This is a preprint of: “Limit cycles of a continuous piecewise differential system formed by a quadratic center and two linear centers”, Maria Elisa Anacleto, Jaume Llibre, Clàudia Valls, Claudio Vidal, Bol. Soc. Mat. Mexicana, vol. 29, 29, 2023. DOI: [10.1007/s40590-023-00501-7] LIMIT CYCLES OF A CONTINUOUS PIECEWISE DIFFERENTIAL SYSTEM FORMED BY A QUADRATIC CENTER AND TWO LINEAR CENTERS. MARIA ELISA ANACLETO1 , JAUME LLIBRE2 , CLAUDIA VALLS3 AND CLAUDIO VIDAL1 Abstract. The study of limit cycles of planar differential systems is one of the main and difficult problems for understanding their dynamics. Thus the objective of this paper is to study the limit cycles of continuous piecewise differential systems in the plane separated by a non-regular line Σ. More precisely, we show that a class of continuous piecewise differential systems formed by an arbitrary quadratic center, an arbitrary linear center and the linear center ẋ = −y, ẏ = x have at most two crossing limit cycles and we find examples of such systems with one crossing limit cycle. So we have solved the extension of the 16th Hilbert problem to this class of piecewise differential systems providing an upper bound for its maximum number of limit cycles. 1. Introduction and statement of the main results The study of the existence of the so-called limit cycles of a planar differential system, i.e. existence of periodic orbits isolated in the set of all periodic orbits of that system is one of the main difficulties for completely understanding (at least qualitatively) its dynamics. In particular to find a limit cycle of a given class of differential systems is very difficult and to provide an upper bound on the maximum number of them is even harder. When such an upper bound exists, additional difficulties arise when trying to prove that such upper bound is achieved. In this paper we shall study the limit cycles of a class of piecewise differential systems. These systems have been studied intensively these last decades due to their applications, see for instance the books [1, 4, 19] and the papers [18, 20]. For planar piecewise differential systems with separation curve Σ = {h−1 (0)} where h : R2 → R (being bivaluated on the separation curve for the vector fields X and Y ) a point p = (x, y) in Σ is a crossing point if Xh(p) · Y h(p) > 0, where · denotes the inner product of two vectors, for more details see Filippov [5]. If there exist a periodic orbit of that piecewise differential system such that all the points of the orbit on Σ are crossing points, then we call it a crossing periodic orbit. A crossing limit cycle is an isolated periodic orbit in the set all crossing periodic orbits of the differential system. 2010 Mathematics Subject Classification. 34C05, 34C07. Key words and phrases. Limit cycles, linear center, quadratic center, continuous piecewise differ- ential systems, first integrals. 1 2 M.E. ANACLETO, J. LLIBRE, C. VALLS AND C. VIDAL The crossing limit cycles of different classes of piecewise differential systems have been studied by many authors during these last years, see for instance [2, 3, 6, 7, 9, 11–17]. In this paper we study the maximum number of crossing limit cycles of the class of planar continuous piecewise differential systems separated by the non-regular line Σ = {(x, y) ∈ R2 : (y = 0) ∨ (x = 0 ∧ y ≥ 0)}. The three components of R2 \ Σ are the positive or first quadrant R1 = {(x, y) ∈ R2 : x ≥ 0 ∧ y ≥ 0}, the second quadrant R2 = {(x, y) ∈ R2 : x ≤ 0 ∧ y ≥ 0}, and the half-plane R3 = {(x, y) ∈ R2 : y ≤ 0}. More precisely, in the region R1 we consider an arbitrary quadratic differential system ẋ = c0 + c1 x + c2 y + c3 x2 + c4 xy + c5 y 2 , (1) ẏ = d0 + d1 x + d2 y + d3 x2 + d4 xy + d5 y 2 , with ci , di ∈ R for i = 0, . . . , 5. In the region R2 we consider an arbitrary linear center ẋ = a0 + a1 x + a2 y, (2) ẏ = b0 + b1 x + b2 y, with ai , bi ∈ R for i = 0, 1, 2, and in the region R3 we consider the linear center ẋ = −y, (3) ẏ = x. Our main result is the following. Theorem 1. Any continuous piecewise differential system in the plane formed by systems (1) in R1 , systems (2) in R2 and systems (3) in R3 separated by the non-regular line Σ has at most two crossing limit cycles. Moreover we provide an example of such a system with one crossing limit cycle. The proof of Theorem 1 is given in Section 2. Note that Theorem 1 provides a positive answer to the extension of the 16th Hilbert problem [8] for the class of continuous piecewise differential systems separated by a non-regular line Σ and formed by the above differential systems. Note that although two is the maximum number of crossing limit cycles that the above mentioned system can have, we are only able to find examples of these piecewise differential systems with one crossing limit cycle. So it remains open if the upper bound of two is reached or not. LIMIT CYCLES CREATED BY A QUADRATIC CENTER 3 2. Proof of Theorem 1 Before proving Theorem 1 we recall that a set of functions {f0 , f1 , . . . , fn } is an extended complete Chebyshev system on R+ if and only if the Wronskians f0 (s) f1 (s) ··· fk (s) f00 (s) f10 (s) ··· fk0 (s) W (f0 , . . . , fk )(s) = .. .. .. .. 6= 0, . . . . f0k (s) f1k (s) ··· fkk (s) on R+ for k = 0, 1, . . . , n. Moreover for an extended complete Chebyshev system in R+ we have the following well-known result, for a proof see for instance [10]. Theorem 2. Assume that the functions f0 , f1 , . . . , fn form an extended complete Chebyshev system in R+ . Then the maximum number of zeros of the function (4) a0 f0 (x) + a1 f1 (x) + . . . + an fn (x) = 0 in R is n. Moreover if the coefficients a0 , a1 , . . . , an are independent there are + functions of the form in (4) having exactly n zeros in R+ . We separate the proof of Theorem 1 in two parts: the part concerning the upper bound and the part providing an example with one crossing limit cycle. First note that in Theorem 1 we are assuming that the piecewise differential system must be continuous, and so systems (1) and (2) must coincide on {x = 0, y ≥ 0}, systems (2) and (3) must coincide on {x ≤ 0, y = 0}, and systems (3) and (1) must coincide on {x ≥ 0, y = 0}. Imposing these three conditions we obtain that a0 = a1 = b0 = c0 = c1 = c3 = c5 = d0 = d3 = d5 = 0, b1 = d1 = 1, a2 = c2 , b2 = d2 . and the continuous piecewise differential system to study is the one formed by the following three differential systems ẋ = c2 y + c4 xy, ẏ = x + d2 y + d4 xy, in R1 with c24 + d24 6= 0. (5) ẋ = c2 y, ẏ = x + d2 y, in R2 with c2 6= 0, ẋ = −y, ẏ = x, in R3 . Note that c24 + d24 6= 0, otherwise the first system in (5) will not be a quadratic system. Moreover, if c2 = 0 then in the second system in (5) we have ẋ = 0, so its solutions live on the straight lines x =constant and then the piecewise differential system cannot have crossing periodic orbits. Hence we have that c2 (c24 + d24 ) 6= 0. The upper bound. We start imposing that the quadratic system in (1) has a center. The equilibrium points of such a quadratic system are   c2 c2 E0 = (0, 0) and E1 = − , . c4 c4 d2 − c2 d4 Since c2 (c24 + d24 ) 6= 0 we have c24 + (c4 d2 − c2 d4 )2 6= 0. We consider different cases: Case 1: c4 (c4 d2 − c2 d4 ) 6= 0. In this case we define T = d4 x + c4 y + d2 , D = −c4 x + (c4 d2 − c2 d4 )y − c2 , 4 M.E. ANACLETO, J. LLIBRE, C. VALLS AND C. VIDAL and ∆ = c24 y 2 + (2c4 d2 − 4(c4 d2 − c2 d4 ))y + d24 x2 + (4c4 + 2d2 d4 )x + 2c4 d4 xy + d22 + 4c2 , where T , D and ∆ are, respectively, the trace, the determinant and the discriminant associated to the linear part of the quadratic system in (5). Then at the equilibrium point E0 we obtain T0 = d2 , D0 = −c2 , ∆0 = 4c2 + d22 , and at the equilibrium point E1 we have c2 c24 + c24 d22 + c22 d24 − 2c2 c4 d2 d4 T1 = , D 1 = c2 , c4 (c4 d2 − c2 d4 ) and  2 −c2 c24 + c22 d24 + c24 d22 − 2c2 c4 d2 d4 ∆1 = . c4 (c4 d2 − c2 d4 ) Observe that E1 cannot be a center because ∆1 ≥ 0. On the other hand E0 is either a weak focus or a center if and only if D0 > 0, ∆0 < 0 and T0 = 0. Thus, c2 < 0 (that we can write as c2 = −c2 with c > 0), d2 = 0 and c4 d4 6= 0. Hence the quadratic system in the region R1 is (6) ẋ = −c2 y + c4 xy, ẏ = x + d4 xy, and the arbitrary linear system in the region R2 is (7) ẋ = −c2 y, ẏ = x. Note that in system (6) we can assume without loss of generality that c4 < 0 and d4 > 0. Indeed, if originally c4 > 0 then doing the change of variables (x, y, t) → (−x, y, −t), c4 becomes negative, and if originally d4 < 0 then doing the change of variables (x, y, t) → (x, −y, −t), d4 becomes positive. The first integrals for systems (6), (7) and (3) are − c2cd24 H1 (x, y) = e−d4 (d4 x+c4 y) (1 + d4 y)−c4 c2 − c4 x 4 in R1 , H2 (x, y) = x2 + c2 y 2 in R2 , 2 2 H3 (x, y) = x + y in R3 , as it is easy to check. The existence of the first integral H1 (x, y) defined in the point (0, 0) forces that the equilibrium (0, 0) of the quadratic system (6) is a center. Now we study the limit cycles of these continuous piecewise differential systems which intersect the non-regular line of discontinuity Σ in the points (x1 , 0), (0, y1 ) and (x2 , 0) with x1 > 0, y1 > 0 and x2 < 0. These points must satisfy e1 = H1 (x1 , 0) − H1 (0, y1 ) = 0, (8) e2 = H2 (x2 , 0) − H2 (0, y1 ) = 0, e3 = H3 (x2 , 0) − H3 (x1 , 0) = 0, LIMIT CYCLES CREATED BY A QUADRATIC CENTER 5 or equivalently 2  c4 − c2cd24 e1 = e−d4 x1 1 − 2 x1 − ec4 d4 y1 (1 + d4 y1 )−c4 = 0, 4 c e2 = x22 − c2 y12 = 0, e3 = x22 − x21 = 0. Solving e2 = 0 and e3 = 0 we obtain x1 x2 = −x1 , y1 = . c Substituting y1 into e1 = 0 we get  − c2cd24  −c4 2 c4 c4 d 4 d4 e−d4 x1 1 − 2 x1 − e c x1 4 x1 + 1 = 0, c c which can be written as  d4  c4  c2 2d24 d4 d x −1 c (9) e 4 1 c4 c 1 − 2 x1 4 − x1 − 1 = 0. c c We note that this last equation in the particular case c = 1 and c4 = −d4 , assumes the form (d4 x1 + 1)(e−2d4 x1 − 1) = 0. Which does not vanish in R+ = (0, ∞) because x1 = −1/d4 < 0 with d4 > 0 and the other is x1 = 0. So there is not limit cycle for the system (8) when c = 1 and c4 = −d4 . We write the equation in (9) as (10) a0 f0 (x1 ) + a1 f1 (x1 ) + a2 f2 (x1 ) = 0, where     c2 d 2 4 d4 x1 d4 − 1c c4 c2 f0 (x1 ) = 1, f1 (x1 ) = x1 , f2 (x1 ) = e c4 1− 2 x1 4 , c and a0 = −1, a1 = −d4 /c, a2 = 1. The functions f0 , f1 and f2 form an extended Chebyshev system on R+ because the Wronskians of these functions are W (f0 )(x1 ) = 1, W (f0 , f1 )(x1 ) = 1, and 2 2   c4  c c2d4  d4 x1 d4 − 1c d24 (cd4 − c4 )x1 1 − c2 x1 4 2 2c + (cd4 − c4 )x1 e c4 W (f0 , f1 , f2 )(x1 ) = 2 . c6 1 − cc42 x1 which does not vanish in R+ = (0, ∞) because from the three zeros of this last 2 Wronskian two are negative (namely c2 /c4 and − cd2c 4 −c4 ) and the other is the 0. In view of Theorem 2 the function (10) has at most two zeros and so the piecewise differential system has at most two limit cycles in this case. The upper bound provided by the theorem is proved in this case. Note that from Theorem 2 we cannot say that that the equation (9) has values of the parameters c, c4 and d4 for which it has exactly two zeros, because the 6 M.E. ANACLETO, J. LLIBRE, C. VALLS AND C. VIDAL coefficients a0 , a1 and a2 are not independent. Moreover we also do not know if the possible zeros of the equation (9) are positive. Case 2: c4 6= 0 and c4 d2 − c2 d4 = 0. We write this condition as c4 6= 0 and d2 = c2 d4 /c4 . In this case, system (5) becomes c2 d4 ẋ = c2 y + c4 xy, ẏ = y + d4 xy + x, in R1 , c4 (11) c2 d4 ẋ = c2 y, ẏ = x + y, in R2 , c4 ẋ = −y, ẏ = x, in R3 . Taking into account that c2 6= 0, the quadratic system in (11) has a unique equilib- rium E0 = (0, 0). So, the trace and the determinant associated to the linear part of the quadratic system in (11) at E0 are c2 d4 /c4 and −c2 , respectively. In order that E0 can be a weak focus or a center, we must have that d4 = 0 and c2 = −c2 < 0 with c > 0. Now system (11) is written as ẋ = −c2 y + c4 xy, ẏ = x, in R1 , 2 ẋ = −c y, ẏ = x, in R2 , ẋ = −y, ẏ = x, in R3 , with first integrals − c22 y2 x H1 (x, y) = c2 − c4 x c4 e 2 − c4 in R1 , (12) 2 H2 (x, y) = x + c y 2 2 in R2 , 2 2 H3 (x, y) = x + y in R3 . The existence of the first integral H1 (x, y) defined in the point (0, 0) forces that the equilibrium (0, 0) of the quadratic system in (11) is a center. Assume that this continuous piecewise differential system has some crossing limit cycle with the points intersecting Σ being (x1 , 0), (0, y1 ) and (x2 , 0) with x1 > 0, y1 > 0, x2 < 0. Then the first integrals given in (12) must satisfy system (8), or equivalently, x  c4 − c22 y12 − 1 c4 e1 = e c4 1 − 2 x1 − e 2 = 0, c e2 = x22 − c2 y12 = 0, e3 = x22 − x21 = 0. x1 From equations e2 = 0 and e3 = 0, we get x2 = −x1 and y1 = . Introducing x2 c and y1 in e1 = 0 we obtain x1  c4 − c22 x21 e− c4 1 − 2 x1 c 4 − e 2c2 = 0. c which can be written as c4 c4 (13) e− c2 (c4 x1 +1)x1 + 2 x1 − 1 = 0. c Note that equation (13) can be written as (10), where c4 f0 (x1 ) = 1, f1 (x1 ) = x1 , f2 (x1 ) = e− c2 (c4 x1 +1)x1 , LIMIT CYCLES CREATED BY A QUADRATIC CENTER 7 and a0 = −1, a1 = c4 /c2 , a2 = 1. The functions f0 , f1 and f2 form an extended Chebyshev system on R+ because the Wronskians of these functions are W (f0 )(x1 ) = 1, W (f0 , f1 )(x1 ) = 1, and  c4 c34 x1 c4 x1 + 2c2 e− c2 (c4 x1 +1)x1 W (f0 , f1 , f2 )(x1 ) = , c8 2 which does not vanish in R+ because its two zeros are one negative (namely − 2c c4 ) and the other is the 0. In view of Theorem 2 the function (13) has at most two zeros. So the piecewise differential system has at most two limit cycles in this case. Hence the upper bound provided by the theorem is proved in this case. Case 3: c4 = 0 and −c2 d4 6= 0. System (5) becomes ẋ = c2 y, ẏ = x + d2 y + d4 xy, in R1 , (14) ẋ = c2 y, ẏ = x + d2 y, in R2 , ẋ = −y, ẏ = x, in R3 . The quadratic system in (14) has a unique equilibrium point E0 = (0, 0). The trace and the determinant associated to the linear part of the quadratic system in (14) are d2 and −c2 , respectively. The point E0 is either a weak focus or a center if and only if d2 = 0 and c2 = −c2 with c > 0. So taking taking d2 = 0 and c2 = −c2 , system (14) is equivalently to ẋ = −c2 y, ẏ = x + d4 xy in R1 , 2 (15) ẋ = −c y, ẏ = x in R2 , ẋ = −y, ẏ = x in R3 , with first integrals 1 2 − y x + 2c d2 H1 (x, y) = (d4 y + 1) 4 e d4 2 in R1 , 2 2 2 H2 (x, y) = x + c y in R2 , 2 2 H3 (x, y) = x + y in R3 . The existence of the first integral H1 (x, y) defined in the point (0, 0) forces that the equilibrium (0, 0) of the quadratic system in (14) is a center. Now repeating the same steps as the ones in the proof of Case 2, we get d4 d x1 (1− 2c4 x1 ) d4 (16) e1 = e c − x1 − 1 = 0, c where d4 d x1 (1− 2c4 x1 ) f0 (x1 ) = 1, f2 (x1 ) = x1 , f3 (x1 ) = e c , and a0 = −1, a1 = −d4 /c, a2 = 1. The Wronskians of these functions are W (f0 )(x1 ) = 1, W (f0 , f1 )(x1 ) = 1 8 M.E. ANACLETO, J. LLIBRE, C. VALLS AND C. VIDAL and d4 d 4 x1 d3 x1 (d4 x1 − 2c)e c (1− 2c )x1 W (f0 , f1 , f2 )(x1 ) = 4 . c4 which does not vanish in R+ because has two solutions, namely d2c4 and 0. So, by Theorem 2 the function (16) has at most two zeros and we conclude that (15) has at most two limit cycles which proves the upper bound in the theorem in this case. The example. The planar continuous piecewise differential system separated by Σ and formed by the quadratic center and the two linear centers 12 ẋ = −36y + 2xy, ẏ = x + xy, in R1 100 (17) ẋ = −36y, ẏ = x, in R2 ẋ = −y, ẏ = x, in R3 with the first integrals e 25 (2y− 25 x) 3 3 H1 = 2 in R1 3 (36 − 2x)162/625 25 y +1 H2 = x2 + 36y 2 in R2 2 2 H3 = x + y in R3 has one crossing limit cycle. Indeed, for this differential system equation (9) is 6172 e−9x1 /625 ex1 /25 − 15625 x1 = 0. (36 − 2x1 )162/625 ( 50 + 1)2 This equation has the approximated solution x1 = 0.0000583439.. and then system (17) has a unique solution (x1 , y1 , x2 ) = (0.0000583439.., 0.00000972399.., −0.0000583439..), which provides the limit cycle of Figure 1. This limit cycle is a crossing limit cycle which is traveled in counterclockwise sense. Acknowledgements The second author is supported by the Agencia Estatal de Investigación grants PID2019-104658GB-I00, and the H2020 European Research Council grant MSCA- RISE-2017-777911. 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Springer, New York, 2012. 1 Departamento de Matemática, Universidad del Bı́o-Bı́o, Concepción, Avda. Collao 1202, Chile E-mail address: manacleto@ubiobio.cl, clvidal@ubiobio.cl 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat 3 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049-001, Lisboa, Portugal E-mail address: cvalls@math.ist.utl.pt