 + 2(8γ 2 + 3)) − 3c 1 (80 2(8γ 2 + 3)c√2 + 1024c22 + 128γ22 + 96γ2 + 27) + 8(32 2(8γ2 + 3)c22 + (−64γ22 − 48γ2 + 9)c2 + 512c32 − 2γ2 (32γ22 + 36γ2 + 9))))]}, z2 = {b1 = 0}, z3 = {β1 = αa11b1 }, z4 = {a1 = 0, b1 = 0}, (40) {z5} z5 = {a1 = 0, α1 = 0}, z6 = {b1 = 0, c2 = − 8γ8√ 2 +3 2 }, z7 = {b1 = 0, β1 = 0}, z8 = {a1 = αβ11b1 , c2 = − 8γ8√ 2 +3 2 }, √ 3β1 − 2b1 (8γ2 +3) z9 = {c1 = 6b1 √ , c2 = − 8γ8√2 +3 }, 1  2 z10 = {c1 = 6 √32c 2 + 2 (8γ 2 + 3) , β 1 = 0}, 8( 2(8γ2 +3)c2 +32c22 −2γ2 (4γ2 +3)) z11 = {c1 = √ 9(16c2 + 2(8γ2 +3)) , β1 = 0}, z12 = {α1 = 0, β1 = 0} and z13 = {c1 = − 8γ3√ 2 +3 2 , c2 = − 8γ8√ 2 +3 2 , β1 = 0}. All real solutions contradict (11) except z1 and z9 of (40). Then we have two different subcases. Subcase 2.1.1: We consider z1 of (40). We solve now the fourth equation of (34) we obtain v1 = {b1 = 0}, v2 = {β1 = αa11b1 }, 3 √ 2 √ v3 = {β1 = −[b1 (243c √ 1 (16 2(8γ2 + 3)c2 + 128c2 + (8γ2 + √ 3)2 ) − 648c21 (24 2(8γ2 + 3)c22 3 2 + √ 2 + 9c32 − 2γ22 (32γ2 + 36γ2 + 9)) + 3c1 (10240 256c 2(8γ2 + 3)c32 + 3072(4γ22 + 3γ2 + 6)c22 4 4 3 2 + 8 2(512γ2 + 576γ2 + 1656γ √ 2 + 567)c42 + 114688c22 + 10240γ2 + 315360γ√2 + 11808γ 2 3 2 + 4536γ2 + 729) − 8(3072 2(8γ2 + 3)c2 + 128(64γ2 + 48γ √ 2 + 81)c2 4+ 16 2(128γ 2 + 144γ2 2 2 5 3 2 + 396γ2 + 135)c2 − 9(32γ2 √ + 24γ2 − 27)c2 + 32768c2 − 2γ2 (1024γ2 + 1920γ√ 2 + 2016γ2 + 1188γ2 + 243)))]/[24(512 2(8γ2 + 3)c32 + 24(256γ22 + 192γ2 + 33)c22 + 2(2048γ23 + 2304γ22 (41) {v6} + 792γ2 + 81)c2 + 2048c42 + γ2 (512γ23 + 768γ22 + 396γ2 + 81))]}, v4 = {a1 = 0, b1 = 0}, v5 = {a1 = 0, α1 = 0}, √ v6 = {b1 = 0, γ2 = − 2c √ 2 }, v7 = {c1 = 8c32 , γ2 = √ − 2c2 }, 3 v8 = {b1 = 0, γ2 = − 2c √ 2 − 4 }3 and 8c2 v9 = {c1 = 3 , γ2 = − 2c2 − 4 }. All real solutions contradict (11) except v3 , v7 and v9 of (41). Then we consider three different subcases. These cases give continuous piecewise differential systems formed by systems (8) and (10). Subcase √ 2.1.1.1: We consider v3 of (41). Now the algebraic system (12), by using only two parameters β2 = − 2 b2 , b2 = 3 b1 /8 and the change of variables y1 + y2 = t and y1 y2 = s, is equivalent to the following algebraic system F1 + F2 s + F3 t = 0, (42) {ste} F4 s + F5 s + (F6 + F7 s) t + (F8 + F9 s) t + F10 t3 + F11 = 0. 2 2 Where F1 = 2(γ1 + 1)(b1 c1 (γ1 + 1) + β1 (γ1 − c21 )), LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 21 F2 = 2β1 (−b1 β1 c1 + b21 (γ1 + 1) + β12 ), F3 = b21 (γ1 + 1)2 + β12 (−c21 + 2γ1 + 1), √ F4 = 324b41 (16c2 + 2(8γ2 + 3)), √ √ F5 = −18b21 (8γ2 + 3)(96(8γ2 + 3)c2 + 256 2c22 + 2(64γ2 (4γ2 + 3) + 45)), √ F6 = −18 2b21 (8γ2 + 3)(−256c22 + 32γ2 (4γ2 + 3) + 9), √ F7 = −324b41 (16c2 + 2(8γ2 + 3)), F8 = 9b1 (8γ2 + 3)2 (−256c22 + 32γ2 (4γ2 + 3) + 9), √ F9 = 54b31 (64 2(8γ2 + 3)c2 + 256c22 + 96γ2 (4γ2 + 3) + 63), F10 = 27b31 (−256c22 + 32γ2 (4γ2 + 3) + 9), and √ √ F11 = 16(8γ2 + 3)3 ((8γ2 + 3)c2 + 16 2c22 − 2γ2 (4γ2 + 3)). Solving the first equation of (42) with respect to s we get  −2b1 c1 (γ1 + 1) 2 + b21 (γ1 + 1) 2 (−t) + β1 c21 (2γ1 + β1 t + 2) − 2γ1 (γ1 + 1) − t (2β1 γ1 + β1 ) s= . (43) {se} 2β1 (−b1 β1 c1 + b21 (γ1 + 1) + β12 ) We invoke parameters γ1 from s4 of (35), and α2 from u1 of (39) replacing them together with this last value of s into the second equation of (42), we get a cubic polynomial in t which has the following form √  ! 16 2 (8γ2 + 3) c2 + 32c22 − 2γ2 (4γ2 + 3) √  + t (G3 t2 + G2 t + G1 ) = 0. (44) {te} 9b1 16c2 + 2 (8γ2 + 3) After replacing remaining parameters, i.e.; a2 from (40) and β1 from (41), coefficients G1 , G2 and G3 are given in Appendix 4. Solving (44) with respect to t we get √ √  16 8 2γ2 c2 + 32c22 + 3 2c2 − 8γ22 − 6γ2 t=− √ √  , (45) {te1} 9b1 8 2γ2 + 16c2 + 3 2 or G3 t2 + G2 t + G1 = 0. (46) {te2} From (45), replacing t into the first equation of (42), we obtain √ √  64 8 2γ2 c2 + 32c22 + 3 2c2 − 8γ22 − 6γ2 2 s= √ √  . 81b21 8 2γ2 + 16c2 + 3 2 2 and since the algebraic system y1 + y2 = t, y1 y2 = s has a unique solution with respect to y1 and y2 except a permutation between y1 , y2 which is given by 1 p  1 p 2  y1 = t − t2 − 4s and y2 = t − 4s + t , 2 2 Then, after replacing by these last values of t and s, we get √ √  8 8 2γ2 c2 + 32c22 + 3 2c2 − 8γ22 − 6γ2 y1 = y2 = − √ √  . 9b1 8 2γ2 + 16c2 + 3 2 The remaining equation (46) gives together with the first equation of (42) two solutions with respect to t and s, the first is  √  p F3 − G22 −4G1 G3 −G2 2 − G2 − 4G1 G3 − G2 − 2G3 − F1 t= and s = , 2G3 F2 LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 22 and the second √  p F3 G22 −4G1 G3 −G2 G22 − 4G1 G3 − G2 − 2G3 − F1 t= and s = . 2G3 F2 Therefore we have two pairs of (y1 , y2 ), the first pair r √  √  √ 8F1 G23 −4F3 G22 −4G1 G3 +G2 G3 −2F2 G1 G3 +F2 G22 −4G1 G3 +G2 G2 p 2G3 F2 G23 + G22 − 4G1 G3 + G2 y1 = − , 4G3 and r √  √  √ 8F1 G23 −4F3 G22 −4G1 G3 +G2 G3 −2F2 G1 G3 +F2 G22 −4G1 G3 +G2 G2 p − 2G3 F2 G23 + G22 − 4G1 G3 + G2 y2 = − . 4G3 The second pair is given as follows r  √   √  2F2 G22 − G22 −4G1 G3 G2 −2G1 G3 +8G3 F3 G22 −4G1 G3 −F3 G2 +2F1 G3 p G3 F2 G23 − G22 − 4G1 G3 + G2 y1 = − , 4G3 and r  √   √  2F2 G22 − G22 −4G1 G3 G2 −2G1 G3 +8G3 F3 G22 −4G1 G3 −F3 G2 +2F1 G3 p G3 F2 G23 + G22 − 4G1 G3 − G2 y2 = . 4G3 Then the continuous piecewise differential systems formed by (8) and (10) can have at most two limit cycles in this case. Numerical example with one limit√cycle. 421680(36883794312444489036 2−36721536410702014105) a1 = 5, a2 = 4801022290403556862615777 , b1 = −1, b2 = − 38 , c1 = 34 , c2 = 0, α1 = √ √ 1265040(4441267076299462989 2−13053293229304909636) 10, α2 = − 4801022290403556862615777 , β1 = 5068448 2+6938793 2698752 , β2 = 4√ 3 2 , γ1 = √ 332937 2+688364 144576 and γ2 = 1. Figure 1 shows the crossing limit cycle in this case. Here the points of intersection with the y−axis are y1 = 12.7503.. and y2 = −1.24732... Subcase 2.1.1.2: We consider v7 of (41). As in subcase 2.1.1.1. the continuous piecewise differential systems formed by (8) and (10) can have at most two limit cycle in this case. Indeed solving the algebraic system (12), we obtain three pairs (y1 , y2 ). The first pair is 8c2 y1 = y2 = − . 3b1 The second pair is √ √ √ √ √ √ √ 3 2b31 + 64 2c2 β1 b21 − 21β1 b21 + 3 2 Sβ1 (2 2b1 + β1 )b21 − 6 2β12 b1 + 32c2 β12 b1 + 3 S1 y1 = − √ , 12b21 β1 (2 2b1 + β1 ) √ √ √ √ √ √ √ −3 2b31 − 64 2c2 β1 b21 + 21β1 b21 + 3 2 Sβ1 (2 2b1 + β1 )b21 + 6 2β12 b1 − 32c2 β12 b1 − 3 S1 y2 = √ . 12b21 β1 (2 2b1 + β1 ) And the third pair is √ √ √ √ √ √ √ −3 2b31 − 3 2β1 (2 2b1 + β1 ) Rb21 − 64 2c2 β1 b21 + 21β1 b21 + 6 2β12 b1 − 32c2 β12 b1 + 3 S1 y1 = √ , 12b21 β1 (2 2b1 + β1 ) √ √ √ √ √ √ √ −3 2b31 − 64 2c2 β1 b21 + 21β1 b21 + 3 2 Rβ1 (2 2b1 + β1 )b21 + 6 2β12 b1 − 32c2 β12 b1 + 3 S1 y2 = √ , 12b21 β1 (2 2b1 + β1 ) with √ √ R1 = −b21 (−2b41 + 14 2β1 b31 + 39β12 b21 + 16 2β13 b1 + 4β14 ), LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 23 Figure 1. The crossing limit cycle of systems (8) and (10). The picture on the left is a zoom of the limit cycle in a neighborhood of the y-axis. {fig1} √ √ S1 = b21 (2b41 − 14 2β1 b31 − 39β12 b21 − 16 2β13 b1 − 4β14 ), √ √ √ √ √ √ −6b51 + 12 2β1 b41 + 31β12 b31 + 2(10β13 − 3 R1 )b21 + (2β14 − 9β1 R1 )b1 − 2 2β12 R1 S= √ b31 β12 (2 2b1 + β1 )2 and √ √ √ √ √ √ −6b51 + 12 2β1 b41 + 31β12 b31 + 2(10β13 + 3 R1 )b21 + (2β14 + 9 R1 β1 )b1 + 2 2 R1 β12 R= √ . b31 β12 (2 2b1 + β1 )2 Since the invariant straight lines for the generalized systems (8) and (10) are given respectively as follows 8c2 α1 x + 1 y=− − , 3b1 β1 which intersects the y−axis at the point y0 1 = − 8c 3b1 − 2 1 β1 , and √ √ √ ! 1 3 2 − 16c2 3x 2a1 b1 + 2a1 β1 + 2α1 β1 − 2α1 b1 y= − , 6 b1 b21 + β12 √ 3 2−16c2 which intersects the y−axis at the point y0 2 = 6b1 . In both pairs of y1 , y2 give y1 < y2 , and in addition we have • If y0 1 < y0 2 we get y0 2 ∈ [y1 , y2 ] of the second pair (y1 , y2 ); • If y0 1 > y0 2 we get y0 2 ∈ [y1 , y2 ] of the third pair (y1 , y2 ); • If y0 1 = y0 2 we get y0 2 ∈ [y1 , y2 ] of the second and the third pair of y1 , y2 . Then the continuous piecewise differential systems formed by (8) and (10) can have at most one limit cycle. Subcase 2.1.1.3: We consider v9 of (41). As in subcase 2.1.1.1. the continuous piecewise differential systems formed by (8) and (10) can have at most two limit cycle in this case. Solving the algebraic system LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 24 (12), we obtain three pairs (y1 , y2 ). The first pair is 8c2 y1 = y2 = − . 3b1 The second pair is √ √ √ √ √ √ 2 S1 3 R1 + β1 b21 (64 2c2 + 21) − 2β12 b1 (16c2 + 3 2) + 3 2b31 y1 = − − √ , 4 12b21 β1 (2 2b1 − β1 ) √ √ √ √ √ S1 3 R1 + β1 b21 (64 2c2 + 21) − 2β12 b1 (16c2 + 3 2) + 3 2b31 y2 = √ − √ . 2 2 12b21 β1 (2 2b1 − β1 ) And the third pair is √ √ √ √ √ √ √ −3 2β1 b21 (2 2b1 − β1 ) S2 − 21β1 b21 + 6 2β12 b1 + 3 R1 − 64 2β1 b21 c2 + 32β12 b1 c2 − 3 2b31 y1 = √ , 12b21 β1 (2 2b1 − β1 ) √ √ √ √ √ √ √ 3 2β1 b21 S2 (2 2b1 − β1 ) − 21β1 b21 + 6 2β12 b1 + 3 R1 − 64 2β1 b21 c2 + 32β12 b1 c2 − 3 2b31 y2 = √ , 12b21 β1 (2 2b1 − β1 ) with √ √ R1 = b21 (14 2β1 b31 − 39β12 b21 + 16 2β13 b1 + 2b41 − 4β14 ), √ √ √ √ √ √ 12 2β1 b41 − 31β12 b31 + 2b21 (3 R1 + 10β13 ) − 2β14 b1 − 9β1 b1 R1 + 2 2β12 R1 + 6b51 S1 = − √ , b31 β12 (β1 − 2 2b1 )2 and √ √ √ √ √ √ −12 2β1 b41 + 31β12 b31 + 2b21 (3 R1 − 10β13 ) + 2β14 b1 − 9β1 b1 R1 + 2 2β12 R1 − 6b51 S2 = √ . b31 β12 (β1 − 2 2b1 )2 The invariant straight lines for the generalized systems (8) and (10) are given respectively by 8c2 α1 x + 1 y=− − , 3b1 β1 which intersects the y−axis at the point y0 1 = − 8c 1 3b1 − β1 , and 2 √ √  √ x 2a1 b1 − 2a1 β1 + 2α1 β1 + 2α1 b1 16c2 + 3 2 y=− − , 2 (b21 + β12 ) 6b1 √ which intersects the y−axis at the point y0 2 = − 16c26b+3 1 2 . The third pair (y1 , y2 ) give y1 < y2 , and in addition we have • If y0 1 < y0 2 we get y0 2 ∈ [y1 , y2 ] of the third pair (y1 , y2 ); • If y0 1 > y0 2 we get y0 2 ∈ [y1 , y2 ] of the secondd pair (y1 , y2 ); • If y0 1 = y0 2 we get y0 2 ∈ [y1 , y2 ] of the second and the third pair of y1 , y2 . Then the continuous piecewise differential systems formed by (8) and (10) can have at most one limit cycle. Subcase 2.1.2: We consider z9 of (40). We solve now the fourth equation of (34) becomes √  −9 2 (α1 b1 − a1 β1 ) b21 (6a1 β1 − 32a2 β1 ) + (3a1 − 32a2 ) β13 + 9α1 β12 b1 + 6α1 b31 = 0. All sets of real solution for this last equation is given as follows 32a2 β1 (b21 +β12 )−3a1 (2b21 β1 +β13 ) e1 = {α1 = 9β12 b1 +6b31 }, α1 b1 e2 = {β1 = a1 }, e3 = {a1 = 0, b1 = 0}, (47) {e8} e4 = {a1 = 0, α1 = 0}, e5 = {a2 = 3a 32 , b1 = 0} and 1 e6 = {b1 = 0, β1 = 0}. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 25 The only allowed solution is e1 . Then we have   32 a2 β1 b21 + β12 − 3 a1 2 b21 β1 + β13 α1 = . 9 β12 b1 + 6 b31 In this case the algebraic system (12) has no real solutions √ √ 2 b1 (8γ2 + 3) + (3 − 3i)β1 2 b1 (8γ2 + 3) + (3 + 3i)β1 y1 = and y 2 = 6b21 6b21 and then the continuous piecewise differential systems formed by (8) and (10) has no limit cycle in this case. Subcase 2.2: We consider u3 of (39). Solving the fifth equation of (34) we obtain all sets of real solutions as follows w1 = {b1 = 0}, w2 = {α1 = 0}, √ √ √ √ 8a2 (32(8γ2 +3)c2 +128 2c22 + 2(64γ22 +48γ2 +27))−27a1 (32(8γ2 +3)c2 +128 2c22 + 2(8γ2 +3)2 ) w3 = {α2 = √ 16(64 2(8γ2 +3)c2 +512c22 +256γ22 +192γ2 +27) }, (48) {w8} 3a1 16γ√2 +9 w4 = {a2 = 8 , c2 = − 16 2 } and 3a1 16γ√2 +3 w5 = {a2 = 8 , c2 = − 16 2 }. The allowed solutions are w3 , w4 and w5 . Then we have three different subcases Subcase 2.2.1: We consider w3 of (48). We solve now the fourth equation of (34) and we obtain e1 = {b1 = 0}, e2 = {α1 = 0}, √ e3 = {α1 = −[(3a1 − 8a2 )(8192(8γ √ 42 +√3)c32 + 192 2(256γ22 + 192γ2 + 87)c22 + 16(2048γ23 + 2304γ22 + 2088γ2 + 567)c2 + 16384 √2c2 + 2(4096γ24 + 6144γ23 + 8352γ 2 √ 23+ 4536γ23+ 729))] 2 2 2 / [54(1536(8γ √ 2 + 3)c2 + 24 2(256γ2 + 192γ2 + 33)c2 + 4096 2c2 + 2048γ2 + 2304γ2 + 792γ2 + 81)]}, e4 = {γ2 = − 2c2 − 38 },√ e5 = {a2 = 3a81 , γ2 = −√2c2 − 16 9 } and 3a1 3 e6 = {a2 = 8 , γ2 = − 2c2 − 16 }. The allowed solution is e3 . The algebraic system (12) in this case has complex solutions and then the continuous piecewise differential systems formed by (8) and (10) cannot have limit cycles. Subcase 2.2.2: We consider w4 of (48). We solve now the fourth equation of (34) which becomes  √  27α1 b1 3 2a1 + 6α1 + 8α2 = 0, we obtain e1 = {b1 = 0}, e2 = {α1 = 0} √and  e3 = {α2 = −3 8 2a1 + 2α1 }. √ √ The allowed solution is e3 . The algebraic system (12) in this case gives y1 = y2 = 4 2γ3b2 +3 1 2 or has complex solutions √ √ √ √ √ √ 8 2γ2 + 6 2 − 3i 10 8 2γ2 + 6 2 + 3i 10 y1 = and y2 = , 6b1 6b1 and then the continuous piecewise differential systems formed by (8) and (10) cannot have limit cycles. Subcase 2.2.3: We consider w5 of (48). We solve now the fourth equation of (34) which becomes  √  27α1 b1 3 2a1 − 6α1 + 8α2 = 0, we obtain e1 = {b1 = 0}, e2 = {α1 = 0} √and  e3 = {α2 = −3 8 2a1 − 2α1 }. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 26 √ 4 2γ2 The allowed solution is e3 . The algebraic system (12) in this case gives y1 = y2 = 3b1 , or has complex solutions √ √ √ √ 8 2γ2 − 3i 10 8 2γ2 + 3i 10 y1 = and y2 = , 6b1 6b1 and then the continuous piecewise differential systems formed by (8) and (10) cannot have limit cycles. Case 3: We consider s5 of (35). This case gives the same results as in the case 2 and all bifurcated cases given in case 2 have the same conclusion as follows: We solve the sixth equation of (34) to get one of the following sets of real solutions s1 = {b2 = 0}, s2 = {b2 = 3b √8 } and 1 s3 = {α2 = 2a2 }. The only allowed real solution is s2 , then b2 = 3 b1 /8. Solving the first equation of (34), this gives all the following sets of real solutions √ √ √ u1 = {α2 = 24β1 (b12 +β 2 ) [9a1 β12 b1 (8 2c2 − 8γ2 − 3) + 24 2a2 β13 − α1 b31 (9c1 (8 2c2 − 8γ2 − 3) 1 1 √ 2 √ +8((8γ2 + 3)c √2 − 16 2c2 + 2γ2 (4γ2 + 3))) √ √ b21 β1√ (9a1 c1 (8 2c2 −√8γ2 − 3) + 8a1 ((8γ2 + 3)c2 − 16 2c22 + 2γ2 (4γ2 + 3)) +24 2a2 + 9α1 (−8 2c2 + 8γ2 + 3))]}, (49) {u6} u2 = {b1 = 0, √β1 = 0}, 8( 2(8γ +3)c −32c2 +2γ (4γ2 +3)) 2 u3 = {c1 = √ 2 2 2 , β1 = 0} and 9( 2(8γ2 +3)−16c2 ) u4 = {α1 = 0, β1 = 0}. All real solutions contradict (11) except u1 and u3 of (49). Then we have two different subcases. Subcase 3.1: We consider u1 of (49). Solving the fifth equation of (34) we obtain √ z1 = {a2 = 72β (b2 +β 2 )(√12(8γ +3)−16c ) [b1 (β1 b1 (a1 (512 2(8γ2 + 3)c22 + 64(16γ22 + 12γ2 − 9)c2 √ 1 1 1 2 2 √ + 54c21 ( 2(8γ√ 2 + 3) − 16c2 ) − 6c1 (80 2(8γ2 + 3)c2 − 1024c 2 2 √2 − 128γ2 − 96γ2 − 27) 3 3 2 − 8192c√2 + 2(−512γ2 − 576γ2 + 72γ2 + 81)) − 3α1 (−136 2(8γ2 + 3)c2 √ + 9c1 (√2(8γ2 + 3) − 16c2 ) + 1280c22 + 448γ22 + 336γ2 + 54)) + 3β12 (a √ 1 (−136 2(8γ2 + 3)c2 + 9c1 ( 2(8γ2 √ + 3) − 16c2 ) + 1280c22 + 448γ22 + √ 336γ 2 + 54) + 9α1 ( 2(8γ2 + 3) − 16c2 )) − 2α1 b21 (27c √ 1 2 ( 2(8γ 2 + 3) − 16c 2 ) + c 1 (−240 2(8γ 2 + 3)c2 √ + 3072c 2 2 2 + 384γ2 + 288γ2 + 81) − 8(−32 2(8γ2 + 3)c22 + (−64γ22 − 48γ2 + 9)c2 + 512c32 + 2γ2 (32γ22 + 36γ2 + 9))))]}, z2 = {b1 = 0}, z3 = {β1 = αa11b1 }, z4 = {a1 = 0, b1 = 0}, (50) {z6} z5 = {a1 = 0, α1 = 0}, z6 = {b1 = 0, c2 = 8γ8√ 2 +3 2 }, z7 = {b1 = 0, β1 = 0}, z8 = {a1 = αβ11b1 , c2 = 8γ8√ 2 +3 2 }, √ 3β1 + 2b1 (8γ2 +3) z9 = {c1 = 6b1 √ , c2 = 8γ8√ 2 +3 }, 1  2 z10 = {c1 = 6 √32c 2 − 2 (8γ 2 + 3) , β 1 = 0}, 8( 2(8γ2 +3)c2 −32c22 +2γ2 (4γ2 +3)) z11 = {c1 = √ 9( 2(8γ2 +3)−16c2 ) , β1 = 0}, z12 = {α1 = 0, β1 = 0} and z13 = {c1 = 8γ3√ 2 +3 2 , c2 = 8γ8√ 2 +3 2 , β1 = 0}. All real solutions contradict (11) except z1 and z9 of (50). Then we have two different subcases. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 27 Subcase 3.1.1: We consider z1 of (50). We solve now the fourth equation of (34) we obtain v1 = {b1 = 0}, v2 = {β1 = αa11b1 }, 3 √ 2 2 2 √ 2 v3 = {β1 = −[b1 (−243c√ 1 (16 2(8γ2 + 3)c2 − 128c2 − (8γ√2 + 3) ) − 648c1 (−24 2(8γ2 + 3)c2 3 2 3 2 + 256c2 + 9c2 + √2γ2 (32γ2 + 36γ2 + 9)) − 3c1 (10240 2(8γ2 + 3)c2 − 3072(4γ2 + 3γ2 + 6)c22 + 8 2(512γ23 + 576γ22 + 1656γ 4 4 3 √ 2 + 567)c4 2 − 114688c2 2 − 10240γ2 − 315360γ2 2 − √ 2 − 4536γ2 − 729) − 8(−3072 2(8γ2 + 3)c 11808γ 2 + 128(64γ2 + 48γ2 + 81)c2√ − 16 2(128γ23 + 144γ22 + 396γ2 + 135)c22 − 9(32γ √ 2 2 + 24γ2 − 27)c2 + 32768c52 + 2γ2 (1024γ24 3 2 + 1920γ2 + 2016γ2√ + 1188γ2 + 243)))]/[24(512 2(8γ2 + 3)c32 − 24(256γ22 + 192γ2 + 33)c2 + 2(2048γ23 + 2304γ22 + 792γ2 + 81)c2 − 2048c42 − γ2 (512γ23 + 768γ22 + 396γ2 + 81))]}, 2 v4 = {a1 = 0, b1 = 0}, v5 = {a1 = 0, α1 = √ 0}, v6 = {b1 = 0, γ2 = 2c √ 2 }, v7 = {c1 = 8c32 , γ2 =√ 2c2 },3 v8 = {b1 = 0, γ2 = 2c √ 2 − 4 } and v9 = {c1 = 8c32 , γ2 = 2c2 − 34 }. (51) {v7} All real solutions contradict (11) except v3 , v7 and v9 of (51). Then we consider three different subcases. These cases give continuous piecewise differential systems formed by systems (8) and (10). Subcase √ 3.1.1.1: We consider v3 of (51). Now the algebraic system (12), by using only two parameters β2 = 2 b2 , b2 = 3 b1 /8 and the change of variables y1 + y2 = t and y1 y2 = s, is equivalent to the following algebraic system E1 + E2 s + E3 t = 0, (52) {ste1} E4 s2 + E5 s + (E6 + E7 s) t2 + (E8 + E9 s) t + E10 t3 + E11 = 0. Where  E1 = 2 (γ1 + 1) b1 c1 (γ1 + 1) + β1 γ1 − c21 ,  E2 = 2β1 −b1 β1 c1 + b21 (γ1 + 1) + β12 ,  E3 = b21 (γ1 + 1) 2 + β12 −c21 + 2γ1 + 1 , √  E4 = 324b41 2 (8γ2 + 3) − 16c2 , √ √  E5 = −18b21 (8γ2 + 3) −96 (8γ2 + 3) c2 + 256 2c22 + 2 (64γ2 (4γ2 + 3) + 45) , √  E6 = −18 2b21 (8γ2 + 3) −256c22 + 32γ2 (4γ2 + 3) + 9 , √  E7 = −324b41 2 (8γ2 + 3) − 16c2 ,  E8 = −9b1 (8γ2 + 3) 2 −256c22 + 32γ2 (4γ2 + 3) + 9 , √  E9 = 54b31 64 2 (8γ2 + 3) c2 − 256c22 − 96γ2 (4γ2 + 3) − 63 ,  E10 = 27b31 256c22 − 32γ2 (4γ2 + 3) − 9 , and √ √  E11 = −16 (8γ2 + 3) 3 (8γ2 + 3) c2 − 16 2c22 + 2γ2 (4γ2 + 3) . Solving the first equation of (52) with respect to s we get  −2b1 c1 (γ1 + 1) 2 + b21 (γ1 + 1) 2 (−t) + β1 c21 (2γ1 + β1 t + 2) − 2γ1 (γ1 + 1) − t (2β1 γ1 + β1 ) s= . (53) {se1} 2β1 (−b1 β1 c1 + b21 (γ1 + 1) + β12 ) LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 28 We invoke parameters γ1 from s5 of (35), and α2 from u1 of (49) replacing them together with this last value of s into the second equation of (52), we get a cubic polynomial in t which has the following form √ √  ! 16 8 2γ2 c2 − 32c22 + 3 2c2 + 8γ22 + 6γ2 √ √  + t (H3 t2 + H2 t + H1 ) = 0. (54) {te0} 9b1 8 2γ2 − 16c2 + 3 2 After replacing remaining parameters, i.e.; a2 from (50) and β1 from (51), coefficients H1 , H2 and H3 are given in Appendix 5. Solving (54) with respect to t we get √ √  16 8 2γ2 c2 − 32c22 + 3 2c2 + 8γ22 + 6γ2 t=− √ √  , (55) {te01} 9b1 8 2γ2 − 16c2 + 3 2 or H3 t2 + H2 t + H1 = 0. (56) {te02} Replacing t, from (55), into the first equation of (52), we obtain √ √  64 8 2γ2 c2 − 32c22 + 3 2c2 + 8γ22 + 6γ2 2 s= √ √  . 81b21 8 2γ2 − 16c2 + 3 2 2 and since the algebraic system y1 + y2 = t, y1 y2 = s has a unique solution with respect to y1 and y2 except a permutation between y1 , y2 which is given by 1 p  1 p 2  y1 = t − t2 − 4s and y2 = t − 4s + t , 2 2 Then, after replacing by these last values of t and s, we get √ √  8 8 2γ2 c2 − 32c22 + 3 2c2 + 8γ22 + 6γ2 y1 = y2 = √ √  . 9 16b1 c2 − 8 2b1 γ2 − 3 2b1 The remaining equation (56) gives together with the first equation of (52) two solutions with respect to t and s, the first is  √  p E3 − H22 −4H1 H3 −H2 2 − H2 − 4H1 H3 − H2 − 2H3 − E1 t= and s = , 2H3 E2 and the second √  p E3 H22 −4H1 H3 −H2 H22 − 4H1 H3 − H2 − 2H3 − E1 t= and s = . 2H3 E2 Therefore we have two pairs of (y1 , y2 ), the first pair r √  √  p √ 8E1 H32 −4E3 H22 −4H1 H3 +H2 H3 −2E2 H1 H3 +E2 H22 −4H1 H3 +H2 H2 2 H2 − 4H1 H3 + H2 + 2 H3 E2 H32 y1 = − , 4H3 and r √  √  p √ 8E1 H32 −4E3 H22 −4H1 H3 +H2 H3 −2E2 H1 H3 +E2 H22 −4H1 H3 +H2 H2 H22 − 4H1 H3 − 2H3 E2 H32 + H2 y2 = − . 4H3 The second pair is given as follows r  √   √  p 2E2 H22 − H22 −4H1 H3 H2 −2H1 H3 +8 E3 H22 −4H1 H3 −E3 H2 +2E1 H3 H3 2 − H2 − 4H1 H3 + H2 + H3 E2 H32 y1 = − , 4H3 LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 29 and r  √   √  p 2E2 H22 − H22 −4H1 H3 H2 −2H1 H3 +8 E3 H22 −4H1 H3 −E3 H2 +2E1 H3 H3 H22 − 4H1 H3 − H2 + E2 H32 H3 y2 = . 4H3 Then the continuous piecewise differential systems formed by (8) and (10) can have at most two limit cycles in this case. Remark. Let (0, y1 ) and (0, y2 ) be with y1 < y2 the two points where a limit cycle intersects the discontinuity line x = 0 for a continuous piecewise diferential system (8)+(10). In the cases 2.1.1.1 and 3.1.1.1 we have found an upper bound of two limit cicles. But all the computations that we have done show that when we have two solutions (0, y11 ) and (0, y21 ), and (0, y12 ) and (0, y22 ), they satisfy y11 < y21 < y12 < y22 . This implies that these two possible limit cicles cannot surround the same equilibrium point of the continuous piecewise differential system, and this is not possible because system (8) has a unique center, and if there exist 2 limit cicles of the continuous piecewise differential system (8)+(10) they must surround the same equilibrium point. Of course those computations do not prove that in the cases 2.1.1.1 and 3.1.1.1 cannot exist two limit cicles, but they provide numerical evidences that probably in these cases the upper bound that we have found is two, but the maximum number of limit cicles can be one. Subcase 3.1.1.2: We consider v7 of (51). As in subcase 3.1.1.1. the continuous piecewise differential systems formed by (8) and (10) can have at most two limit cycle in this case. Indeed solving the algebraic system (12), we obtain three pairs (y1 , y2 ). The first pair is 8c2 y1 = y2 = − . 3b1 The second pair is √ √ √ √ 1√ p 3 R1 + β1 b21 (64 2c2 + 21) − 2β12 b1 (16c2 + 3 2) + 3 2b31 y1 = − 2 R2 − √ , 4 12b21 β1 (2 2b1 − β1 ) √ √ √ √ √ R2 3 R1 + β1 b21 (64 2c2 + 21) − 2β12 b1 (16c2 + 3 2) + 3 2b31 y2 = √ − √ . 2 2 12b21 β1 (2 2b1 − β1 ) And the third pair is √ √ √ √ √ √ √ −3 2β1 b21 (2 2b1 − β1 ) R3 − 21β1 b21 + 6 2β12 b1 + 3 R1 − 64 2β1 b21 c2 + 32β12 b1 c2 − 3 2b31 y1 = √ , 12b21 β1 (2 2b1 − β1 ) √ √ √ √ √ √ √ 3 2β1 b21 R3 (2 2b1 − β1 ) − 21β1 b21 + 6 2β12 b1 + 3 R1 − 64 2β1 b21 c2 + 32β12 b1 c2 − 3 2b31 y2 = √ , 12b21 β1 (2 2b1 − β1 ) with √ √ R1 = b21 (14 2β1 b31 − 39β12 b21 + 16 2β13 b1 + 2b41 − 4β14 ), √ √ √ √ √ √ 12 2β1 b41 − 31β12 b31 + 2b21 (3 R1 + 10β13 ) − 2β14 b1 − 9β1 b1 R1 + 2 2β12 R1 + 6b51 R2 = − √ , b31 β12 (β1 − 2 2b1 )2 and √ √ √ √ √ √ −12 2β1 b41 + 31β12 b31 + 2b21 (3 R1 − 10β13 ) + 2β14 b1 − 9β1 b1 R1 + 2 2β12 R1 − 6b51 R3 = √ . b31 β12 (β1 − 2 2b1 )2 The invariant straight lines for the generalized systems (8) and (10) are given respectively by 8c2 α1 x + 1 y=− − , 3b1 β1 which intersects the y−axis at the point y0 1 = − 8c 1 3b1 − β1 , and 2 √ √  √ x 2a1 b1 − 2a1 β1 + 2α1 β1 + 2α1 b1 16c2 + 3 2 y=− − , 2 (b21 + β12 ) 6b1 LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 30 √ which intersects the y−axis at the point y0 2 = − 16c26b+3 1 2 . The third pair (y1 , y2 ) give y1 < y2 , and in addition we have • If y0 1 < y0 2 we get y0 2 ∈ [y1 , y2 ] of the third pair (y1 , y2 ) • If y0 1 > y0 2 we get y0 2 ∈ [y1 , y2 ] of the secondd pair (y1 , y2 ); • If y0 1 = y0 2 we get y0 2 ∈ [y1 , y2 ] of the second and the third pair of y1 , y2 . Then the continuous piecewise differential systems formed by (8) and (10) can have at most one limit cycle Subcase 3.1.1.3: We consider v9 of (51). As in subcase 3.1.1.1. the continuous piecewise differential systems formed by (8) and (10) can have at most two limit cycle in this case. Solving the algebraic system (12), we obtain three pairs (y1 , y2 ). The first pair is 8c2 y1 = y2 = − . 3b1 The second pair is √ √ √ √ √ √ √ −21β1 b21 + 3 2β1 b21 S2 (2 2b1 + β1 ) − 6 2β12 b1 + 3 S4 + 64 2β1 b21 c2 + 32β12 b1 c2 + 3 2b31 y1 = − √ , 12b21 β1 (2 2b1 + β1 ) √ √ √ √ √ √ √ 21β1 b21 + 3 2β1 b21 S2 (2 2b1 + β1 ) + 6 2β12 b1 − 3 S4 − 64 2β1 b21 c2 − 32β12 b1 c2 − 3 2b31 y2 = √ . 12b21 β1 (2 2b1 + β1 ) And the third pair is √ √ √ √ √ √ √ −3 2β1 b21 (2 2b1 + β1 ) S3 + 21β1 b21 + 6 2β12 b1 + 3 S4 − 64 2β1 b21 c2 − 32β12 b1 c2 − 3 2b31 y1 = √ , 12b21 β1 (2 2b1 + β1 ) √ √ √ √ √ √ √ 21β1 b21 + 3 2β1 b21 S3 (2 2b1 + β1 ) + 6 2β12 b1 + 3 S4 − 64 2β1 b21 c2 − 32β12 b1 c2 − 3 2b31 y2 = √ . 12b21 β1 (2 2b1 + β1 ) With √ √ S1 = −b21 (14 2β1 b31 + 39β12 b21 + 16 2β13 b1 − 2b41 + 4β14 ), √ √ √ √ √ √ 12 2β1 b41 + 31β12 b31 + 2b21 (10β13 − 3 S1 ) + b1 (2β14 − 9β1 S1 ) − 2 2β12 S1 − 6b51 S2 = √ , b31 β12 (2 2b1 + β1 )2 √ √ √ √ √ √ 12 2β1 b41 + 31β12 b31 + 2b21 (3 S1 + 10β13 ) + b1 (9β1 S1 + 2β14 ) + 2 2β12 S1 − 6b51 S3 = √ , b31 β12 (2 2b1 + β1 )2 and √ √ S4 = b21 (−14 2β1 b31 − 39β12 b21 − 16 2β13 b1 + 2b41 − 4β14 ). Since the invariant straight lines for the generalized systems (8) and (10) are given respectively as follows 8c2 α1 x + 1 y=− − , 3b1 β1 which intersects the y−axis at the point y0 1 = − 8c 1 3b1 − β1 , and 2 √ √ √ ! 1 3 2 − 16c2 3x 2a1 b1 + 2a1 β1 + 2α1 β1 − 2α1 b1 y= − , 6 b1 b21 + β12 √ 3 2−16c2 which intersects the y−axis at the point y0 2 = 6b1 . In both pairs of y1 , y2 give y1 < y2 , and in addition we have • If y0 1 < y0 2 we get y0 2 ∈ [y1 , y2 ] of the second pair (y1 , y2 ); • If y0 1 > y0 2 we get y0 2 ∈ [y1 , y2 ] of the third pair (y1 , y2 ); • If y0 1 = y0 2 we get y0 2 ∈ [y1 , y2 ] of the second and the third pair of y1 , y2 . LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 31 Then the continuous piecewise differential systems formed by (8) and (10) can have at most one limit cycle. Subcase 3.1.2: We consider z9 of (50). The fourth equation of (34) becomes √  9 2 (α1 b1 − a1 β1 ) b21 (6a1 β1 − 32a2 β1 ) + (3a1 − 32a2 ) β13 + 9α1 β12 b1 + 6α1 b31 = 0. All sets of real solution for this last equation is given as follows 32a2 β1 (b21 +β12 )−3a1 (2b21 β1 +β13 ) e1 = {α1 = 9β12 b1 +6b31 , α1 b1 e2 = {β1 = a1 }, e3 = {a1 = 0, b1 = 0}, (57) {e9} e4 = {a1 = 0, α1 = 0}, e5 = {a2 = 3a 32 , b1 = 0} and 1 e6 = {b1 = 0, β1 = 0}. The only allowed solution is e1 . Then we have   32a2 β1 b21 + β12 − 3a1 2b21 β1 + β13 α1 = . 9β12 b1 + 6b31 In this case the algebraic system (12) has no real solutions √ √ − 2b1 (8γ2 + 3) + (3 + 3i)β1 2b1 (8γ2 + 3) + (−3 + 3i)β1 y1 = and y2 = − 6b21 6b21 and then the continuous piecewise differential systems formed by (8) and (10) has no limit cycle in this case. Subcase 3.2: We consider u3 of (49). Solving the fifth equation of (34) we obtain all sets of real solutions as follows w1 = {b1 = 0}, w2 = {α1 = 0}, √ √ √ √ 8a2 (−32(8γ2 +3)c2 +128 2c22 + 2(64γ22 +48γ2 +27))−27a1 (−32(8γ2 +3)c2 +128 2c22 + 2(8γ2 +3)2 ) w3 = {α2 = − √ 16(−64 2(8γ2 +3)c2 +512c2 +256γ2 +192γ2 +27) 2 2 , (58) {w9} 3a1 16γ√ 2 +3 w4 = {a2 = 8 , c2 = 16 2 } and 3a1 16γ√ 2 +9 w5 = {a2 = 8 , c2 = 16 2 }. The allowed solutions are w3 , w4 and w5 . Then we have three different subcases Subcase 3.2.1: We consider w3 of (58). We solve now the fourth equation of (34) and we obtain e1 = {b1 = 0}, e2 = {α1 = 0}, √ 3 e3 = {α1 = [(3a1 − 8a2 )(−8192(8γ2 + 3)c √2 + 192 √ 2(256γ22 + 192γ2 + 87)c22 − 16(2048γ23 2 4 + 2304γ2 + 2088γ2 + 567)c√ 2 + 16384 2c2 + 2(4096γ24 + 6144γ 3 2 √ 23+ 8352γ23+ 4536γ22+ 729))] 2 2 √ 2 + 3)c2 − 24 2(256γ2 + 192γ2 + 33)c2 − 4096 2c2 + 2048γ2 + 2304γ2 + 792γ2 + 81)]}, / [54(1536(8γ e4 = {γ2 = 2c2 − 38 }, √ e5 = {a2 = 3a81 , γ2 = √2c2 − 16 9 } and 3a1 3 e6 = {a2 = 8 , γ2 = 2c2 − 16 }. The allowed solution is e3 . The algebraic system (12) in this case has complex solutions and then the continuous piecewise differential systems formed by (8) and (10) cannot have limit cycles. Subcase 3.2.2: We consider w4 of (58). We solve now the fourth equation of (34) which becomes  √  −27α1 b1 3 2a1 + 6α1 − 8α2 = 0, we obtain e1 = {b1 = 0}, e2 = {α1 = 0} √and  e3 = {α2 = 38 2a1 + 2α1 }. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 32 √ The allowed solution is e3 . The algebraic system (12) in this case gives y1 = y2 = − 4 3b2γ 1 2 or has complex solutions √ √  √ √  i 3 10 − 8i 2γ2 i 3 10 + 8i 2γ2 y1 = − and y2 = , 6b1 6b1 and then the continuous piecewise differential systems formed by (8) and (10) cannot have limit cycles. Subcase 3.2.3: We consider w5 of (58). We solve now the fourth equation of (34) which becomes  √  27α1 b1 −3 2a1 + 6α1 + 8α2 = 0, we obtain e1 = {b1 = 0}, e2 = {α1 = 0} √and  e3 = {α2 = 38 2a1 − 2α1 }. √ √ The allowed solution is e3 . The algebraic system (12) in this case gives y1 = y2 = −4 2γ2 −3 2 3b1 , or has complex solutions √ √ √  √ √ √ i −8i 2γ2 − 6i 2 + 3 10 −8 2γ2 − 6 2 + 3i 10 y1 = − and y2 = , 6b1 6b1 and then the continuous piecewise differential systems formed by (8) and (10) cannot have limit cycles. Case 4: We consider s8 of (35). We solve now the remaining equations of (34) which are equivalent to the system   a1 b2 (8γ2 + 3) b2 c21 − γ1 + β1 c2 = 0,   a1 (8γ2 + 3) a1 b2 3b2 c1 (γ1 + 1) + β1 γ2 (4γ2 + 3) − 16β1 c22 − 3 a2 β12 c2 + α1 b22 c21 − γ1 = 0, a1 b2 (8γ2 + 3) (a1 b2 (3c1 − 32c2 ) − 3a2 β1 + 3α1 b2 ) = 0, 16a1 b32 (8γ2 + 3) = 0. we get one of the following sets of real solutions s1 = {a1 = 0}, s2 = {γ2 = − 83 }, s3 = {a2 = 0, b2 = 0}, s4 = {b2 = 0, c2 = 0} and s5 = {b2 = 0, β1 = 0}. No allowed solution exists. Then no continuous piecewise differential systems formed by (8) and (10). √  Case 5: We consider s10 of (35). We solve now the sixth equation of (34) which is equivalent to −8b22 2a2 + α2 = 0, and we obtain one of the following sets of real solutions s1 = {b2 = 0}√ and s2 = {α2 = − 2a2 }. No solution is allowed, and hence no continuous piecewise differential systems formed by (8) and (10). Case 6: We consider s12 of (35). Solving the four remaing equations of (34), and we obtain one of the following sets of real solutions n o α2 a1 = 0, a2 = √ , n 2o α a2 = √2 , b2 = 0 , 2 {b2 = 0, β1 = 0} , o n α2 a2 = √ 2 , β1 = 0 and {a2 = 0, b2 = 0, α2 = 0} . No solution is allowed, and hence no continuous piecewise differential systems formed by (8) and (10).  LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 33 3. Conclusion In order to complete the remaining statements for Theorem 1, we have considered the planar continuous piecewise differential systems formed by two a quadratic isochronous centers separated by the y−axis, having in x > 0 the quadratic isochronous center (8), and when x < 0 one of the systems (8), (9), (10) or (10). These four continuous piecewise differential systems have two as an upper bound of the number of crossing limit cycles, and there are a realized examples having one crossing limit cycle. See Theorem 1. In conclusion for these four different continuous piecewise differential systems, we have proved the extension of the 16th Hilbert problem to them. 4. Appendix 1 {app1} The coefficients G1 , G2 and G3 in Subcase 2.1.1.1 in the proof of statement (g) of Theorem 1 are given as follows √ √ G1 = b31 (59049(4503599627370496c14 13 12 2 +3940649673949184 2(8γ2 +3)c2 +35184372088832(1528γ2 (4γ2 +3)+873)c2 +8796093022208 2(8γ2 + √ 3)(1672γ2 (4γ2 + 3) + 981)c11 2 + 1099511627776(γ2 (4γ2 + 3)(83072γ2 (4γ2 + 3) + 97101) + 28350)c 10 + 137438953472 2(8γ + 2 2 3)(γ2 (4γ2 + 3)(95744γ2 (4γ2 + 3) + 114057) + 33939)c92 + 805306368(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(111584γ2 (4γ2 + 3) + 198423) + √ 1880577) + 11874195)c82 + 402653184 2(8γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(18656γ2 (4γ2 + 3) + 33669) + 323919) + 2076435)c72 + √ 176160768(8γ2 +3) (16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(10912γ2 (4γ2 +3)+20007)+195615)+637389)c62 +1048576 2(8γ2 +3)3 (16γ2 (4γ2 + 2 3)(16γ2 (4γ2 +3)(89056γ2 (4γ2 +3)+166077)+1652157)+5479893)c52 +32768(8γ2 +3)4 (64γ2 (4γ2 +3)(8γ2 (4γ2 +3)(103312γ2 (4γ2 + √ 3) + 196209) + 994113) + 13442031)c42 + 8192 2(8γ2 + 3)5 (128γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(10832γ2 (4γ2 + 3) + 20979) + 54189) + √ 1495179)c32 +512(8γ2 +3)6 (8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(24832γ2 (4γ2 +3)+49113)+517833)+911979)c22 +64 2(8γ2 +3)7 (8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(2176γ2 (4γ2 + 3) + 4401) + 47385) + 85293)c2 + (8γ2 + 3)8 (32γ2 (4γ2 + 3) + 27)(256γ2 (4γ2 + 3)(22γ2 (4γ2 + √ 3) + 27) + 2187))c61 − 52488(63050394783186944c15 14 2 + 53620983063379968 2(8γ2 + 3)c2 + 1583296743997440(448γ2 (4γ2 + 3) + √ 257)c13 2 + 1099511627776 2(8γ2 + 3)(170672γ2 (4γ2 + 3) + 101367)c 12 + 1649267441664(4γ (4γ + 3)(170176γ (4γ + 3) + 2 2 2 2 2 √ 201501)+238221)c11 2 +19327352832 2(8γ2 +3)(128γ2 (4γ2 +3)(62568γ2 (4γ2 +3)+76261)+2968965)c 10 +4294967296(2γ (4γ + 2 2 2 √ 3)(32γ2 (4γ2 + 3)(1864192γ2 (4γ2 + 3) + 3402369) + 66117303) + 26720037)c92 + 2415919104 2(8γ2 + 3)(γ2 (4γ2 + 3)(320γ2 (4γ2 + 3)(51568γ2 (4γ2 +3)+97083)+19447857)+4048380)c82 +226492416(32γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(39776γ2 (4γ2 +3)+ √ 100493)+1517967)+10156995)+50813001)c72 +262144 2(8γ2 +3)(64γ2 (4γ2 +3)(4γ2 (4γ2 +3)(128γ2 (4γ2 +3)(358204γ2 (4γ2 +3)+ 951381)+120327525)+209920653)+2184071607)c62 +196608(8γ2 +3)2 (16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(64γ2 (4γ2 +3)(53152γ2 (4γ2 + √ 3)+154773)+10561995)+78664203)+216637659)c52 +36864 2(8γ2 +3)3 (8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(128γ2 (4γ2 +3)(4036γ2 (4γ2 + 3) + 15251) + 2456757) + 20609559) + 31121739)c42 − 512(8γ2 + 3)4 (16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(20γ2 (4γ2 + 3)(3392γ2 (4γ2 + √ 3) − 8631) − 500823) − 22910283) − 82570185)c32 − 24 2(8γ2 + 3)5 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(256γ2 (4γ2 + 3)(1964γ2 (4γ2 + 3) + 2817) + 116721) − 1677429) − 19860147)c22 − 36(8γ2 + 3)6 (8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(768γ2 (4γ2 + 3) + 1447) + √ 26487) + 44469) − 137781)c2 − 2γ2 (4γ2 + 3)(8γ2 + 3)7 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(1504γ2 (4γ2 + 3) + 1863) + 9477))c51 + √ 4374(3891110078048108544c16 15 2 + 3211066534315163648 2(8γ2 + 3)c2 + 1688849860263936(24320γ2 (4γ2 + 3) + 14091)c2 + 14 √ 316659348799488 2(8γ2 + 3)(33040γ2 (4γ2 + 3) + 20249)c13 2 + 2199023255552(512γ 2 (4γ2 + 3)(53185γ 2 (4γ2 + 3) + 65277) + √ 10204623)c122 + 206158430208 2(8γ2 + 3)(64γ2 (4γ2 + 3)(592304γ2 (4γ2 + 3) + 773493) + 16001199)c 11 + 77309411328(256γ (4γ + 2 2 2 √ 3)(2γ2 (4γ2 + 3)(607552γ2 (4γ2 + 3) + 1211461) + 1588041) + 87802137)c10 2 + 8589934592 2(8γ2 + 3)(γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(3061520γ2 (4γ2 +3)+6789231)+309823137)+71758386)c92 +1207959552(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(529760γ2 (4γ2 + √ 3) + 1723731) + 31276287) + 240995979) + 674964333)c82 + 226492416 2(8γ2 + 3)(64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(20680γ2 (4γ2 + 3) + 96791) + 8365815) + 17964909) + 217464831)c72 + 2097152(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(256γ2 (4γ2 + √ 3)(4γ2 (4γ2 + 3)(46816γ2 (4γ2 + 3) + 1316943) + 12078153) + 2755209573) + 8807046813) + 21181880133)c62 − 393216 2(8γ2 + 3)(8γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(90352γ2 (4γ2 + 3) − 273921) − 32568183) − 66914181) − 3691662561) − 4672803249)c52 − 147456(8γ2 + 3)2 (8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(256γ2 (4γ2 + 3)(2770γ2 (4γ2 + 3) − 3289) − 4904829) − √ 87842313)−634426101)−830747259)c42 −256 2(8γ2 +3)3 (16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(256γ2 (4γ2 +3)(15860γ2 (4γ2 + 3) − 28989) − 35660169) − 633167847) − 4615420743) − 12271149837)c32 − 96(8γ2 + 3)4 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + √ 3)(8γ2 (4γ2 + 3)(4160γ2 (4γ2 + 3) − 38703) − 953991) − 62963001) − 225271935) − 1198753749)c22 + 288 2(8γ2 + 3)5 (γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(176γ2 (4γ2 + 3) + 1847) + 266697) + 2012769) + 27903933) + 4605822)c2 + (8γ2 + 3)6 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(1472γ2 (4γ2 + 3)(8γ22 + 6γ2 + 9) + 18225) + 164025) + 531441))c41 − √ 1296(35740566642812256256c17 16 2 + 28570836036038426624 2(8γ2 + 3)c2 + 2251799813685248(156496γ2 (4γ2 + 3) + 92349)c2 + 15 √ 14 70368744177664 2(8γ2 + 3)(1222960γ2 (4γ2 + 3) + 794349)c2 + 35184372088832(8γ2 (4γ2 + 3)(1662640γ2 (4γ2 + 3) + 2186937) + √ 5656959)c132 + 549755813888 2(8γ2 + 3)(64γ2 (4γ2 + 3)(1624276γ2 (4γ2 + 3) + 2428641) + 55619379)c 12 + 34359738368(16γ (4γ + 2 2 2 11 √ 3)(32γ2 (4γ2 +3)(9093536γ2 (4γ2 +3)+21828825)+520907193)+1981526247)c2 +1073741824 2(8γ2 +3)(16γ2 (4γ2 +3)(64γ2 (4γ2 + LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 34 3)(8729776γ2 (4γ2 + 3) + 27712233) + 1538398251) + 6476856633)c10 2 + 268435456(32γ2√ (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(4922720γ2 (4γ2 +3)+30752613)+722209851)+6445072233)+39815710623)c92 +8388608 2(8γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(152680γ2 (4γ2 + 3) + 11672919) + 2756410479) + 27369608805) + 90671150115)c82 − 2097152(64γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(503624γ2 (4γ2 + 3) − 3493251) − 688213503) − 10972936431) − 37593631143) − √ 380949604335)c72 − 65536 2(8γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(1663984γ2 (4γ2 + 3) − 7804791) − 882556155) − 14978195109) − 107632778535) − 568326843585)c62 − 16384(32γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(124856γ2 (4γ2 + 3) − 950589) − 32569047) − 2979587025) − 16289473653) − 348760104867) − 1473838416567)c52 + √ 512 2(8γ2 +3)(16γ2 (4γ2 +3)(128γ2 (4γ2 +3)(4γ2 (4γ2 +3)(16γ2 (4γ2 +3)(512γ2 (4γ2 +3)(5735γ2 (4γ2 +3)+107514)+191405025)+ 4420770453) + 12615799167) + 566991745695) + 1255878519237)c42 + 256(8γ2 + 3)2 (8γ2 (4γ2 + 3)(256γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(γ2 (4γ2 +3)(32γ2 (4γ2 +3)(122080γ2 (4γ2 +3)+455139)+39605355)+61267347)+778442967)+77082859845)+92650892499)c32 + √ 8 2(8γ2 + 3)3 (8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(18184γ2 (4γ2 + 3) + 8397) − 510057) + 51484167) + 674752923) + 23163918867) + 33705582543)c22 + 2(8γ2 + 3)4 (64γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(8γ2 (4γ2 + √ 3)(16γ2 (4γ2 + 3)(3232γ2 (4γ2 + 3) − 23679) − 1084023) − 7433613) − 125951517) + 43046721) + 1420541793)c2 − 2γ2 (4γ2 + 3)(8γ2 + 3)5 (32γ2 (4γ2 + 3) + 27)(32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(416γ2 (4γ2 + 3) + 8721) + 238383) + 570807) + √ 3720087))c31 +9(7821419487252849885184c18 17 2 +6045920370158305542144 2(8γ2 +3)c2 +864691128455135232(82928γ2 (4γ2 +3)+ 16 √ 15 50403)c2 + 27021597764222976 2(8γ2 + 3)(618544γ2 (4γ2 + 3) + 440529)c2 + 13510798882111488(4γ2 (4γ2 + 3)(1586320γ2 (4γ2 + √ 3) + 2336103) + 3277179)c142 + 211106232532992 2(8γ2 + 3)(64γ2 (4γ2 + 3)(715316γ2 (4γ2 + 3) + 1328763) + 34660737)c 13 + 12 √ 2 39582418599936(64γ2 (4γ2 +3)(8γ2 (4γ2 +3)(1178888γ2 (4γ2 +3)+3863173)+27006867)+454358565)c2 +412316860416 2(8γ2 + 3)(16γ2 (4γ2 + 3)(256γ2 (4γ2 + 3)(665252γ2 (4γ2 + 3) + 3892149) + 1057756563) + 4992478497)c11 2 + 51539607552(512γ 2 (4γ2 + √ 3)(γ2 (4γ2 +3)(56γ2 (4γ2 +3)(313984γ2 (4γ2 +3)+9974967)+1028916351)+644559201)+68959891323)c10 2 −1073741824 2(8γ2 + 3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(32γ2 (4γ2 +3)(6971360γ2 (4γ2 +3)−89007381)−6526225485)−73004332293)−264178894635)c92 − 201326592(64γ2 (4γ2 +3)(4γ2 (4γ2 +3)(32γ2 (4γ2 +3)(16γ2 (4γ2 +3)(5293024γ2 (4γ2 +3)−37997325)−2230608699)−80476063245)− √ 152131081419)−1672746272145)c82 −25165824 2(8γ2 +3)(32γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(32γ2 (4γ2 +3)(489776γ2 (4γ2 + 3)−4812207)−711677367)−14737165839)−122830333389)−727740287271)c72 +6291456(8γ2 (4γ2 +3)(128γ2 (4γ2 +3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(286496γ2 (4γ2 + 3) + 2406567) + 43182909) + 2860677891) + 39373245531) + 1978479398061) + √ 2353970611251)c62 + 196608 2(8γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(206512γ2 (4γ2 + 3) − 579483) + 79925697) + 4748690691) + 39533298939) + 1099633802355) + 2798449794657)c52 + 12288(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(32γ2 (4γ2 +3)(16γ2 (4γ2 +3)(124240γ2 (4γ2 +3)−1558569)−18976923)+3132532683)+47747692809)+ √ 1106031289113) + 5920657166601) + 24501375705501)c42 − 9216 2(8γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 +3)(512γ2 (4γ2 +3)(358γ2 (4γ2 +3)+17059)+9683307)−301807539)−5539520097)−134870218911)−742484192715)− 784612583667)c32 −1536(8γ2 +3)2 (16γ2 (4γ2 +3)(32γ2 (4γ2 +3)(2γ2 (4γ2 +3)(8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(37888γ2 (4γ2 + √ 3)+417195)+4524579)−322339743)−4945963923)−7236632097)−78732452709)−166203389781)c22 −192 2(8γ2 +3)3 (2γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(3808γ2 (4γ2 + 3) + 20067) − 869697) − 44761329) − 2073859929) − 11120402925) − 58242213513) − 15109399071)c2 − (8γ2 + 3)4 (32γ2 (4γ2 + 3) + 27)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4γ2 (4γ2 +3)(4γ2 (4γ2 +3)(64γ2 (4γ2 +3)(248γ2 (4γ2 +3)−1053)−742851)−5885217)−19663317)−243931419)−1162261467))c21 − √ 24(2361183241434822606848c19 18 2 + 1761664059039262179328 2(8γ2 + 3)c2 + 6917529027641081856(2896γ2 (4γ2 + 3) + 1839)c2 + 17 √ 16 972777519512027136 2(8γ2 + 3)(4560γ2 (4γ2 + 3) + 3703)c2 + 81064793292668928(8γ2 (4γ2 + 3)(32816γ2 (4γ2 + 3) + 56963) + √ 176373)c15 2 +105553116266496 2(8γ2 +3)(32γ2 (4γ2 +3)(644456γ2 (4γ2 +3)+1621131)+24413913)c 14 +131941395333120(256γ (4γ + 2 2 2 13 √ 3)(γ2 (4γ2 + 3)(134176γ2 (4γ2 + 3) + 692919) + 714663) + 53030619)c2 + 824633720832 2(8γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(619184γ2 (4γ2 + 3) + 9820401) + 203726853) + 1080500715)c12 2 − 103079215104(16γ2 (4γ 2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + √ 3)(428032γ2 (4γ2 +3)−10945131)−402746013)−4583251431)−16786701225)c11 2 −536870912 2(8γ2 +3)(32γ2 (4γ2 +3)(8γ2 (4γ2 + 3)(1280γ2 (4γ2 + 3)(133364γ2 (4γ2 + 3) − 1377279) − 5492737251) − 36416968281) − 295893771363)c10 2 − 536870912(4γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(1512896γ2 (4γ2 + 3) − 14556051) − 2830869891) − 15952623165) − 553059750879) − √ 417619151433)c92 + 25165824 2(8γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(70064γ2 (4γ2 + 3) + 721365) + 1335076965)+18911826621)+366397455051)+594764458695)c82 +37748736(8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(64γ2 (4γ2 +3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(39872γ2 (4γ2 + 3) − 261411) + 15486417) + 756152577) + 47558442951) + 320706318627) + 399559605273)c72 + √ 294912 2(8γ2 +3)(32γ2 (4γ2 +3)(64γ2 (4γ2 +3)(8γ2 (4γ2 +3)(8γ2 (4γ2 +3)(64γ2 (4γ2 +3)(16296γ2 (4γ2 +3)−287443)+65935737)+ 1944233847) + 16216223733) + 454429891251) + 2335727304603)c62 − 49152(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(2γ2 (4γ2 +3)(512γ2 (4γ2 +3)(586γ2 (4γ2 +3)+24033)−51587199)−467180379)−82693752687)−867239849367)−8907538402305)− √ 9049152548457)c52 −3072 2(8γ2 +3)(4γ2 (4γ2 +3)(32γ2 (4γ2 +3)(512γ2 (4γ2 +3)(2γ2 (4γ2 +3)(4γ2 (4γ2 +3)(32γ2 (4γ2 +3)(25712γ2 (4γ2 + 3) + 46845) − 63747243) − 845437095) − 2203435305) − 731126235663) − 7550590965129) − 3884579149761)c42 − 768(8γ2 + 3)2 (16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(4864γ2 (4γ2 + 3) − 121263) − 3478869) − √ 549289593) − 10551682323) − 107299532223) − 278373578769) − 581776434315)c32 + 24 2(8γ2 + 3)3 (128γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(1520γ2 (4γ2 + 3) + 58353) + 14301441) + 434161053) + 3795256377) + LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 35 18986792607) + 12526595811) + 215793212373)c22 + 16(8γ2 + 3)4 (2γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(1984γ2 (4γ2 + 3) + 32355) + 2565351) + 13443489) + 199998963) + 1018772397) + 11751754833) + 3486784401)c2 + √ 2γ2 (4γ2 + 3)(8γ2 + 3)5 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(352γ2 (4γ2 + 3) − √ 135)−3645)−19683))c1 +64(295147905179352825856c20 19 2 +212137556847659843584 2(8γ2 +3)c2 +57646075230342348800(40γ2 (4γ2 + 18 √ 17 3) + 27)c2 + 324259173170675712 2(8γ2 + 3)(1488γ2 (4γ2 + 3) + 1435)c2 + 3377699720527872(32γ2 (4γ2 + 3)(19832γ2 (4γ2 + 3) + √ 42855)+591327)c16 2 +316659348799488 2(8γ2 +3)(32γ2 (4γ2 +3)(19160γ2 (4γ2 +3)+70893)+1249263)c 15 +13194139533312(32γ (4γ + 2 2 2 14 √ 3)(8γ2 (4γ2 + 3)(91904γ2 (4γ2 + 3) + 890301) + 8779455) + 90548361)c2 − 2473901162496 2(8γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(720γ2 (4γ2 + 3) − 428701) − 11373093) − 69244551)c13 2 − 51539607552(32γ√ 2 (4γ2 + 3)(8γ2 (4γ2 + 3)(256γ2 (4γ2 + 3)(11474γ2 (4γ2 + 3) − 169119) − 136525257) − 909075879) − 7356711519)c12 2 − 536870912 2(8γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(265424γ2 (4γ2 +3)−2876715)−1023792291)−8229027087)−74254320891)c11 2 −67108864(32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(64γ 2 (4γ2 + √ 3)(8γ2 (4γ2 +3)(71104γ2 (4γ2 +3)−1284057)−275672079)−30805712397)−147688077195)−949653593883)c10 2 +8388608 2(8γ 2+ 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(118960γ2 (4γ2 + 3) − 1110987) + 920049759) + 15602051439) + 326967992685)+558593560719)c92 +18874368(8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(18400γ2 (4γ2 + √ 3) − 312821) + 33449643) + 1759740957) + 29141622639) + 205215463197) + 265387168395)c82 − 98304 2(8γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(340γ2 (4γ2 + 3) + 57027) − 57169611) − 1636300035) − 29161648269) − 436621985685) − 2378454806709)c72 − 331776(128γ2 (4γ2 + 3)(256γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(64γ2 (4γ2 + √ 3)(20γ2 (4γ2 + 3) − 29) − 97911) − 57047463) − 369417105) − 277475625) − 25428395529) − 450976758219)c62 − 3072 2(8γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(2512γ2 (4γ2 + 3) − 75729) + 301401) − 43466139) − 4711338189) − 152635110561) − 2059034673717) − 1265415759423)c52 + 1152(8γ2 + 3)2 (16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(8γ2 (4γ2 +3)(4γ2 (4γ2 +3)(16γ2 (4γ2 +3)(128γ2 (4γ2 +3)(76γ2 (4γ2 +3)+2827)−2456271)−80394525)−120535047)+741950685)+ √ 39013615251)+120286887381)c42 +24 2(8γ2 +3)3 (32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(8γ2 (4γ2 +3)(8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(128γ2 (4γ2 + 3)(280γ2 (4γ2 + 3) + 369) − 10363059) − 352188891) − 2035937349) − 3714595443) + 4424246325) + 62891259381)c32 + (8γ2 + 3)4 (32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(512γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(64γ2 (4γ2 + 3) − 23823) − 338013) − √ 345692529) − 1062232461) − 1353580227) − 3486784401) + 15109399071)c22 − 2γ2 (4γ2 + 3)(8γ2 + 3)5 (16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(112γ2 (4γ2 + 3) + 2529) + 245673) + 925101) + 1948617)c2 − 4γ22 (4γ2 + 3)2 (8γ2 + 3)6 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)2 )), √ √ G2 = −6b41 (531441(8γ2 + 3)(8796093022208 2c12 2 + 13194139533312(8γ 11 2 + 3)c2 + 103079215104 2(704γ2 (4γ2 + 3) + 399)c2 + 10 9 √ 2 8 3 42949672960(8γ2 +3)(704γ2 (4γ2 +3)+405)c2 +12079595520 2(8γ2 +3) (352γ2 (4γ2 +3)+207)c2 +2415919104(8γ2 +3) (352γ2 (4γ2 + √ √ 3) + 213)c72 + 88080384 2(8γ2 + 3)4 (704γ2 (4γ2 + 3) + 441)c62 + 9437184(8γ2 + 3)5 (704γ2 (4γ2 + 3) + 459)c52 + 11796480 2(8γ2 + √ 3)6 (22γ2 (4γ2 + 3) + 15)c42 + 163840(8γ2 + 3)7 (88γ2 (4γ2 + 3) + 63)c32 + 384 2(8γ2 + 3)8 (704γ2 (4γ2 + 3) + 531)c22 + 96(8γ2 + √ √ 3)9 (64γ2 (4γ2 + 3) + 51)c2 + 2(8γ2 + 3)10 (32γ2 (4γ2 + 3) + 27))c61 − 78732(−140737488355328c14 2 + 615726511554560 2(8γ2 + √ 3)c13 2 12 2 + 981863883603968(8γ2 + 3) c2 + 137438953472 √ 2(8γ2 + 3)(39136γ2 (4γ2 + 3) + 22311)c11 2 + 2147483648(128γ2 (4γ2 + 3)(127160γ2 (4γ2 + 3) + 146079) + 5367951)c102 + 268435456 2(8γ 2 + 3)(64γ 2 (4γ2 + 3)(274384γ 2 (4γ2 + 3) + 322587) + 6060015)c92 + 8 √ 201326592(4γ2 (4γ2 + 3)(320γ2 (4γ2 + 3)(55088γ2 (4γ2 + 3) + 98037) + 18575649) + 14640021)c2 + 100663296 2(8γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 +3)(73568γ2 (4γ2 +3)+135621)+2654775)+2156625)c72 +5505024(8γ2 +3)2 (16γ2 (4γ2 +3)(32γ2 (4γ2 +3)(31504γ2 (4γ2 + √ 3) + 61371) + 1261413) + 4281903)c62 + 32768 2(8γ2 + 3)3 (16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(164560γ2 (4γ2 + 3) + 353673) + 7871499) + √ 28554201)c52 + 8192(8γ2 + 3)4 (4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(89584γ2 (4γ2 + 3) + 245133) + 6374133) + 6461127)c42 − 512 2(8γ2 + 3)5 (16γ2 (4γ2 + 3)(80γ2 (4γ2 + 3)(160γ2 (4γ2 + 3) − 3699) − 389529) − 1976319)c32 − 8(8γ2 + 3)6 (16γ2 (4γ2 + 3)(64γ2 (4γ2 + √ 3)(10672γ2 (4γ2 + 3) + 7209) − 280179) − 2932767)c22 − 2(8γ2 + 3)7 (16γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(872γ2 (4γ2 + 3) + 1053) + 26973) − 124659)c2 − γ2 (4γ2 + 3)(8γ2 + 3)8 (32γ2 (4γ2 + 3) + 27)(800γ2 (4γ2 + 3) + 459))c51 + 4374(−24769797950537728c15 2 + √ √ 46443371157258240 2(8γ2 + 3)c14 2 + 52776558133248(24080γ 2 (4γ2 + 3) + 13533)c 13 + 2199023255552 2(8γ + 3)(195760γ (4γ + 2 2 2 2 √ 3) + 113283)c12 11 2 + 103079215104(1280γ2 (4γ2 + 3)(20428γ2 (4γ2 + 3) + 24147) + 9113391)c2 + 12884901888 2(8γ2 + 3)(32γ2 (4γ2 + 3)(829664γ2 (4γ2 + 3) + 1048173) + 10486611)c10 2 + 2147483648(10γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(682352γ2 (4γ2 + 3) + 1354977) + √ 56302533) + 120045159)c92 + 603979776 2(8γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(346720γ2 (4γ2 + 3) + 809037) + 9406611) + 34930035)c82 + 37748736(64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(256γ2 (4γ2 + 3)(23870γ2 (4γ2 + 3) + 92931) + 29075517) + 57924153) + √ 657490203)c72 + 1048576 2(8γ2 + 3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(160γ2 (4γ2 + 3)(28336γ2 (4γ2 + 3) + 351171) + 90000639) + 409445037)+1261923057)c62 −196608(8γ2 +3)2 (8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(512γ2 (4γ2 +3)(7381γ2 (4γ2 +3)−28071)−34174791)− √ 354478437) − 586966743)c52 − 12288 2(8γ2 + 3)3 (32γ2 (4γ2 + 3)(80γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(10384γ2 (4γ2 + 3) − 15549) − 838269) − 47273463) − 327934089)c42 − 128(8γ2 + 3)4 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(55880γ2 (4γ2 + 3) − 101187) − √ 41737437) − 238169403) − 1680672321)c32 − 48 2(8γ2 + 3)5 (64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4960γ2 (4γ2 + 3) − 28917) − 2209923) − 6042681) − 84853413)c22 + 48(8γ2 + 3)6 (γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(2560γ2 (4γ2 + 3)(14γ2 (4γ2 + 3) + 195) + √ 908091) + 9310059) + 2007666)c2 + 2(8γ2 + 3)7 (32γ2 (4γ2 + 3) + 27)(32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(136γ2 (4γ2 + 3) + 567) + √ 2187) + 19683))c41 − 5832(−72057594037927936c16 15 2 + 75435293758455808 2(8γ2 + 3)c2 + 211106232532992(10640γ2 (4γ2 + 3) + √ 5949)c14 2 + 9851624184872960 2(8γ2 + 3)(76γ2 (4γ2 + 3) + 45)c 13 + 5497558138880(16γ (4γ + 3)(50824γ (4γ + 3) + 63351) + 2 2 2 2 2 LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 36 √ 312903)c12 2 + 51539607552 2(8γ2 + 3)(64γ2 (4γ2 + 3)(158896γ2 (4γ2 + 3) + 231651) + 5147469)c 11 + 1073741824(16γ (4γ + 2 2 2 10 √ 3)(64γ2 (4γ2 + 3)(1143472γ2 (4γ2 + 3) + 2892825) + 140222259) + 533154879)c2 + 134217728 2(8γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(399520γ2 (4γ2 +3)+1644417)+48560877)+414357039)c92 +75497472(16γ2 (4γ2 +3)(80γ2 (4γ2 +3)(1024γ2 (4γ2 +3)(209γ2 (4γ2 + √ 3)+4266)+7305471)+338267907)+1053453843)c82 −2097152 2(8γ2 +3)(32γ2 (4γ2 +3)(40γ2 (4γ2 +3)(64γ2 (4γ2 +3)(13816γ2 (4γ2 + 3) − 106101) − 14430069) − 367467759) − 2430686475)c72 − 65536(224γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + √ 3)(39952γ2 (4γ2 + 3) − 166383) − 18179397) − 293994765) − 2008656711) − 70545308859)c62 − 24576 2(8γ2 + 3)(64γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(8224γ2 (4γ2 + 3) − 55431) − 2926773) − 195330447) − 692743185) − 7211949615)c52 + 512(8γ2 + 3)2 (256γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(11120γ2 (4γ2 + 3) + 301887) + 27684909) + 237446235) + √ 442270449) + 19361281365)c42 + 64 2(8γ2 + 3)3 (16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(80γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(1112γ2 (4γ2 + 3) + 4095)+ 1280043) + 119921229) + 991609857) + 2966680809)c32 + 24(8γ2 + 3)4 (8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(80γ2 (4γ2 +3)(64γ2 (4γ2 + √ 3)(496γ2 (4γ2 + 3) + 315) + 36693) + 8085339) + 99536931) + 185118615)c22 + 2(8γ2 + 3)5 (32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(416γ2 (4γ2 + 3) − 1791) − 122229) − 1791153) + 767637) + 23914845)c2 − γ2 (4γ2 + 3)(8γ2 + 3)6 (32γ2 (4γ2 + 3) + √ 27)(16γ2 (4γ2 +3)(8γ2 (4γ2 +3)(32γ2 (4γ2 +3)+1431)+12879)+59049))c31 +27(−29975959119778021376c17 2 +19167320014088830976 2(8γ2 + √ 3)c16 15 2 + 4503599627370496(140912γ2 (4γ2 + 3) + 77589)c2 + 562949953421312 2(8γ2 + 3)(368000γ 14 2 (4γ2 + 3) + 223659)c2 + 13 √ 35184372088832(320γ2 (4γ2 +3)(103684γ2 (4γ2 +3)+139959)+14693157)c2 +35184372088832 2(8γ2 +3)(4γ2 (4γ2 +3)(878716γ2 (4γ2 + 3) + 1601523) + 2521935)c12 2 + 68719476736(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(3467696γ2 (4γ2 + 3) + 13236129) + 762612975) + √ 3207437433)c112 +549755813888 2(8γ2 +3)(γ2 (4γ2 +3)(4γ2 (4γ2 +3)(2628032γ2 (4γ2 +3)+33238593)+151400421)+45039807)c2 − 10 536870912(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(64γ2 (4γ2 +3)(1183600γ2 (4γ2 +3)−16309341)−2169499383)−22400960589)−75380975811)c92 − √ 134217728 2(8γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(20γ2 (4γ2 + 3)(743776γ2 (4γ2 + 3) − 5008437) − 259275897) − 12023978013) − 22000273029)c82 −4194304(32γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(32γ2 (4γ2 +3)(844976γ2 (4γ2 +3)−6479235)−881545599)− √ 17001899109) − 133103892735) − 744344826705)c72 + 1048576 2(8γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(155456γ2 (4γ2 +3)+1761759)+26852067)+5215346919)+46528965189)+141939328995)c62 +32768(16γ2 (4γ2 +3)(512γ2 (4γ2 + 3)(γ2 (4γ2 +3)(16γ2 (4γ2 +3)(64γ2 (4γ2 +3)(163448γ2 (4γ2 +3)−396495)+105558957)+6094259937)+6109485102)+1305941324319)+ √ 3187241401437)c52 + 8192 2(8γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(112400γ2 (4γ2 + 3) − 1293327) − 948915) + 512247159) + 10635689367) + 313465690881) + 404175229407)c42 − 512(8γ2 + 3)2 (8γ2 (4γ2 + 3)(512γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(640γ2 (4γ2 + 3)(1112γ2 (4γ2 + 3) + 69525) + 32785317) − 659801133) − 989306946) − 243884120751) − √ 322520382639)c32 − 64 2(8γ2 + 3)3 (16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(33344γ2 (4γ2 + 3) + 370791) + 2622213) − 221125383) − 4764722859) − 17912218905) − 47050066053)c22 − 32(8γ2 + 3)4 (2γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(3728γ2 (4γ2 + 3) + 20385) − 1188999) − 112160295) − 122231430) − 6882692391) − √ 2195382771)c2 − 2(8γ2 + 3)5 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(26γ2 (4γ2 + 3) − √ 27) − 43011) − 1082565) − 4782969) − 14348907))c21 − 12(−64563604257983430656c18 2 + 25940733853654056960 2(8γ2 + 3)c2 + 17 √ 5188146770730811392(190γ2 (4γ2 +3)+101)c16 2 +3377699720527872 2(8γ2 +3)(92848γ 2 (4γ2 +3)+57621)c 15 +316659348799488(160γ (4γ + 2 2 2 √ 3)(32536γ2 (4γ2 +3)+48819)+2740311)c14 2 +39582418599936 2(8γ2 +3)(544γ 2 (4γ2 +3)(7112γ 2 (4γ2 +3)+17575)+4256721)c 13 + 2 √ 824633720832(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(1038064γ2 (4γ2 +3)+7470357)+126380763)+579754989)c12 2 −154618822656 2(8γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(35072γ2 (4γ2 + 3) − 1683393) − 74672793) − 397657593)c11 2 − 4831838208(32γ 2 (4γ2 + 3)(8γ2 (4γ2 + √ 3)(256γ2 (4γ2 +3)(91300γ2 (4γ2 +3)−808369)−549450027)−3264549723)−24286529295)c10 2 −67108864 2(8γ2 +3)(256γ2 (4γ2 + 3)(γ2 (4γ2 +3)(320γ2 (4γ2 +3)(239888γ2 (4γ2 +3)−1900737)−2573984061)−2317559526)−155008434213)c92 +75497472(8γ2 (4γ2 + 3)(32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(64γ2 (4γ2 +3)(5456γ2 (4γ2 +3)+405189)+496835775)+6550265475)+119844675981)+184552669701)c82 + √ 9437184 2(8γ2 +3)(128γ2 (4γ2 +3)(4γ2 (4γ2 +3)(32γ2 (4γ2 +3)(128γ2 (4γ2 +3)(3658γ2 (4γ2 +3)−22987)+19991403)+1378335123)+ 3481961337) + 90973454751)c72 + 98304(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(84728γ2 (4γ2 + 3) − √ 1259427) + 249961491) + 7318577043) + 117711257481) + 1582571087703) + 7791361373061)c62 − 36864 2(8γ2 + 3)(128γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(512γ2 (4γ2 + 3)(241γ2 (4γ2 + 3) + 12486) − 27433917) − 183363183) − 11751942915) − 39893681547) − 796810217499)c52 − 9216(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(160γ2 (4γ2 + 3)(2896γ2 (4γ2 + 3) + 5555) − 33107157) − 105492132) − 16851439341) − 334277156637) − 3300035502159) − 1622387867769)c42 − √ 384 2(8γ2 + 3)(32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4672γ2 (4γ2 + 3) − 104547) − 23537709)−887214627)−8089785171)−155946224187)−191658350799)−758956737951)c32 +72(8γ2 +3)2 (32γ2 (4γ2 +3)(8γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(848γ2 (4γ2 + 3) + 35427) + 8681067) + 253595043) + 2086955685) + √ 9755662437)+24034419225)+96725982087)c22 +9 2(8γ2 +3)3 (16γ2 (4γ2 +3)(64γ2 (4γ2 +3)(8γ2 (4γ2 +3)(2γ2 (4γ2 +3)(16γ2 (4γ2 + 3)(1024γ2 (4γ2 +3)(13γ2 (4γ2 +3)+218)+579717)+12313539)+21434787)+97253703)+1994498073)+4261625379)c2 +γ2 (4γ2 + 3)(8γ2 + 3)4 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3) + 27)(184γ2 (4γ2 + 3) − √ 135) − 19683))c1 + 32(−9223372036854775808c19 18 2 + 2305843009213693952 2(8γ2 + 3)c2 + 108086391056891904(976γ2 (4γ2 + 3) + √ 483)c17 2 +6755399441055744 2(8γ2 +3)(4816γ 2 (4γ2 +3)+2997)c 16 +105553116266496(32γ (4γ +3)(46568γ (4γ +3)+79563)+ 2 2 2 2 2 √ 947673)c15 2 + 13194139533312 2(8γ2 + 3)(64γ2 (4γ2 + 3)(14492γ2 (4γ2 + 3) + 52887) + 1657449)c 14 + 2473901162496(16γ (4γ + 2 2 2 √ 3)(16γ2 (4γ2 + 3)(13328γ2 (4γ2 + 3) + 296067) + 5726295) + 28169937)c13 2 − 1236950581248 2(8γ2 + 3)(8γ2 (4γ2 + 3)(8γ2 (4γ2 + LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 37 3)(14512γ2 (4γ2 + 3) − 203129) − 2768895) − 8254791)c12 2 − 1610612736(32γ2√(4γ2 + 3)(8γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(61400γ2 (4γ2 + 3) − 511953) − 247821849) − 1720462599) − 14085493785)c11 2 − 67108864 2(8γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(103136γ2 (4γ2 + 3) − 1037943) − 423575973) − 3754547559) − 34924275171)c10 2 + 8388608(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ 2 (4γ2 + √ 3)(64γ2 (4γ2 +3)(151184γ2 (4γ2 +3)−1193967)+755240679)+12633531363)+256609639521)+423514131201)c92 +12582912 2(8γ2 + 3)(4γ2 (4γ2 +3)(16γ2 (4γ2 +3)(32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(44384γ2 (4γ2 +3)−637989)+22563819)+1890504765)+21223105083)+ 18784569465)c82 + 98304(32γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(376γ2 (4γ2 + 3) − 166491) + √ 42349635) + 1469145681) + 6786075105) + 408571427457) + 2209108652001)c72 − 12288 2(8γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 +3)(4γ2 (4γ2 +3)(32γ2 (4γ2 +3)(6832γ2 (4γ2 +3)−11217)−4553847)−134931825)−12856946535)−112423646979)− 671532567687)c62 − 3072(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4592γ2 (4γ2 + √ 3) − 120471) + 10676205) + 57287007) − 1120551003) − 129612495951) − 2026708180569) − 1315665631737)c52 + 3456 2(8γ2 + 3 3) (16γ2 (4γ2 +3)(8γ2 (4γ2 +3)(128γ2 (4γ2 +3)(8γ2 (4γ2 +3)(8γ2 (4γ2 +3)(80γ2 (4γ2 +3)+3101)−234009)−2853927)−102296925)+ 368249247) + 2382450003)c42 + 72(8γ2 + 3)2 (64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + √ 3)(336γ2 (4γ2 + 3) + 349) − 6596181) − 226708713) − 1371063105) − 3095525727) − 74933181) + 22685621967)c32 + 3 2(8γ2 + 3)3 (16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3) − 7305) − 3400623) − 108267435) − 82924479) − 1713897225) − 2625849981) + 2711943423)c22 − γ2 (4γ2 + 3)(8γ2 + 3)4 (16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + √ 81) + 729)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(208γ2 (4γ2 + 3) + 4869) + 467289) + 1712421) + 3365793)c2 − 4 2γ22 (4γ2 + 3)2 (8γ2 + 3)5 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)2 )), and √ √ G3 = 9b51 (531441(8796093022208c12 11 2 +6597069766656 2(8γ2 +3)c2 +103079215104(704γ2√ (4γ2 +3)+399)c10 2 +21474836480 2(8γ2 + 3)(704γ2 (4γ2 + 3) + 405)c92 + 12079595520(8γ2 + 3)2 (352γ2 (4γ2 + 3) + 207)c82 + 1207959552 2(8γ2 + 3)3 (352γ2 (4γ2 + 3) + 213)c72 + √ 88080384(8γ2 + 3)4 (704γ2 (4γ2 + 3) + 441)c62 + 4718592 2(8γ2 + 3)5 (704γ2 (4γ2 + 3) + 459)c52 + 11796480(8γ2 + 3)6 (22γ2 (4γ2 + 3) + √ √ 15)c42 +81920 2(8γ2 +3)7 (88γ2 (4γ2 +3)+63)c32 +384(8γ2 +3)8 (704γ2 (4γ2 +3)+531)c22 +48 2(8γ2 +3)9 (64γ2 (4γ2 +3)+51)c2 + √ 10 6 13 (8γ2 +3) (32γ2 (4γ2 +3)+27))c1 −944784(61572651155456c2 +45079976738816 2(8γ2 +3)c12 2 +103079215104(4672γ2 (4γ2 +3)+ √ 2661)c11 2 +34359738368 2(8γ 2 +3)(2816γ 2 (4γ2 +3)+1647)c 10 +268435456(80γ (4γ +3)(19360γ (4γ +3)+22707)+531927)c9 + 2 2 2 2 2 2 √ 1207959552 2(8γ2 + 3)(4γ2 (4γ2 + 3)(8096γ2 (4γ2 + 3) + 9813) + 11853)c82 + 12582912(8γ2 + 3)2 (16γ2 (4γ2 + 3)(26048γ2 (4γ2 + √ 3) + 33129) + 167265)c72 + 352321536 2(8γ2 + 3)3 (γ2 (4γ2 + 3)(704γ2 (4γ2 + 3) + 963) + 324)c62 + 589824(8γ2 + 3)4 (8γ2 (4γ2 + √ 3)(3344γ2 (4γ2 + 3) + 5169) + 15363)c52 + 4096 2(8γ2 + 3)5 (80γ2 (4γ2 + 3)(880γ2 (4γ2 + 3) + 1773) + 62613)c42 + 128(8γ2 + √ 3)6 (16γ2 (4γ2 + 3)(704γ2 (4γ2 + 3) + 7407) + 76707)c32 − 384 2(8γ2 + 3)7 (8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3) − 3) − 297)c22 − (8γ2 + √ 3)8 (16γ2 (4γ2 +3)(544γ2 (4γ2 +3)+333)−1215)c2 −2 2γ2 (4γ2 +3)(8γ2 +3)9 (32γ2 (4γ2 +3)+27))c51 +13122(22517998136852480c14 2 + √ √ 16008889300418560 2(8γ2 + 3)c13 12 2 + 2199023255552(74960γ2 (4γ2 + 3) + 43299)c2 + 137438953472 2(8γ√ 2 + 3)(229760γ2 (4γ2 + 3) + 141363)c11 10 2 + 8589934592(16γ2 (4γ2 + 3)(925760γ2 (4γ2 + 3) + 1159713) + 5758533)c2 + 42949672960 2(8γ2 + 3)(8γ2 (4γ2 + 3)(31768γ2 (4γ2 + 3) + 44433) + 120447)c92 + 4026531840(64γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(78496γ2 (4γ2 + 3) + 178281) + 127332) + √ 1867941)c82 +50331648 2(8γ2 +3)(32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(56320γ2 (4γ2 +3)+175203)+1180737)+9630333)c72 +22020096(8γ2 + √ 3) (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(3520γ2 (4γ2 + 3) + 29241) + 251019) + 2312631)c62 − 131072 2(8γ2 + 3)3 (16γ2 (4γ2 + 3)(16γ2 (4γ2 + 2 5 4 3)(25520γ2 (4γ2 + 3) − 139509) − 3235545) − 16707951)c2 − 40960(8γ2 + 3) (16γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(1298γ2 (4γ2 + 3) − √ 2781) − 660231) − 3692385)c42 − 2560 2(8γ2 + 3)5 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(3968γ2 (4γ2 + 3) − 8775) − 135351) − 1565163)c32 − √ 32(8γ2 + 3)6 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(6080γ2 (4γ2 + 3) − 30753) − 410427) − 4721733)c22 + 16 2(8γ2 + 3)7 (32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(80γ2 (4γ2 + 3) + 1809) + 9963) + 111537)c2 + (8γ2 + 3)8 (8γ2 (4γ2 + 3) + 27)(32γ2 (4γ2 + 3) + 27)(40γ2 (4γ2 + √ 3) + 27))c41 − 69984(11258999068426240c15 14 2 + 7740561859543040 2(8γ2 + 3)c2 + 21990232555520(3472γ2 (4γ2 + 3) + 2061)c2 + 13 √ 549755813888 2(8γ2 +3)(25120γ2 (4γ2 +3)+17073)c2 +17179869184(176γ2 (4γ2 +3)(16960γ2 (4γ2 +3)+24399)+1476873)c11 12 2 + √ 2147483648 2(8γ2 + 3)(16γ2 (4γ2 + 3)(112640γ2 (4γ2 + 3) + 214281) + 1386477)c10 2 + 12884901888(5γ2 (4γ2 + 3)(8γ2 (4γ2 + √ 3)(10912γ2 (4γ2 + 3) + 41805) + 301563) + 396819)c92 + 201326592 2(8γ2 + 3)(20γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(704γ2 (4γ2 + 3) + 9423) + 338985) + 1988469)c82 − 6291456(8γ2 + 3)2 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(7040γ2 (4γ2 + 3) − 71199) − 784161) − 8070273)c72 − √ 262144 2(8γ2 + 3)3 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(12320γ2 (4γ2 + 3) − 68877) − 871641) − 9692055)c62 − 16384(8γ2 + 3)4 (16γ2 (4γ2 + √ 3)(32γ2 (4γ2 + 3)(4840γ2 (4γ2 + 3) − 36261) − 1972917) − 11844063)c52 + 2048 2(8γ2 + 3)5 (80γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 4 3) + 2781) + 79461) + 2645541)c2 + 64(8γ2 + 3) (160γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(704γ2 (4γ2 + 3) + 2457) + 40095) + 3243321)c32 + 6 √ 8 2(8γ2 + 3)7 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(640γ2 (4γ2 + 3) + 81) + 10449) + 303993)c22 + 4(8γ2 + 3)8 (4γ2 (4γ2 + 3)(8γ2 (4γ2 + √ 3)(160γ2 (4γ2 + 3) − 621) − 2673) + 6561)c2 − 27 2γ2 (4γ2 + 3)(8γ2 + 3)9 (32γ2 (4γ2 + 3) + 27))c31 + 81(14411518807585587200c162 + √ √ 9547631210025451520 2(8γ2 +3)c15 14 2 +6755399441055744(13280γ2 (4γ2 +3)+8241)c2 +70368744177664 2(8γ 2 +3)(215600γ2 (4γ2 + √ 3)+171459)c13 12 2 +10995116277760(32γ2 (4γ2 +3)(144632γ2 (4γ2 +3)+261405)+3275559)c2 +824633720832 2(8γ2 +3)(16γ2 (4γ2 + 3)(242720γ2 (4γ2 +3)+755493)+5864859)c11 2 +8589934592(32γ2 (4γ2 +3)(2920γ2 (4γ2 +3)(3520γ2 (4γ2 +3)+36603)+118670589)+ √ 1113354315)c102 −1073741824 2(8γ2 +3)(32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(344960γ2 (4γ2 +3)−6820533)−75439107)−793854027)c2 − 9 603979776(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(149600γ2 (4γ2 + 3) − 1142751) − 44930457) − 507514491) − 1822681521)c82 − √ 2147483648 2(8γ2 + 3)(4γ2 (4γ2 + 3)(40γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(1078γ2 (4γ2 + 3) − 9333) − 925587) − 29953881) − 29596671)c72 + 524288(128γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(24640γ2 (4γ2 + 3) + 940671) + 21925485) + 992114325) + LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 38 √ 4274484939) + 102025393641)c62 + 196608 2(8γ2 + 3)(64γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(20γ2 (4γ2 + 3)(3616γ2 (4γ2 + 3) − 6009) + 1133487) + 39062007) + 793170225) + 10379849733)c52 + 4096(8γ2 + 3)2 (32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + √ 3)(128γ2 (4γ2 +3)(2645γ2 (4γ2 +3)−26136)+1629477)+179334729)+4360250331)+30926854701)c42 −2048 2(8γ2 +3)3 (16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(800γ2 (4γ2 + 3) + 115497) + 33939) − 30192993) − 409347351) − 1516201173)c32 − 384(8γ2 + 3)4 (32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(160γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(704γ2 (4γ2 + 3) + 8331) − 351) − 11982573) − 39503781) − √ 291761109)c22 − 16 2(8γ2 + 3)5 (16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(1280γ2 (4γ2 + 3) + 7227) − 15795) − 1948617) − 22851963) − 81310473)c2 − (8γ2 + 3)6 (32γ2 (4γ2 + 3) + 27)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(320γ22 (4γ2 + 3)2 − 6561) − √ 59049)−531441))c21 −144(6341068275337658368c17 16 2 +4035225266123964416 2(8γ2 +3)c2 +4503599627370496(7976γ2 (4γ2 +3)+ √ 5283)c15 2 +140737488355328 2(8γ2 +3)(39920γ2 (4γ2 +3)+39411)c 14 +2199023255552(32γ (4γ +3)(235480γ (4γ +3)+589131)+ 2 2 2 2 2 √ 8438823)c13 2 + 274877906944 2(8γ2 + 3)(256γ2 (4γ2 + 3)(11147γ2 (4γ2 + 3) + 69507) + 10267641)c 12 − 8589934592(32γ (4γ + 2 2 2 √ 3)(8γ2 (4γ2 + 3)(29504γ2 (4γ2 + 3) − 6545907) − 70427961) − 737840583)c11 2 − 2147483648 2(8γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(149248γ2 (4γ2 +3)−1696977)−24709455)−304505487)c10 2 −67108864(16γ2√ (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(487520γ2 (4γ2 + 3) − 4492647) − 267871779) − 3708547659) − 15154801191)c92 − 33554432 2(8γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(19360γ2 (4γ2 + 3) − 696699) − 56124009) − 1932030063) − 2203465923)c82 + 524288(512γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(64γ2 (4γ2 + √ 3)(2γ2 (4γ2 + 3)(88000γ2 (4γ2 + 3) − 423261) + 9452133) + 1213136919) + 1493145819) + 153420226065)c72 + 131072 2(8γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(160γ2 (4γ2 + 3)(1232γ2 (4γ2 + 3) − 16839) + 11239641) + 430173423) + 4499461629) + 29985731739)c62 −4096(16γ2 (4γ2 +3)(32γ2 (4γ2 +3)(128γ2 (4γ2 +3)(40γ2 (4γ2 +3)(4γ2 (4γ2 +3)(496γ2 (4γ2 +3)+70011)−1175877)− √ 156493701) − 19858237749) − 266410133271) − 656224941123)c52 − 1024 2(8γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(5360γ2 (4γ2 +3)+22491)−6841665)−146805291)−4481641953)−59743357191)−73834692453)c42 − 128(8γ2 + 3)2 (128γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(5440γ2 (4γ2 + 3) − 89019) − 11718675) − √ 197419761) − 2836576179) − 1163324349) − 22972600107)c32 + 16 2(8γ2 + 3)3 (8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(16γ2 (4γ2 +3)(256γ2 (4γ2 +3)+12555)+771525)+5181003)+69264477)+1784047437)+2195382771)c22 +(8γ2 +3)4 (16γ2 (4γ2 + 3)(16γ2 (4γ2 +3)(64γ2 (4γ2 +3)(γ2 (4γ2 +3)(16γ2 (4γ2 +3)(608γ2 (4γ2 +3)+10287)+406539)+518319)+24977727)+157837977)+ √ 387420489)c2 +256 2γ23 (4γ2 +3)3 (8γ2 +3)5 (32γ2 (4γ2 +3)+27)(16γ2 (4γ2 +3)(8γ2 (4γ2 +3)+81)+729))c1 +64(4611686018427387904c18 2 + √ √ 2810246167479189504 2(8γ2 +3)c17 16 2 +13510798882111488(1744γ2 (4γ2 +3)+1263)c2 +20266198323167232 2(8γ2 +3)(166γ2 (4γ2 + √ 3) + 215)c15 14 2 + 52776558133248(16γ2 (4γ2 + 3)(9920γ2 (4γ2 + 3) + 37989) + 308529)c2 + 824633720832 2(8γ2 + 3)(32γ2 (4γ2 + 3)(8456γ2 (4γ2 +3)+151845)+3372381)c13 2 −51539607552(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(45616γ2 (4γ2 +3)−861873)−23305671)− √ 138661875)c122 − 3221225472 2(8γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(73984γ2 (4γ2 + 3) − 809253) − 18279837) − 268027785)c2 − 11 201326592(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(54208γ2 (4γ2 +3)−704625)−102240387)−1730283687)−7827786945)c10 2 + √ 16777216 2(8γ2 +3)(64γ2 (4γ2 +3)(64γ2 (4γ2 +3)(γ2 (4γ2 +3)(36080γ2 (4γ2 +3)−230949)+4718331)+391378959)+7796915253)c92 + 25165824(2γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(51040γ2 (4γ2 + 3) − 694773) + 19038645) + 2764517445) + √ 14576882067) + 6169498569)c82 + 196608 2(8γ2 + 3)(128γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(10γ2 (4γ2 + 3)(256γ2 (4γ2 + 3) − 21633) + 1508571) + 122552919) + 1397464569) + 40192784415)c72 − 36864(16γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(2240γ2 (4γ2 + 3) + 5573) − 3293451) − 23760945) − 880465059) − 53802187983) − 147097377243)c62 − √ 49152 2(8γ2 + 3)(4γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(680γ2 (4γ2 + 3) − 14049) − 399411) − 7680015)−1704147579)−3962955537)−3030808023)c52 +384(16γ2 (4γ2 +3)(4γ2 (4γ2 +3)(32γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 + 3)(1792γ2 (4γ2 + 3)(5γ2 (4γ2 + 3) + 276) − 2337255) − 70005141) − 147515337) + 19515753549) + 47397628467) + 130130237583)c42 + √ 48 2(8γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(512γ2 (4γ2 + 3) + 2589) − 1006749) − 66568149) − 179423667) − 524886561) + 2319739965) + 11751754833)c32 + 3(8γ2 + 3)2 (256γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(16γ2 (4γ2 +3)(2γ2 (4γ2 +3)(32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(64γ2 (4γ2 +3)−6183)−770391)−48208041)−70996581)−336402153)− √ 43046721) + 1937102445)c22 − 12 2γ2 (4γ2 + 3)(8γ2 + 3)3 (16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 195) + 4617) + 32805) + 59049)c2 − 2γ22 (4γ2 + 3)2 (8γ2 + 3)4 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)2 )). 5. Appendix 2 {app2} The coefficients H1 , H2 and H3 in Subcase 3.1.1.1 in the proof of statement (g) of Theorem 1 are given as follows √ H1 = −b31 (−59049(−4503599627370496c14 13 2 + 3940649673949184 2(8γ2 + 3)c2 − 35184372088832(1528γ2 (4γ2 + 3) + 873)c2 + 12 √ 8796093022208 2(8γ2 + 3)(1672γ2 (4γ2 + 3) + 981)c11 2 − 1099511627776(γ 2 (4γ2 + 3)(83072γ2 (4γ2 + 3) + 97101) + 28350)c 10 + 2 √ 137438953472 2(8γ2 +3)(γ2 (4γ2 +3)(95744γ2 (4γ2 +3)+114057)+33939)c92 −805306368(32γ2 (4γ2 +3)(16γ2 (4γ2 +3)(111584γ2 (4γ2 + √ 3)+198423)+1880577)+11874195)c82 +402653184 2(8γ2 +3)(32γ2 (4γ2 +3)(16γ2 (4γ2 +3)(18656γ2 (4γ2 +3)+33669)+323919)+ √ 2076435)c72 −176160768(8γ2 +3)2 (16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(10912γ2 (4γ2 +3)+20007)+195615)+637389)c62 +1048576 2(8γ2 + 3)3 (16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(89056γ2 (4γ2 + 3) + 166077) + 1652157) + 5479893)c52 − 32768(8γ2 + 3)4 (64γ2 (4γ2 + 3)(8γ2 (4γ2 + LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 39 √ 3)(103312γ2 (4γ2 + 3) + 196209) + 994113) + 13442031)c42 + 8192 2(8γ2 + 3)5 (128γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(10832γ2 (4γ2 + 3) + 20979) + 54189) + 1495179)c32 − 512(8γ2 + 3)6 (8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(24832γ2 (4γ2 + 3) + 49113) + 517833) + 911979)c22 + √ 64 2(8γ2 +3)7 (8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(2176γ2 (4γ2 +3)+4401)+47385)+85293)c2 −(8γ2 +3)8 (32γ2 (4γ2 +3)+27)(256γ2 (4γ2 + √ 3)(22γ2 (4γ2 +3)+27)+2187))c61 +52488(−63050394783186944c15 14 2 +53620983063379968 2(8γ2 +3)c2 −1583296743997440(448γ2 (4γ2 + 13 √ 12 3) + 257)c2 + 1099511627776 2(8γ2 + 3)(170672γ2 (4γ2 + 3) + 101367)c2 − 1649267441664(4γ2 (4γ2 + 3)(170176γ2 (4γ2 + 3) + √ 201501)+238221)c11 2 +19327352832 2(8γ2 +3)(128γ2 (4γ2 +3)(62568γ2 (4γ2 +3)+76261)+2968965)c 10 −4294967296(2γ (4γ + 2 2 2 9 √ 3)(32γ2 (4γ2 + 3)(1864192γ2 (4γ2 + 3) + 3402369) + 66117303) + 26720037)c2 + 2415919104 2(8γ2 + 3)(γ2 (4γ2 + 3)(320γ2 (4γ2 + 3)(51568γ2 (4γ2 +3)+97083)+19447857)+4048380)c82 −226492416(32γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(39776γ2 (4γ2 +3)+ √ 100493)+1517967)+10156995)+50813001)c72 +262144 2(8γ2 +3)(64γ2 (4γ2 +3)(4γ2 (4γ2 +3)(128γ2 (4γ2 +3)(358204γ2 (4γ2 +3)+ 951381)+120327525)+209920653)+2184071607)c62 −196608(8γ2 +3)2 (16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(64γ2 (4γ2 +3)(53152γ2 (4γ2 + √ 3)+154773)+10561995)+78664203)+216637659)c52 +36864 2(8γ2 +3)3 (8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(128γ2 (4γ2 +3)(4036γ2 (4γ2 + 3) + 15251) + 2456757) + 20609559) + 31121739)c2 + 512(8γ2 + 3)4 (16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(20γ2 (4γ2 + 3)(3392γ2 (4γ2 + 4 √ 3) − 8631) − 500823) − 22910283) − 82570185)c32 − 24 2(8γ2 + 3)5 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(256γ2 (4γ2 + 3)(1964γ2 (4γ2 + 3) + 2817) + 116721) − 1677429) − 19860147)c22 + 36(8γ2 + 3)6 (8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(768γ2 (4γ2 + 3) + √ 1447) + 26487) + 44469) − 137781)c2 − 2γ2 (4γ2 + 3)(8γ2 + 3)7 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(1504γ2 (4γ2 + 3) + 1863) + √ 9477))c51 − 4374(−3891110078048108544c16 15 2 + 3211066534315163648 2(8γ2 + 3)c2 − 1688849860263936(24320γ2 (4γ2 + 3) + 14 √ 13 14091)c2 +316659348799488 2(8γ2 +3)(33040γ2 (4γ2 +3)+20249)c2 −2199023255552(512γ2 (4γ2 +3)(53185γ2 (4γ2 +3)+65277)+ √ 10204623)c122 + 206158430208 2(8γ2 + 3)(64γ2 (4γ2 + 3)(592304γ2 (4γ2 + 3) + 773493) + 16001199)c 11 − 77309411328(256γ (4γ + 2 2 2 √ 3)(2γ2 (4γ2 + 3)(607552γ2 (4γ2 + 3) + 1211461) + 1588041) + 87802137)c10 2 + 8589934592 2(8γ 2 + 3)(γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(3061520γ2 (4γ2 +3)+6789231)+309823137)+71758386)c92 −1207959552(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(529760γ2 (4γ2 + √ 3) + 1723731) + 31276287) + 240995979) + 674964333)c82 + 226492416 2(8γ2 + 3)(64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(64γ2 (4γ2 + 7 3)(20680γ2 (4γ2 + 3) + 96791) + 8365815) + 17964909) + 217464831)c2 − 2097152(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(256γ2 (4γ2 + √ 3)(4γ2 (4γ2 + 3)(46816γ2 (4γ2 + 3) + 1316943) + 12078153) + 2755209573) + 8807046813) + 21181880133)c62 − 393216 2(8γ2 + 3)(8γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(90352γ2 (4γ2 + 3) − 273921) − 32568183) − 66914181) − 3691662561) − 4672803249)c52 + 147456(8γ2 + 3)2 (8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(256γ2 (4γ2 + 3)(2770γ2 (4γ2 + 3) − 3289) − 4904829) − √ 87842313)−634426101)−830747259)c42 −256 2(8γ2 +3)3 (16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(256γ2 (4γ2 +3)(15860γ2 (4γ2 + 3) − 28989) − 35660169) − 633167847) − 4615420743) − 12271149837)c32 + 96(8γ2 + 3)4 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + √ 3)(8γ2 (4γ2 + 3)(4160γ2 (4γ2 + 3) − 38703) − 953991) − 62963001) − 225271935) − 1198753749)c22 + 288 2(8γ2 + 3)5 (γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(176γ2 (4γ2 + 3) + 1847) + 266697) + 2012769) + 27903933) + 4605822)c2 − (8γ2 + 3)6 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(1472γ2 (4γ2 + 3)(8γ22 + 6γ2 + 9) + 18225) + 164025) + 531441))c41 + √ 1296(−35740566642812256256c17 16 2 + 28570836036038426624 2(8γ2 + 3)c2 − 2251799813685248(156496γ2 (4γ2 + 3) + 92349)c2 + 15 √ 70368744177664 2(8γ2 + 3)(1222960γ2 (4γ2 + 3) + 794349)c14 2 − 35184372088832(8γ 2 (4γ2 + 3)(1662640γ 2 (4γ2 + 3) + 2186937) + √ 5656959)c132 + 549755813888 2(8γ 2 + 3)(64γ 2 (4γ2 + 3)(1624276γ2 (4γ2 + 3) + 2428641) + 55619379)c 12 − 34359738368(16γ (4γ + 2 2 2 √ 3)(32γ2 (4γ2 +3)(9093536γ2 (4γ2 +3)+21828825)+520907193)+1981526247)c11 2 +1073741824 2(8γ2 +3)(16γ2 (4γ2 +3)(64γ2 (4γ2 + 3)(8729776γ2 (4γ2 + 3) + 27712233) + 1538398251) + 6476856633)c10 2 − 268435456(32γ2√ (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(4922720γ2 (4γ2 +3)+30752613)+722209851)+6445072233)+39815710623)c92 +8388608 2(8γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(152680γ2 (4γ2 + 3) + 11672919) + 2756410479) + 27369608805) + 90671150115)c82 + 2097152(64γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(503624γ2 (4γ2 + 3) − 3493251) − 688213503) − 10972936431) − 37593631143) − √ 380949604335)c72 − 65536 2(8γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(1663984γ2 (4γ2 + 3) − 7804791) − 882556155) − 14978195109) − 107632778535) − 568326843585)c62 + 16384(32γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(124856γ2 (4γ2 + 3) − 950589) − 32569047) − 2979587025) − 16289473653) − 348760104867) − 1473838416567)c52 + √ 512 2(8γ2 +3)(16γ2 (4γ2 +3)(128γ2 (4γ2 +3)(4γ2 (4γ2 +3)(16γ2 (4γ2 +3)(512γ2 (4γ2 +3)(5735γ2 (4γ2 +3)+107514)+191405025)+ 4420770453) + 12615799167) + 566991745695) + 1255878519237)c42 − 256(8γ2 + 3)2 (8γ2 (4γ2 + 3)(256γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(γ2 (4γ2 +3)(32γ2 (4γ2 +3)(122080γ2 (4γ2 +3)+455139)+39605355)+61267347)+778442967)+77082859845)+92650892499)c32 + √ 8 2(8γ2 + 3)3 (8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(18184γ2 (4γ2 + 3) + 8397) − 510057) + 51484167) + 674752923) + 23163918867) + 33705582543)c22 − 2(8γ2 + 3)4 (64γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(8γ2 (4γ2 + √ 3)(16γ2 (4γ2 +3)(3232γ2 (4γ2 +3)−23679)−1084023)−7433613)−125951517)+43046721)+1420541793)c2 − 2γ2 (4γ2 +3)(8γ2 + 3)5 (32γ2 (4γ2 + 3) + 27)(32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(416γ2 (4γ2 + 3) + 8721) + 238383) + 570807) + 3720087))c31 − √ 9(−7821419487252849885184c18 17 2 +6045920370158305542144 2(8γ2 +3)c2 −864691128455135232(82928γ2 (4γ2 +3)+50403)c2 + 16 √ 15 27021597764222976 2(8γ2 +3)(618544γ2 (4γ2 +3)+440529)c2 −13510798882111488(4γ2 (4γ2 +3)(1586320γ2 (4γ2 +3)+2336103)+ √ 3277179)c142 +211106232532992 2(8γ2 +3)(64γ2 (4γ2 +3)(715316γ2 (4γ2 +3)+1328763)+34660737)c 13 −39582418599936(64γ (4γ + 2 2 2 √ 3)(8γ2 (4γ2 +3)(1178888γ2 (4γ2 +3)+3863173)+27006867)+454358565)c12 2 +412316860416 2(8γ 2 +3)(16γ2 (4γ2 +3)(256γ2 (4γ2 + 3)(665252γ2 (4γ2 +3)+3892149)+1057756563)+4992478497)c11 2 −51539607552(512γ 2 (4γ2 +3)(γ2 (4γ2 +3)(56γ2 (4γ2 +3)(313984γ2 (4γ2 + √ 3) + 9974967) + 1028916351) + 644559201) + 68959891323)c10 2 − 1073741824 2(8γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 40 3)(6971360γ2 (4γ2 + 3) − 89007381) − 6526225485) − 73004332293) − 264178894635)c92 + 201326592(64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 +3)(16γ2 (4γ2 +3)(5293024γ2 (4γ2 +3)−37997325)−2230608699)−80476063245)−152131081419)−1672746272145)c82 − √ 25165824 2(8γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(489776γ2 (4γ2 + 3) − 4812207) − 711677367) − 14737165839) − 122830333389) − 727740287271)c72 − 6291456(8γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(2γ2 (4γ2 + √ 3)(286496γ2 (4γ2 +3)+2406567)+43182909)+2860677891)+39373245531)+1978479398061)+2353970611251)c62 +196608 2(8γ2 + 3)(16γ2 (4γ2 +3)(64γ2 (4γ2 +3)(8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(32γ2 (4γ2 +3)(206512γ2 (4γ2 +3)−579483)+79925697)+4748690691)+ 39533298939) + 1099633802355) + 2798449794657)c52 − 12288(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(124240γ2 (4γ2 + 3) − 1558569) − 18976923) + 3132532683) + 47747692809) + 1106031289113) + 5920657166601) + √ 24501375705501)c42 − 9216 2(8γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(512γ2 (4γ2 + 3)(358γ2 (4γ2 + 3) + 17059) + 9683307) − 301807539) − 5539520097) − 134870218911) − 742484192715) − 784612583667)c32 + 1536(8γ2 + 3)2 (16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(37888γ2 (4γ2 + 3) + 417195) + √ 4524579)−322339743)−4945963923)−7236632097)−78732452709)−166203389781)c22 −192 2(8γ2 +3)3 (2γ2 (4γ2 +3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(3808γ2 (4γ2 + 3) + 20067) − 869697) − 44761329) − 2073859929) − 11120402925) − 58242213513) − 15109399071)c2 + (8γ2 + 3)4 (32γ2 (4γ2 + 3) + 27)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(248γ2 (4γ2 + 3) − 1053) − 742851) − 5885217) − 19663317) − 243931419) − 1162261467))c21 + √ 24(−2361183241434822606848c19 18 2 +1761664059039262179328 2(8γ2 +3)c2 −6917529027641081856(2896γ2 (4γ2 +3)+1839)c2 + 17 √ 16 972777519512027136 2(8γ2 + 3)(4560γ2 (4γ2 + 3) + 3703)c2 − 81064793292668928(8γ2 (4γ2 + 3)(32816γ2 (4γ2 + 3) + 56963) + √ 176373)c15 2 +105553116266496 2(8γ2 +3)(32γ2 (4γ2 +3)(644456γ2 (4γ2 +3)+1621131)+24413913)c 14 −131941395333120(256γ (4γ + 2 2 2 √ 3)(γ2 (4γ2 + 3)(134176γ2 (4γ2 + 3) + 692919) + 714663) + 53030619)c13 2 + 824633720832 2(8γ 2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(619184γ2 (4γ2 + 3) + 9820401) + 203726853) + 1080500715)c12 2 + 103079215104(16γ2 (4γ 2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + √ 3)(428032γ2 (4γ2 +3)−10945131)−402746013)−4583251431)−16786701225)c11 2 −536870912 2(8γ2 +3)(32γ2 (4γ2 +3)(8γ2 (4γ2 + 3)(1280γ2 (4γ2 + 3)(133364γ2 (4γ2 + 3) − 1377279) − 5492737251) − 36416968281) − 295893771363)c10 2 + 536870912(4γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(1512896γ2 (4γ2 + 3) − 14556051) − 2830869891) − 15952623165) − 553059750879) − √ 417619151433)c92 + 25165824 2(8γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(70064γ2 (4γ2 + 3) + 721365) + 1335076965)+18911826621)+366397455051)+594764458695)c82 −37748736(8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(64γ2 (4γ2 +3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(39872γ2 (4γ2 + 3) − 261411) + 15486417) + 756152577) + 47558442951) + 320706318627) + 399559605273)c72 + √ 294912 2(8γ2 +3)(32γ2 (4γ2 +3)(64γ2 (4γ2 +3)(8γ2 (4γ2 +3)(8γ2 (4γ2 +3)(64γ2 (4γ2 +3)(16296γ2 (4γ2 +3)−287443)+65935737)+ 1944233847) + 16216223733) + 454429891251) + 2335727304603)c62 + 49152(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(2γ2 (4γ2 +3)(512γ2 (4γ2 +3)(586γ2 (4γ2 +3)+24033)−51587199)−467180379)−82693752687)−867239849367)−8907538402305)− √ 9049152548457)c52 −3072 2(8γ2 +3)(4γ2 (4γ2 +3)(32γ2 (4γ2 +3)(512γ2 (4γ2 +3)(2γ2 (4γ2 +3)(4γ2 (4γ2 +3)(32γ2 (4γ2 +3)(25712γ2 (4γ2 + 3) + 46845) − 63747243) − 845437095) − 2203435305) − 731126235663) − 7550590965129) − 3884579149761)c42 + 768(8γ2 + 3)2 (16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(4864γ2 (4γ2 + 3) − 121263) − 3478869) − √ 549289593) − 10551682323) − 107299532223) − 278373578769) − 581776434315)c32 + 24 2(8γ2 + 3)3 (128γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(1520γ2 (4γ2 + 3) + 58353) + 14301441) + 434161053) + 3795256377) + 18986792607) + 12526595811) + 215793212373)c22 − 16(8γ2 + 3)4 (2γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(1984γ2 (4γ2 + 3) + 32355) + 2565351) + 13443489) + 199998963) + 1018772397) + 11751754833) + 3486784401)c2 + √ 2γ2 (4γ2 + 3)(8γ2 + 3)5 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(352γ2 (4γ2 + 3) − √ 135)−3645)−19683))c1 +64(295147905179352825856c20 19 2 −212137556847659843584 2(8γ2 +3)c2 +57646075230342348800(40γ2 (4γ2 + 18 √ 17 3) + 27)c2 − 324259173170675712 2(8γ2 + 3)(1488γ2 (4γ2 + 3) + 1435)c2 + 3377699720527872(32γ2 (4γ2 + 3)(19832γ2 (4γ2 + 3) + √ 42855)+591327)c16 2 −316659348799488 2(8γ2 +3)(32γ2 (4γ2 +3)(19160γ2 (4γ2 +3)+70893)+1249263)c 15 +13194139533312(32γ (4γ + 2 2 2 √ 3)(8γ2 (4γ2 + 3)(91904γ2 (4γ2 + 3) + 890301) + 8779455) + 90548361)c14 2 + 2473901162496 2(8γ 2 + 3)(16γ 2 (4γ2 + 3)(16γ2 (4γ2 + 3)(720γ2 (4γ2 + 3) − 428701) − 11373093) − 69244551)c13 2 − 51539607552(32γ√2 (4γ2 + 3)(8γ2 (4γ2 + 3)(256γ2 (4γ2 + 3)(11474γ2 (4γ2 + 3) − 169119) − 136525257) − 909075879) − 7356711519)c12 2 + 536870912 2(8γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(265424γ2 (4γ2 +3)−2876715)−1023792291)−8229027087)−74254320891)c11 2 −67108864(32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(64γ 2 (4γ2 + √ 3)(8γ2 (4γ2 +3)(71104γ2 (4γ2 +3)−1284057)−275672079)−30805712397)−147688077195)−949653593883)c10 2 −8388608 2(8γ 2+ 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(118960γ2 (4γ2 + 3) − 1110987) + 920049759) + 15602051439) + 326967992685)+558593560719)c92 +18874368(8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(18400γ2 (4γ2 + √ 3) − 312821) + 33449643) + 1759740957) + 29141622639) + 205215463197) + 265387168395)c82 + 98304 2(8γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(340γ2 (4γ2 + 3) + 57027) − 57169611) − 1636300035) − 29161648269) − 436621985685) − 2378454806709)c72 − 331776(128γ2 (4γ2 + 3)(256γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(64γ2 (4γ2 + √ 3)(20γ2 (4γ2 + 3) − 29) − 97911) − 57047463) − 369417105) − 277475625) − 25428395529) − 450976758219)c62 + 3072 2(8γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(2512γ2 (4γ2 + 3) − 75729) + 301401) − 43466139) − 4711338189) − 152635110561) − 2059034673717) − 1265415759423)c52 + 1152(8γ2 + 3)2 (16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(8γ2 (4γ2 +3)(4γ2 (4γ2 +3)(16γ2 (4γ2 +3)(128γ2 (4γ2 +3)(76γ2 (4γ2 +3)+2827)−2456271)−80394525)−120535047)+741950685)+ LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 41 √ 39013615251)+120286887381)c42 −24 2(8γ2 +3)3 (32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(8γ2 (4γ2 +3)(8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(128γ2 (4γ2 + 3)(280γ2 (4γ2 + 3) + 369) − 10363059) − 352188891) − 2035937349) − 3714595443) + 4424246325) + 62891259381)c32 + (8γ2 + 3)4 (32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(512γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(64γ2 (4γ2 + 3) − 23823) − 338013) − √ 345692529) − 1062232461) − 1353580227) − 3486784401) + 15109399071)c22 + 2γ2 (4γ2 + 3)(8γ2 + 3)5 (16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(112γ2 (4γ2 + 3) + 2529) + 245673) + 925101) + 1948617)c2 − 4γ22 (4γ2 + 3)2 (8γ2 + 3)6 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)2 )), √ √ H2 = −6b41 (531441(8γ2 + 3)(8796093022208 2c12 2 − 13194139533312(8γ 11 2 + 3)c2 + 103079215104 2(704γ2 (4γ2 + 3) + 399)c2 − 10 √ 42949672960(8γ2 +3)(704γ2 (4γ2 +3)+405)c92 +12079595520 2(8γ2 +3)2 (352γ2 (4γ2 +3)+207)c82 −2415919104(8γ2 +3)3 (352γ2 (4γ2 + √ √ 3) + 213)c72 + 88080384 2(8γ2 + 3)4 (704γ2 (4γ2 + 3) + 441)c62 − 9437184(8γ2 + 3)5 (704γ2 (4γ2 + 3) + 459)c52 + 11796480 2(8γ2 + √ 3)6 (22γ2 (4γ2 + 3) + 15)c42 − 163840(8γ2 + 3)7 (88γ2 (4γ2 + 3) + 63)c32 + 384 2(8γ2 + 3)8 (704γ2 (4γ2 + 3) + 531)c22 − 96(8γ2 + √ √ 3)9 (64γ2 (4γ2 + 3) + 51)c2 + 2(8γ2 + 3)10 (32γ2 (4γ2 + 3) + 27))c61 − 78732(140737488355328c14 2 + 615726511554560 2(8γ2 + 13 2 12 √ 11 3)c2 − 981863883603968(8γ2 + 3) c2 + 137438953472 2(8γ2 + 3)(39136γ2 (4γ2 + 3) + 22311)c2 − 2147483648(128γ2 (4γ2 + √ 3)(127160γ2 (4γ2 + 3) + 146079) + 5367951)c102 + 268435456 2(8γ2 + 3)(64γ2 (4γ2 + 3)(274384γ2 (4γ2 + 3) + 322587) + 6060015)c92 − 8 √ 201326592(4γ2 (4γ2 + 3)(320γ2 (4γ2 + 3)(55088γ2 (4γ2 + 3) + 98037) + 18575649) + 14640021)c2 + 100663296 2(8γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 +3)(73568γ2 (4γ2 +3)+135621)+2654775)+2156625)c72 −5505024(8γ2 +3)2 (16γ2 (4γ2 +3)(32γ2 (4γ2 +3)(31504γ2 (4γ2 + √ 3) + 61371) + 1261413) + 4281903)c62 + 32768 2(8γ2 + 3)3 (16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(164560γ2 (4γ2 + 3) + 353673) + 7871499) + √ 28554201)c52 − 8192(8γ2 + 3)4 (4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(89584γ2 (4γ2 + 3) + 245133) + 6374133) + 6461127)c42 − 512 2(8γ2 + 3)5 (16γ2 (4γ2 + 3)(80γ2 (4γ2 + 3)(160γ2 (4γ2 + 3) − 3699) − 389529) − 1976319)c32 + 8(8γ2 + 3)6 (16γ2 (4γ2 + 3)(64γ2 (4γ2 + √ 3)(10672γ2 (4γ2 + 3) + 7209) − 280179) − 2932767)c22 − 2(8γ2 + 3)7 (16γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(872γ2 (4γ2 + 3) + 1053) + 26973) − 124659)c2 + γ2 (4γ2 + 3)(8γ2 + 3)8 (32γ2 (4γ2 + 3) + 27)(800γ2 (4γ2 + 3) + 459))c51 + 4374(24769797950537728c15 2 + √ √ 46443371157258240 2(8γ2 + 3)c14 2 − 52776558133248(24080γ 2 (4γ2 + 3) + 13533)c13 + 2199023255552 2(8γ + 3)(195760γ (4γ + 2 2 2 2 √ 3) + 113283)c12 2 − 103079215104(1280γ 2 (4γ2 + 3)(20428γ2 (4γ2 + 3) + 24147) + 9113391)c 11 + 12884901888 2(8γ + 3)(32γ (4γ + 2 2 2 2 3)(829664γ2 (4γ2 + 3) + 1048173) + 10486611)c10 2 − 2147483648(10γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(682352γ2 (4γ2 + 3) + 1354977) + √ 56302533) + 120045159)c92 + 603979776 2(8γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(346720γ2 (4γ2 + 3) + 809037) + 9406611) + 34930035)c82 − 37748736(64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(256γ2 (4γ2 + 3)(23870γ2 (4γ2 + 3) + 92931) + 29075517) + 57924153) + √ 657490203)c72 + 1048576 2(8γ2 + 3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(160γ2 (4γ2 + 3)(28336γ2 (4γ2 + 3) + 351171) + 90000639) + 409445037)+1261923057)c62 +196608(8γ2 +3)2 (8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(512γ2 (4γ2 +3)(7381γ2 (4γ2 +3)−28071)−34174791)− √ 354478437) − 586966743)c52 − 12288 2(8γ2 + 3)3 (32γ2 (4γ2 + 3)(80γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(10384γ2 (4γ2 + 3) − 15549) − 838269) − 47273463) − 327934089)c42 + 128(8γ2 + 3)4 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(55880γ2 (4γ2 + 3) − 101187) − √ 41737437) − 238169403) − 1680672321)c32 − 48 2(8γ2 + 3)5 (64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4960γ2 (4γ2 + 3) − 28917) − 2209923) − 6042681) − 84853413)c22 − 48(8γ2 + 3)6 (γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(2560γ2 (4γ2 + 3)(14γ2 (4γ2 + 3) + 195) + √ 908091) + 9310059) + 2007666)c2 + 2(8γ2 + 3)7 (32γ2 (4γ2 + 3) + 27)(32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(136γ2 (4γ2 + 3) + 567) + √ 2187) + 19683))c1 − 5832(72057594037927936c16 4 15 2 + 75435293758455808 2(8γ2 + 3)c2 − 211106232532992(10640γ2 (4γ2 + 3) + 14 √ 13 5949)c2 + 9851624184872960 2(8γ2 + 3)(76γ2 (4γ2 + 3) + 45)c2 − 5497558138880(16γ2 (4γ2 + 3)(50824γ2 (4γ2 + 3) + 63351) + √ 312903)c122 + 51539607552 2(8γ2 + 3)(64γ2 (4γ2 + 3)(158896γ2 (4γ2 + 3) + 231651) + 5147469)c 11 − 1073741824(16γ (4γ + 2 2 2 10 √ 3)(64γ2 (4γ2 + 3)(1143472γ2 (4γ2 + 3) + 2892825) + 140222259) + 533154879)c2 + 134217728 2(8γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(399520γ2 (4γ2 +3)+1644417)+48560877)+414357039)c92 −75497472(16γ2 (4γ2 +3)(80γ2 (4γ2 +3)(1024γ2 (4γ2 +3)(209γ2 (4γ2 + √ 3)+4266)+7305471)+338267907)+1053453843)c82 −2097152 2(8γ2 +3)(32γ2 (4γ2 +3)(40γ2 (4γ2 +3)(64γ2 (4γ2 +3)(13816γ2 (4γ2 + 3) − 106101) − 14430069) − 367467759) − 2430686475)c72 + 65536(224γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + √ 3)(39952γ2 (4γ2 + 3) − 166383) − 18179397) − 293994765) − 2008656711) − 70545308859)c62 − 24576 2(8γ2 + 3)(64γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(8224γ2 (4γ2 + 3) − 55431) − 2926773) − 195330447) − 692743185) − 7211949615)c52 − 512(8γ2 + 3)2 (256γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(11120γ2 (4γ2 + 3) + 301887) + 27684909) + 237446235) + √ 442270449) + 19361281365)c42 + 64 2(8γ2 + 3)3 (16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(80γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(1112γ2 (4γ2 + 3) + 4095)+ 1280043) + 119921229) + 991609857) + 2966680809)c32 − 24(8γ2 + 3)4 (8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(80γ2 (4γ2 +3)(64γ2 (4γ2 + √ 3)(496γ2 (4γ2 + 3) + 315) + 36693) + 8085339) + 99536931) + 185118615)c22 + 2(8γ2 + 3)5 (32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(416γ2 (4γ2 + 3) − 1791) − 122229) − 1791153) + 767637) + 23914845)c2 + γ2 (4γ2 + 3)(8γ2 + 3)6 (32γ2 (4γ2 + 3) + √ 27)(16γ2 (4γ2 +3)(8γ2 (4γ2 +3)(32γ2 (4γ2 +3)+1431)+12879)+59049))c31 +27(29975959119778021376c17 2 +19167320014088830976 2(8γ2 + √ 3)c16 15 2 − 4503599627370496(140912γ2 (4γ2 + 3) + 77589)c2 + 562949953421312 2(8γ2 + 3)(368000γ 14 2 (4γ2 + 3) + 223659)c2 − 13 √ 35184372088832(320γ2 (4γ2 +3)(103684γ2 (4γ2 +3)+139959)+14693157)c2 +35184372088832 2(8γ2 +3)(4γ2 (4γ2 +3)(878716γ2 (4γ2 + 3) + 1601523) + 2521935)c12 2 − 68719476736(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(3467696γ2 (4γ2 + 3) + 13236129) + 762612975) + √ 3207437433)c112 +549755813888 2(8γ2 +3)(γ2 (4γ2 +3)(4γ2 (4γ2 +3)(2628032γ2 (4γ2 +3)+33238593)+151400421)+45039807)c2 + 10 536870912(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(64γ2 (4γ2 +3)(1183600γ2 (4γ2 +3)−16309341)−2169499383)−22400960589)−75380975811)c92 − √ 134217728 2(8γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(20γ2 (4γ2 + 3)(743776γ2 (4γ2 + 3) − 5008437) − 259275897) − 12023978013) − 22000273029)c82 +4194304(32γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(32γ2 (4γ2 +3)(844976γ2 (4γ2 +3)−6479235)−881545599)− LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 42 √ 17001899109) − 133103892735) − 744344826705)c72 + 1048576 2(8γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(155456γ2 (4γ2 +3)+1761759)+26852067)+5215346919)+46528965189)+141939328995)c62 −32768(16γ2 (4γ2 +3)(512γ2 (4γ2 + 3)(γ2 (4γ2 +3)(16γ2 (4γ2 +3)(64γ2 (4γ2 +3)(163448γ2 (4γ2 +3)−396495)+105558957)+6094259937)+6109485102)+1305941324319)+ √ 3187241401437)c52 + 8192 2(8γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(112400γ2 (4γ2 + 3) − 1293327) − 948915) + 512247159) + 10635689367) + 313465690881) + 404175229407)c42 + 512(8γ2 + 3)2 (8γ2 (4γ2 + 3)(512γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(640γ2 (4γ2 + 3)(1112γ2 (4γ2 + 3) + 69525) + 32785317) − 659801133) − 989306946) − 243884120751) − √ 322520382639)c32 − 64 2(8γ2 + 3)3 (16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(33344γ2 (4γ2 + 3) + 370791) + 2622213) − 221125383) − 4764722859) − 17912218905) − 47050066053)c22 + 32(8γ2 + 3)4 (2γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(3728γ2 (4γ2 + 3) + 20385) − 1188999) − 112160295) − 122231430) − 6882692391) − √ 2195382771)c2 − 2(8γ2 + 3)5 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(26γ2 (4γ2 + 3) − √ 27) − 43011) − 1082565) − 4782969) − 14348907))c21 − 12(64563604257983430656c18 2 + 25940733853654056960 2(8γ2 + 3)c2 − 17 16 √ 15 5188146770730811392(190γ2 (4γ2 +3)+101)c2 +3377699720527872 2(8γ2 +3)(92848γ2 (4γ2 +3)+57621)c2 −316659348799488(160γ2 (4γ2 + √ 3)(32536γ2 (4γ2 +3)+48819)+2740311)c14 2 +39582418599936 2(8γ2 +3)(544γ2 (4γ2 +3)(7112γ2 (4γ2 +3)+17575)+4256721)c 13 − 2 12 √ 824633720832(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(1038064γ2 (4γ2 +3)+7470357)+126380763)+579754989)c2 −154618822656 2(8γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(35072γ2 (4γ2 + 3) − 1683393) − 74672793) − 397657593)c11 2 + 4831838208(32γ 2 (4γ2 + 3)(8γ2 (4γ2 + √ 3)(256γ2 (4γ2 +3)(91300γ2 (4γ2 +3)−808369)−549450027)−3264549723)−24286529295)c10 2 −67108864 2(8γ2 +3)(256γ2 (4γ2 + 3)(γ2 (4γ2 +3)(320γ2 (4γ2 +3)(239888γ2 (4γ2 +3)−1900737)−2573984061)−2317559526)−155008434213)c92 −75497472(8γ2 (4γ2 + 3)(32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(64γ2 (4γ2 +3)(5456γ2 (4γ2 +3)+405189)+496835775)+6550265475)+119844675981)+184552669701)c82 + √ 9437184 2(8γ2 +3)(128γ2 (4γ2 +3)(4γ2 (4γ2 +3)(32γ2 (4γ2 +3)(128γ2 (4γ2 +3)(3658γ2 (4γ2 +3)−22987)+19991403)+1378335123)+ 3481961337) + 90973454751)c72 − 98304(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(84728γ2 (4γ2 + 3) − √ 1259427) + 249961491) + 7318577043) + 117711257481) + 1582571087703) + 7791361373061)c62 − 36864 2(8γ2 + 3)(128γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(512γ2 (4γ2 + 3)(241γ2 (4γ2 + 3) + 12486) − 27433917) − 183363183) − 11751942915) − 39893681547) − 796810217499)c52 + 9216(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(160γ2 (4γ2 + 3)(2896γ2 (4γ2 + 3) + 5555) − 33107157) − 105492132) − 16851439341) − 334277156637) − 3300035502159) − 1622387867769)c42 − √ 384 2(8γ2 + 3)(32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4672γ2 (4γ2 + 3) − 104547) − 23537709)−887214627)−8089785171)−155946224187)−191658350799)−758956737951)c32 −72(8γ2 +3)2 (32γ2 (4γ2 +3)(8γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(848γ2 (4γ2 + 3) + 35427) + 8681067) + 253595043) + 2086955685) + √ 9755662437)+24034419225)+96725982087)c22 +9 2(8γ2 +3)3 (16γ2 (4γ2 +3)(64γ2 (4γ2 +3)(8γ2 (4γ2 +3)(2γ2 (4γ2 +3)(16γ2 (4γ2 + 3)(1024γ2 (4γ2 +3)(13γ2 (4γ2 +3)+218)+579717)+12313539)+21434787)+97253703)+1994498073)+4261625379)c2 −γ2 (4γ2 + 3)(8γ2 + 3)4 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3) + 27)(184γ2 (4γ2 + 3) − √ 135) − 19683))c1 + 32(9223372036854775808c19 18 2 + 2305843009213693952 2(8γ2 + 3)c2 − 108086391056891904(976γ2 (4γ2 + 3) + √ 483)c17 2 +6755399441055744 2(8γ 2 +3)(4816γ 2 (4γ 2 +3)+2997)c 16 −105553116266496(32γ (4γ +3)(46568γ (4γ +3)+79563)+ 2 2 2 2 2 √ 947673)c15 2 + 13194139533312 2(8γ 2 + 3)(64γ2 (4γ2 + 3)(14492γ2 (4γ2 + 3) + 52887) + 1657449)c 14 − 2473901162496(16γ (4γ + 2 2 √ 2 3)(16γ2 (4γ2 + 3)(13328γ2 (4γ2 + 3) + 296067) + 5726295) + 28169937)c13 2 − 1236950581248 2(8γ2 + 3)(8γ 2 (4γ2 + 3)(8γ2 (4γ2 + 3)(14512γ2 (4γ2 + 3) − 203129) − 2768895) − 8254791)c12 2 + 1610612736(32γ 2 (4γ2 + 3)(8γ 2 (4γ 2 + 3)(128γ 2 (4γ2 + 3)(61400γ 2 (4γ 2 + √ 3) − 511953) − 247821849) − 1720462599) − 14085493785)c11 2 − 67108864 2(8γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(103136γ2 (4γ2 + 3) − 1037943) − 423575973) − 3754547559) − 34924275171)c10 2 − 8388608(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + √ 3)(64γ2 (4γ2 +3)(151184γ2 (4γ2 +3)−1193967)+755240679)+12633531363)+256609639521)+423514131201)c92 +12582912 2(8γ2 + 3)(4γ2 (4γ2 +3)(16γ2 (4γ2 +3)(32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(44384γ2 (4γ2 +3)−637989)+22563819)+1890504765)+21223105083)+ 18784569465)c82 − 98304(32γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(376γ2 (4γ2 + 3) − 166491) + √ 42349635) + 1469145681) + 6786075105) + 408571427457) + 2209108652001)c72 − 12288 2(8γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 +3)(4γ2 (4γ2 +3)(32γ2 (4γ2 +3)(6832γ2 (4γ2 +3)−11217)−4553847)−134931825)−12856946535)−112423646979)− 671532567687)c62 + 3072(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4592γ2 (4γ2 + √ 3) − 120471) + 10676205) + 57287007) − 1120551003) − 129612495951) − 2026708180569) − 1315665631737)c52 + 3456 2(8γ2 + 3 3) (16γ2 (4γ2 +3)(8γ2 (4γ2 +3)(128γ2 (4γ2 +3)(8γ2 (4γ2 +3)(8γ2 (4γ2 +3)(80γ2 (4γ2 +3)+3101)−234009)−2853927)−102296925)+ 368249247) + 2382450003)c42 − 72(8γ2 + 3)2 (64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + √ 3)(336γ2 (4γ2 + 3) + 349) − 6596181) − 226708713) − 1371063105) − 3095525727) − 74933181) + 22685621967)c32 + 3 2(8γ2 + 3)3 (16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3) − 7305) − 3400623) − 108267435) − 82924479) − 1713897225) − 2625849981) + 2711943423)c22 + γ2 (4γ2 + 3)(8γ2 + 3)4 (16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + √ 81) + 729)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(208γ2 (4γ2 + 3) + 4869) + 467289) + 1712421) + 3365793)c2 − 4 2γ22 (4γ2 + 3)2 (8γ2 + 3)5 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)2 )) and √ √ H3 = −9b51 (−531441(−8796093022208c12 11 2 +6597069766656 2(8γ2 +3)c2 −103079215104(704γ2 (4γ 10 2 +3)+399)c2 +21474836480 2(8γ2 + √ 3)(704γ2 (4γ2 + 3) + 405)c92 − 12079595520(8γ2 + 3)2 (352γ2 (4γ2 + 3) + 207)c82 + 1207959552 2(8γ2 + 3)3 (352γ2 (4γ2 + 3) + √ 213)c72 − 88080384(8γ2 + 3)4 (704γ2 (4γ2 + 3) + 441)c62 + 4718592 2(8γ2 + 3)5 (704γ2 (4γ2 + 3) + 459)c52 − 11796480(8γ2 + LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 43 √ √ 3)6 (22γ2 (4γ2 + 3) + 15)c42 + 81920 2(8γ2 + 3)7 (88γ2 (4γ2 + 3) + 63)c32 − 384(8γ2 + 3)8 (704γ2 (4γ2 + 3) + 531)c22 + 48 2(8γ2 + √ 3)9 (64γ2 (4γ2 + 3) + 51)c2 − (8γ2 + 3)10 (32γ2 (4γ2 + 3) + 27))c61 + 944784(−61572651155456c13 2 + 45079976738816 2(8γ2 + √ 3)c12 11 2 − 103079215104(4672γ2 (4γ2 + 3) + 2661)c2 + 34359738368 2(8γ2 + 3)(2816γ2 (4γ2 + 3) + 1647)c102 − 268435456(80γ2 (4γ2 + √ 3)(19360γ2 (4γ2 +3)+22707)+531927)c92 +1207959552 2(8γ2 +3)(4γ2 (4γ2 +3)(8096γ2 (4γ2 +3)+9813)+11853)c82 −12582912(8γ2 + √ 3)2 (16γ2 (4γ2 + 3)(26048γ2 (4γ2 + 3) + 33129) + 167265)c72 + 352321536 2(8γ2 + 3)3 (γ2 (4γ2 + 3)(704γ2 (4γ2 + 3) + 963) + √ 324)c62 − 589824(8γ2 + 3)4 (8γ2 (4γ2 + 3)(3344γ2 (4γ2 + 3) + 5169) + 15363)c52 + 4096 2(8γ2 + 3)5 (80γ2 (4γ2 + 3)(880γ2 (4γ2 + 3) + √ 1773) + 62613)c42 − 128(8γ2 + 3)6 (16γ2 (4γ2 + 3)(704γ2 (4γ2 + 3) + 7407) + 76707)c32 − 384 2(8γ2 + 3)7 (8γ2 (4γ2 + 3)(64γ2 (4γ2 + √ 3) − 3) − 297)c22 + (8γ2 + 3)8 (16γ2 (4γ2 + 3)(544γ2 (4γ2 + 3) + 333) − 1215)c2 − 2 2γ2 (4γ2 + 3)(8γ2 + 3)9 (32γ2 (4γ2 + 3) + √ 27))c51 − 13122(−22517998136852480c14 13 2 + 16008889300418560 2(8γ2 + 3)c2 − 2199023255552(74960γ2 (4γ2 + 3) + 43299)c2 + 12 √ 137438953472 2(8γ2 +3)(229760γ2 (4γ2 +3)+141363)c11 2 −8589934592(16γ 2 (4γ 2 +3)(925760γ2 (4γ2 +3)+1159713)+5758533)c 10 + 2 √ 42949672960 2(8γ2 +3)(8γ2 (4γ2 +3)(31768γ2 (4γ2 +3)+44433)+120447)c92 −4026531840(64γ2 (4γ2 +3)(γ2 (4γ2 +3)(78496γ2 (4γ2 + √ 3) + 178281) + 127332) + 1867941)c82 + 50331648 2(8γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(56320γ2 (4γ2 + 3) + 175203) + 1180737) + √ 9630333)c72 − 22020096(8γ2 + 3)2 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(3520γ2 (4γ2 + 3) + 29241) + 251019) + 2312631)c62 − 131072 2(8γ2 + 3)3 (16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(25520γ2 (4γ2 +3)−139509)−3235545)−16707951)c52 +40960(8γ2 +3)4 (16γ2 (4γ2 +3)(128γ2 (4γ2 + √ 3)(1298γ2 (4γ2 + 3) − 2781) − 660231) − 3692385)c42 − 2560 2(8γ2 + 3)5 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(3968γ2 (4γ2 + 3) − 8775) − √ 135351) − 1565163)c32 + 32(8γ2 + 3)6 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(6080γ2 (4γ2 + 3) − 30753) − 410427) − 4721733)c22 + 16 2(8γ2 + 3)7 (32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(80γ2 (4γ2 + 3) + 1809) + 9963) + 111537)c2 − (8γ2 + 3)8 (8γ2 (4γ2 + 3) + 27)(32γ2 (4γ2 + 3) + √ 27)(40γ2 (4γ2 + 3) + 27))c41 + 69984(−11258999068426240c15 14 2 + 7740561859543040 2(8γ2 + 3)c2 − 21990232555520(3472γ2 (4γ2 + √ 3) + 2061)c132 + 549755813888 2(8γ2 + 3)(25120γ 2 (4γ 2 + 3) + 17073)c 12 − 17179869184(176γ (4γ + 3)(16960γ (4γ + 3) + 2 2 2 2 2 √ 24399)+1476873)c11 2 +2147483648 2(8γ2 +3)(16γ2 (4γ2 +3)(112640γ2 (4γ2 +3)+214281)+1386477)c 10 −12884901888(5γ (4γ + 2 2 2 √ 3)(8γ2 (4γ2 +3)(10912γ2 (4γ2 +3)+41805)+301563)+396819)c92 +201326592 2(8γ2 +3)(20γ2 (4γ2 +3)(32γ2 (4γ2 +3)(704γ2 (4γ2 + 3)+9423)+338985)+1988469)c82 +6291456(8γ2 +3)2 (32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(7040γ2 (4γ2 +3)−71199)−784161)−8070273)c72 − √ 262144 2(8γ2 + 3)3 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(12320γ2 (4γ2 + 3) − 68877) − 871641) − 9692055)c62 + 16384(8γ2 + 3)4 (16γ2 (4γ2 + √ 3)(32γ2 (4γ2 + 3)(4840γ2 (4γ2 + 3) − 36261) − 1972917) − 11844063)c52 + 2048 2(8γ2 + 3)5 (80γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3) + 2781) + 79461) + 2645541)c42 − 64(8γ2 + 3)6 (160γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(704γ2 (4γ2 + 3) + 2457) + 40095) + 3243321)c32 + √ 8 2(8γ2 + 3)7 (32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(640γ2 (4γ2 + 3) + 81) + 10449) + 303993)c22 − 4(8γ2 + 3)8 (4γ2 (4γ2 + 3)(8γ2 (4γ2 + √ 3)(160γ2 (4γ2 + 3) − 621) − 2673) + 6561)c2 − 27 2γ2 (4γ2 + 3)(8γ2 + 3)9 (32γ2 (4γ2 + 3) + 27))c31 − 81(−14411518807585587200c16 2 + √ √ 9547631210025451520 2(8γ2 +3)c15 14 2 −6755399441055744(13280γ2 (4γ2 +3)+8241)c2 +70368744177664 2(8γ 2 +3)(215600γ2 (4γ2 + √ 3)+171459)c13 12 2 −10995116277760(32γ2 (4γ2 +3)(144632γ2 (4γ2 +3)+261405)+3275559)c2 +824633720832 2(8γ2 +3)(16γ2 (4γ2 + 3)(242720γ2 (4γ2 +3)+755493)+5864859)c11 2 −8589934592(32γ2 (4γ2 +3)(2920γ2 (4γ2 +3)(3520γ2 (4γ2 +3)+36603)+118670589)+ √ 1113354315)c10 2 −1073741824 2(8γ2 +3)(32γ 2 (4γ2 +3)(8γ2 (4γ2 +3)(344960γ2 (4γ2 +3)−6820533)−75439107)−793854027)c2 + 9 603979776(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(149600γ2 (4γ2 + 3) − 1142751) − 44930457) − 507514491) − 1822681521)c82 − √ 2147483648 2(8γ2 + 3)(4γ2 (4γ2 + 3)(40γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(1078γ2 (4γ2 + 3) − 9333) − 925587) − 29953881) − 29596671)c72 − 524288(128γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(24640γ2 (4γ2 + 3) + 940671) + 21925485) + 992114325) + √ 4274484939) + 102025393641)c62 + 196608 2(8γ2 + 3)(64γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(20γ2 (4γ2 + 3)(3616γ2 (4γ2 + 3) − 6009) + 1133487) + 39062007) + 793170225) + 10379849733)c52 − 4096(8γ2 + 3)2 (32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + √ 3)(128γ2 (4γ2 +3)(2645γ2 (4γ2 +3)−26136)+1629477)+179334729)+4360250331)+30926854701)c42 −2048 2(8γ2 +3)3 (16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(800γ2 (4γ2 + 3) + 115497) + 33939) − 30192993) − 409347351) − 1516201173)c32 + 384(8γ2 + 3)4 (32γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(160γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(704γ2 (4γ2 + 3) + 8331) − 351) − 11982573) − 39503781) − √ 291761109)c22 − 16 2(8γ2 + 3)5 (16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(1280γ2 (4γ2 + 3) + 7227) − 15795) − 1948617)−22851963)−81310473)c2 +(8γ2 +3)6 (32γ2 (4γ2 +3)+27)(32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(320γ22 (4γ2 +3)2 −6561)−59049)− √ 531441))c21 + 144(−6341068275337658368c17 16 2 + 4035225266123964416 2(8γ2 + 3)c2 − 4503599627370496(7976γ2 (4γ2 + 3) + √ 5283)c15 2 +140737488355328 2(8γ 2 +3)(39920γ 2 (4γ 2 +3)+39411)c 14 −2199023255552(32γ (4γ +3)(235480γ (4γ +3)+589131)+ 2 2 2 2 2 √ 8438823)c13 2 + 274877906944 2(8γ 2 + 3)(256γ2 (4γ2 + 3)(11147γ2 (4γ2 + 3) + 69507) + 10267641)c 12 + 8589934592(32γ (4γ + 2 2 2 √ 3)(8γ2 (4γ2 + 3)(29504γ2 (4γ2 + 3) − 6545907) − 70427961) − 737840583)c11 2 − 2147483648 2(8γ2 + 3)(32γ 2 (4γ2 + 3)(8γ2 (4γ 2+ 3)(149248γ2 (4γ2 +3)−1696977)−24709455)−304505487)c10 2 +67108864(16γ 2 (4γ2 +3)(16γ2 (4γ 2 +3)(16γ 2 (4γ2 +3)(487520γ 2 (4γ 2+ √ 3) − 4492647) − 267871779) − 3708547659) − 15154801191)c92 − 33554432 2(8γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(19360γ2 (4γ2 + 3) − 696699) − 56124009) − 1932030063) − 2203465923)c82 − 524288(512γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(64γ2 (4γ2 + √ 3)(2γ2 (4γ2 + 3)(88000γ2 (4γ2 + 3) − 423261) + 9452133) + 1213136919) + 1493145819) + 153420226065)c72 + 131072 2(8γ2 + 3)(32γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(160γ2 (4γ2 + 3)(1232γ2 (4γ2 + 3) − 16839) + 11239641) + 430173423) + 4499461629) + 29985731739)c62 +4096(16γ2 (4γ2 +3)(32γ2 (4γ2 +3)(128γ2 (4γ2 +3)(40γ2 (4γ2 +3)(4γ2 (4γ2 +3)(496γ2 (4γ2 +3)+70011)−1175877)− √ 156493701) − 19858237749) − 266410133271) − 656224941123)c52 − 1024 2(8γ2 + 3)(8γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 +3)(16γ2 (4γ2 +3)(5360γ2 (4γ2 +3)+22491)−6841665)−146805291)−4481641953)−59743357191)−73834692453)c42 + 128(8γ2 + 3)2 (128γ2 (4γ2 + 3)(γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(5440γ2 (4γ2 + 3) − 89019) − 11718675) − LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 44 √ 197419761) − 2836576179) − 1163324349) − 22972600107)c32 + 16 2(8γ2 + 3)3 (8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(16γ2 (4γ2 +3)(256γ2 (4γ2 +3)+12555)+771525)+5181003)+69264477)+1784047437)+2195382771)c22 −(8γ2 +3)4 (16γ2 (4γ2 + 3)(16γ2 (4γ2 +3)(64γ2 (4γ2 +3)(γ2 (4γ2 +3)(16γ2 (4γ2 +3)(608γ2 (4γ2 +3)+10287)+406539)+518319)+24977727)+157837977)+ √ 387420489)c2 +256 2γ23 (4γ2 +3)3 (8γ2 +3)5 (32γ2 (4γ2 +3)+27)(16γ2 (4γ2 +3)(8γ2 (4γ2 +3)+81)+729))c1 +64(4611686018427387904c18 2 − √ √ 2810246167479189504 2(8γ2 +3)c17 16 2 +13510798882111488(1744γ2 (4γ2 +3)+1263)c2 −20266198323167232 2(8γ2 +3)(166γ2 (4γ2 + √ 3) + 215)c15 14 2 + 52776558133248(16γ2 (4γ2 + 3)(9920γ2 (4γ2 + 3) + 37989) + 308529)c2 − 824633720832 2(8γ2 + 3)(32γ2 (4γ2 + 3)(8456γ2 (4γ2 +3)+151845)+3372381)c13 2 −51539607552(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(45616γ2 (4γ2 +3)−861873)−23305671)− √ 138661875)c122 + 3221225472 2(8γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(73984γ2 (4γ2 + 3) − 809253) − 18279837) − 268027785)c2 − 11 201326592(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 +3)(54208γ2 (4γ2 +3)−704625)−102240387)−1730283687)−7827786945)c10 2 − √ 16777216 2(8γ2 +3)(64γ2 (4γ2 +3)(64γ2 (4γ2 +3)(γ2 (4γ2 +3)(36080γ2 (4γ2 +3)−230949)+4718331)+391378959)+7796915253)c92 + 25165824(2γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(64γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(51040γ2 (4γ2 + 3) − 694773) + 19038645) + 2764517445) + √ 14576882067) + 6169498569)c82 − 196608 2(8γ2 + 3)(128γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(10γ2 (4γ2 + 3)(256γ2 (4γ2 + 3) − 21633) + 1508571) + 122552919) + 1397464569) + 40192784415)c72 − 36864(16γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(2240γ2 (4γ2 + 3) + 5573) − 3293451) − 23760945) − 880465059) − 53802187983) − 147097377243)c62 + √ 49152 2(8γ2 + 3)(4γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(128γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(680γ2 (4γ2 + 3) − 14049) − 399411) − 7680015)−1704147579)−3962955537)−3030808023)c52 +384(16γ2 (4γ2 +3)(4γ2 (4γ2 +3)(32γ2 (4γ2 +3)(16γ2 (4γ2 +3)(16γ2 (4γ2 + 3)(1792γ2 (4γ2 + 3)(5γ2 (4γ2 + 3) + 276) − 2337255) − 70005141) − 147515337) + 19515753549) + 47397628467) + 130130237583)c42 − √ 48 2(8γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(4γ2 (4γ2 + 3)(32γ2 (4γ2 + 3)(8γ2 (4γ2 + 3)(512γ2 (4γ2 + 3) + 2589) − 1006749) − 66568149) − 179423667) − 524886561) + 2319739965) + 11751754833)c32 + 3(8γ2 + 3)2 (256γ2 (4γ2 + 3)(2γ2 (4γ2 + 3)(16γ2 (4γ2 +3)(2γ2 (4γ2 +3)(32γ2 (4γ2 +3)(8γ2 (4γ2 +3)(64γ2 (4γ2 +3)−6183)−770391)−48208041)−70996581)−336402153)− √ 43046721) + 1937102445)c22 + 12 2γ2 (4γ2 + 3)(8γ2 + 3)3 (16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 195) + 4617) + 32805) + 59049)c2 − 2γ22 (4γ2 + 3)2 (8γ2 + 3)4 (32γ2 (4γ2 + 3) + 27)(16γ2 (4γ2 + 3)(8γ2 (4γ2 + 3) + 81) + 729)2 )). Acknowledgements The first author is partially supported by the university Mohammed El Bachir El Ibrahimi, Bordj Bou Arreridj, Algerian Ministry of Higher Education and Scientific Research. The second author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grant PID2019-104658GB-I00, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The third author is partially supported by the university Mohammed El Bachir El Ibrahimi, Bordj Bou Arreridj, Algerian Ministry of Higher Education and Scientific Research. References [1] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966. [2] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Appl. Math. Sci. Series 163, Springer-Verlag, London, 2008. [3] H. Dulac, Détermination et integration d’une certaine classe d’équations différentielle ayant par point singulier un centre, Bull. Sci. Math. Sér. (2) 32 (1908), 230–252. [4] B. Ghermoul, J. Llibre and T. Salhi, Limit cycles of continuous piecewise differential systems formed by linear and quadratic isochronous centers I, to appear in Int. J. Bifurcation and Chaos, (2021). [5] D. Hilbert, Mathematische Probleme, Lecture, Second Internat. Congr. Math. (Paris, 1900), Nachr. Ges. Wiss. G”ottingen Math. Phys. KL. (1900), 253–297; English transl., Bull. Amer. Math. Soc. 8 (1902), 437–479; Bull. (New Series) Amer. Math. Soc. 37 (2000), 407–436. [6] C. Huygens, Horologium Oscillatorium, translated by Ian Bruce (July 2007), (PDF). 17thcenturymaths.com. Retrieved 2009-03-01. [7] Yu. Ilyashenko, Centennial history of Hilbert’s 16th problem, Bull. (New Series) Amer. Math. Soc. 39 (2002), 301–354. [8] J. Li, Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 47–106. [9] J. Llibre, D.D. Novaes and M.A. Teixeira, Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dyn. 82 (2015), 1159–1175. [10] J. Llibre and M.A. Teixeira, Piecewise linear differential systems with only centers can create limit cycles?, Nonlinear Dynamics 91 (2018), 249–255. [11] W.S. Loud, Behavior of the periodic of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations 3 (1964), 21–36. LIMIT CYCLES OF CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS 45 [12] H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, Journal de Mathématiques 37 (1881), 375–422; Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951, pp 3–84. [13] H. Poincaré, Sur les courbes définies par une équation différentielle, Oevres complètes, Vol. 1, 1928. [14] D.J.W. Simpson, Bifurcations in piecewise–smooth continuous systems, World Scientific Series on Nonlinear Science A, vol. 69, 2010. 1 Departement of Mathematics, University Mohamed El Bachir El-Ibrahimi, Bordj Bou Arreridj 34265, El- Anasser, Algeria Email address: bilal.ghermoul@univ-bba.dz, t.salhi@univ-bba.dz 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat