This is a preprint of: “Non-existence and uniqueness of limit cycles for a class of three-dimensional piecewise linear differential systems”, Ting Chen, Lihong Huang, Jaume Llibre, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 33(6), 2350075 (13 pages), 2023. DOI: [10.1142/S021812742350075X] NON-EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF 3-DIMENSIONAL PIECEWISE LINEAR DIFFERENTIAL SYSTEMS TING CHEN1 , LIHONG HUANG1,2 AND JAUME LLIBRE3 Abstract. In this paper we study the non-symmetric limit cycles for a family of 3-dimensional piecewise linear differential systems with three zeros separated by two parallel planes. For a class of these differential systems we study the non-existence, existence and uniqueness of their limit cycles. 1. Introduction and statement of the main results As we know, the continuous and discontinuous piecewise smooth differential systems play an important role inside many disciplines, such as control theory, electrical engineering, mechanics, biology and economics, see for instance the papers [1, 5, 12, 13, 15] and the references quoted there. The maximum number of limit cycles, i.e. the periodic orbits isolated in the set of all periodic orbits, for differential systems is the second part of Hilbert’s 16th problem. In the last decades there have been extensively studied on the limit cycles of continuous and discontinuous piecewise differential systems in R2 , see [2, 3, 4, 6, 8, 9, 10, 11, 14, 16]. Recently the authors of [17, 18, 19, 21] considered the limit cycles in several cases of the 3-dimensional (3D) piecewise linear differential systems. Freire et al. [7] considered the birth of limit cycles in 3D piecewise linear systems for the relevant case of symmetrical oscillators with the three zones separated by two planes. Llibre et al. [20] studied a one-parameter family of symmetric 3D piecewise linear differential systems, and proved that it has at most 2 limit cycles. But there are no results about the non-symmetric limit cycles in 3D piecewise linear differential systems with three zones. Consider the piecewise linear differential systems ẋ = Ax in P+ ∪ R, (1) ẋ = Bx in P− ∪ C ∪ P+ , ẋ = Cx in L ∪ P− , 2010 Mathematics Subject Classification. Primary: 34C25, 37G15. Key words and phrases. Limit cycles, periodic orbits, 3-dimensional, piecewise linear differen- tial systems. 1 2 TING CHEN, LIHONG HUANG AND JAUME LLIBRE in R3 , so that we have x(t) = (x(t), y(t), z(t)) ∈ R3 . And systems (1) are divided into three zones L, C and R separated by the two parallel planes P± , where L = {(x, y, z) ∈ R3 : x < −1}, C = {(x, y, z) ∈ R3 : −1 < x < 1}, R = {(x, y, z) ∈ R3 : x > 1}, P± = {(x, y, z) ∈ R3 : x = ±1}. In this paper we consider a three–parametric family of piecewise linear differential systems (1), with one continuous parameter α ∈ R and two discrete parameters r, s ∈ Z. More precisely, we consider the three–parameter family of systems (1) with     −6 −1 0 0 1 0 (2) A =  11 0 −1  , B= 0 0 0  −6 0 0 0 0 0 and   (1 + r + s)α −1 0 (3) C =  (r + s + rs)α2 0 −1  . rsα3 0 0 Therefore the eigenvalues of A are −1, −2 and −3, while the eigenvalues of C are α, rα and sα. The equilibrium points of systems (1)–(3) are the strip {(x, 0, z)|x ∈ [−1, 1]} if rsα 6= 0, and the strip {(x, 0, z)|x ∈ [−1, 1]} union the half plane {(x, (1 + r + s)αx, (r + s + rs)α2 x)|x ∈ (−∞, −1]} if rsα = 0. The differential system ẋ = Bx has no limit cycles, because it is linear in its domain of definition P− ∪ C ∪ P+ and has no pure imaginary eigenvalues. Hence one limit cycle must intersect the three zones L, C and R, and has at least two points on the plane P+ . Our main result is to study the non-symmetric limit cycles in the class of piece- wise linear systems (1)–(3) by the algebraic approach, which is based on the closing equations for periodic orbits, for more details see [1]. In fact we determine for every value of α ∈ R the non-existence and the uniqueness of limit cycles having two points in the plane P+ , as studied in the following results. Theorem 1. For r = 2 and s = 3 the piecewise linear differential systems (1) satisfying (2) and (3) have no limit cycles for all α ∈ R. Theorem 2. For r = 2 and s = −3 the following statements hold for the piecewise linear differential systems (1) satisfying (2) and (3): (a) If α < αN1 ≈ 2.936 systems (1) have no limit cycles having exactly two points in the plane P+ . (b) If αN1 < α < αN2 = 3 systems (1) have one non-symmetric limit cycle having exactly two points in the plane P+ . (c) If α > αN2 systems (1) have no limit cycles having exactly two points in the plane P+ . The above results are proved in Section 3, where algebraic expressions of deter- mining the non-existence and uniqueness of the limit cycles are provided. From The- orem 2 we obtain that the non-symmetric limit cycles are organized in one branch NON-EXISTENCE AND UNIQUENESS OF LIMIT CYCLES 3 which exist for different ranges of parameter α. However, this three–parametric family of piecewise linear differential systems (1) is not yet complete. We conjec- ture that in this bifurcation there also appear limit cycles for other values of r and s, as the following. Conjecture For all α ∈ R and (r − 1)(s − 1)(r − s) 6= 0 for all r, s ∈ Z such that the non-symmetric piecewise linear differential systems (1)–(3) satisfying α = 0 or rs ≥ 0 have no limit cycles, and satisfying rs < 0 have a unique non-symmetric limit cycle. We note that if we fix the integers r and s then we can prove this conjecture. For instance in the above theorems we prove the conjecture for some cases r = 2 and s = 3, or r = 2 and s = −3. The arguments are given in that proof can be used for proving the conjecture for other values of r and s, but of course the computations increase when r and s increase. On the other hand, we can prove that systems (1)–(3) satisfying (r − 1)(s − 1)(r − s) = 0 have no limit cycles for all α ∈ R. Since solving the polynomial equations based on the closing equations becomes extremely complex in this case, we do not provide it here. 2. Preliminaries In this section we provide the algebraic procedure which helps us to determine the limit cycles for the 3D piecewise linear differential systems (1)–(3). The solution (x(t), y(t), z(t)) of systems (1) satisfying (2) in the region P+ ∪ R starting at the point (1, y0 , z0 ) when t = 0 is (4) (x(t), y(t), z(t))T = eAt (1, y0 , z0 )T , where   1 − 8u + 9u2 (u − 1)(−1 + 3u) (u − 1)2 u (5) eAt = (1 − u)(27u − 5) −5 + 16u − 9u2 (1 − u)(3u − 5)  , 2 6(u − 1)(3u − 1) 6(u − 1)2 2(3 − 3u + u2 ) and u = e−t . Note that for all positive values of t we will have 0 < u < 1. Regarding the solution (x(τ ), y(τ ), z(τ )) of systems (1) satisfying (2) in the region P− ∪ C ∪ P+ starting at the point (1, y0 , z0 ) when τ = 0, and it satisfies (6) (x(τ ), y(τ ), z(τ ))T = eBτ (1, y0 , z0 )T . In other wards, any points (1, y0 , z0 ) of system (1) in P+ will map to the point (−1, y0 , z0 ) in P− . While the solution (X(T ), Y (T ), Z(T )) of systems (1) satisfying (3) in the region P− ∪ C starting at the point (−1, Y0 , Z0 ) when T = 0 satisfies (7) (X(T ), Y (T ), Z(T ))T = eCT (−1, Y0 , Z0 )T . Assuming that there exists one of the periodic orbits for systems (1)–(3) having two points in the plane P+ . Let (1, y0 , z0 ) ∈ P+ be the point where this periodic orbit exists the zone C enter the zone P+ ∪ R and let (1, Y0 , Z0 ) ∈ P+ be the point where this periodic orbit enters into the zone C. Then this periodic orbit will enter into the zone L ∪ P− through the point (−1, Y0 , Z0 ) ∈ P− . Let t be the time for this periodic orbit in going from the point (1, y0 , z0 ) to the point (1, Y0 , Z0 ), and 4 TING CHEN, LIHONG HUANG AND JAUME LLIBRE y−z (−1, Y (T ), Z(T )) (1, y0 , z0 ) 0 x (−1, Y0 , Z0 ) (1, y(t), z(t)) x = −1 x=1 Figure 1. The periodic orbits passing through the points (1, y0 , z0 ) and (1, Y0 , Z0 ) in the plant P+ . let T be the elapsed time for this periodic orbit to go from the point (−1, Y0 , Z0 ) to the point (−1, y0 , z0 ), see Figure 1. Hence we have the closing equations (8) (x(t), y(t), z(t)) = (1, Y0 , Z0 ), (X(T ), Y (T ), Z(T )) = (−1, y0 , z0 ). The specific case α = 0 of systems (1)–(3) will be considered with the above way. Now we assume α 6= 0 and we can integrate backwards in time the solution from the point (−1, y0 , z0 ) to (−1, Y0 , Z0 ) within L ∪ P− , by defining (9) (X(−T ), Y (−T ), Z(−T ))T = e−CT (−1, y0 , z0 )T . We have that the exponential e−CT is the matrix   b11 b12 b13 1  b21 (10) b22 b23  , (r − 1)(r − s)(s − 1)α2 b31 b32 b33 where b11 =((r − s)v + r2 v r (s − 1) + (1 − r)s2 v s )α2 , b12 = − ((r − s)v + (s − 1)rv r + (1 − r)sv s )α, b13 =(r − s)v + (s − 1)v r + (1 − r)v s , b21 =((r2 − s2 )v + (s2 − 1)r2 v r + (1 − r2 )s2 v s )α3 , (11) b22 =((s2 − r2 )v + (1 − s2 )rv r + (r2 − 1)sv s )α2 , b23 = − ((s2 − r2 )v + (1 − s2 )v r + (r2 − 1)v s )α, b31 =((r − s)v + (s − 1)rv r + (1 − r)sv s )rsα4 , b32 = − ((r − s)v + (s − 1)v r + (1 − r)v s )rsα3 , b33 = − ((s − r)rsv + (1 − s)sv r + (r − 1)rv s )α2 , and v = e−αT . Hence we must have (X(−T ), Y (−T ), Z(−T )) = (−1, Y0 , Z0 ). NON-EXISTENCE AND UNIQUENESS OF LIMIT CYCLES 5 y−z (−1, y0 , z0 ) (1, y0 , z0 ) 0 x (−1, Y (−T ), Z(−T )) (1, y(t), z(t)) x = −1 x=1 Figure 2. The periodic orbits passing through the points (1, y0 , z0 ) and (1, Y0 , Z0 ) in the plane P+ . Therefore the periodic orbits that we are looking for must satisfy the following four equations (12) x(t) − 1 = 0, X(−T ) + 1 = 0, y(t) − Y (−T ) = 0, z(t) − Z(−T ) = 0, see Figure 2. Then we change the variables (t, T ) by the variables (u, v), through t = − ln u > 0 and T = −(ln v)/α > 0, the four equations (12) become (13) ei (y0 , z0 , u, v) = 0 for i = 1, 2, 3, 4. For every periodic orbits having two points (1, y0 , z0 ) and (1, Y0 , Z0 ) in the plane P+ we can associate one solution (y0 , z0 , u, v) of equations (13) with 0 < u < 1 and v > 1 (0 < v < 1) if α < 0 (α > 0). Also since ẋ = −6 − y when x = 1, the two conditions −6 − y0 > 0 and −6 − Y0 < 0 must be fulfilled. In fact the parameterization x(u) is a cubic polynomial in its parameter, then the cubic equation x(u) = 1 can not have four roots with y0 and z0 given. Thus these periodic orbits can not have at least other two points in P+ , i.e. they have two unique points in P+ . Similarly to Proposition 2 of [20] we give the following result to establish an equivalence relation between solutions of equations (13) and periodic orbits of systems (1)–(3) with two points in P+ . Proposition 3. Let (y0 , z0 , u, v) be one solution of corresponding equations (13) and let Y0 = y(t), Z0 = z(t) computed from (4). If α < 0 (α > 0) and the four conditions −6 − y0 > 0,−6 − Y0 < 0, 0 < u < 1 and v > 1 (0 < v < 1) are fulfilled, then systems (1)–(3) have a periodic orbit passing through the points (1, y0 , z0 ) and (1, Y0 , Z0 ) of P+ . 3. The proof of Theorems 1 and 2 Now we study the non-existence, existence and uniqueness of limit cycles for the piecewise linear systems (1)–(3) in the cases α = 0 and α(r − 1)(s − 1)(r − s) 6= 0, as the following subsections. 3.1. The case α = 0. For the solution (x(t), y(t), z(t)) of systems (1)–(2) in the region P+ ∪ R passing through the point (1, y0 , z0 ) with y0 < −6, we have that ẋ = −6 − y0 > 0. Then we remain for a time t in the right zone until arriving 6 TING CHEN, LIHONG HUANG AND JAUME LLIBRE at (1, Y0 , Z0 ), where from (4) we have (1, Y0 , Z0 )T = eAt (1, y0 , z0 )T . The solu- tion (X(T ), Y (T ), Z(T )) of systems (1) in the region L ∪ P− starting at the point (−1, Y0 , Z0 ) is      2   X(T ) −1 1 −T T2 −1 (14)  Y (T )  = eC0 T  Y0  =  0 1 −T   Y0  . Z(T ) Z0 0 0 1 Z0 Using the flow in the left zone and starting from (−1, Y0 , Z0 ), after a time T we have a periodic orbit if the equations (15) (1, Y0 , Z0 )T = eAt (1, y0 , z0 )T , (−1, y0 , z0 )T = eC0 T (−1, Y0 , Z0 )T hold. After eliminating Y0 = y0 + z0 T and Z0 = z0 , by using the last two equations of (15), we obtain the algebraic equations (16) − 2y0 − T z0 = 0, 2 + u + 9u2 + (−u + 3u2 )y0 + (−u + u2 )z0 = 0, − 3u + 9u2 + (−3u + 3u2 )y0 + (1 − 2u + u2 )z0 = 0, − 5u + 32u2 − 27u3 − (2 + 5u − 16u2 + 9u3 )y0 − (2T + 5u − 8u2 + 3u3 )z0 = 0, where u = e−t . From the first equation of (16) we have T = −2y0 /z0 . Then we observe that the second and third equations of (16) are linear in the variables (y0 , z0 ) and do not depend on T . And the determinant of this linear system is −(u − 1)2 u 6= 0, so it has one solution y0 = (2 − u + 5u2 )/((1 − u)u) and z0 = 6(1 + u2 )/(u − 1)2 . Therefore substituting T , y0 and z0 in the forth equations of (16) we have 4(u − 1)2 /u = 0, but this is in contradiction with 0 < u < 1. Hence we get no solutions from the algebraic equations (16). This proves the parts of Theorems 1 and 2 when α = 0 and the conjecture when α = 0. 3.2. The case α(r − 1)(s − 1)(r − s) 6= 0. Equivalently, we can get a dual approach to write the closing equations. We define the solution of systems (1)–(3) in the region P+ ∪ R backwards in time starting from the point (1, Y0 , Z0 ) when t = 0 as (17) (x(t), y(t), z(t))T = e−At (1, Y0 , Z0 )T , where the matrix e−At has the same expression that eAt in (5) if we change u by U = 1/u = et . Then we can take ȳ(t) = y0 and z̄(t) = z0 to write the closing equations in the four unknowns (Y0 , Z0 , U, V ) (18) x̄(t) − 1 = 0, X(T ) + 1 = 0, y(t) − Y (T ) = 0, z(t) − Z(T ) = 0, where (X(T ), Y (T ), Z(T ))T = eCT (1, Y0 , Z0 )T . The matrix eCT comes from e−CT in (10) by V = 1/v = eαT . We will have U > 1 for all t > 0 and 0 < V < 1 (V > 1) for α < 0 (α > 0) and T > 0. Remark 4. We can obtain four equations of the form ei (Y0 , Z0 , U, V ) = 0, which are exactly the same that in (13) if we take U = 1/u = et and V = 1/v = eT , from equations (18). Thus, if for α(r − 1)(s − 1)(r − s) 6= 0 there exists a periodic orbit with two points (1, y0 , z0 ) and (1, Y0 , Z0 ) at P+ , then both (y0 , z0 , u, v) and (Y0 , Z0 , U, V ) are solutions of equations (13). From Remark 4, note that if (u, v) are both different from zero and satisfy the polynomial systems (13) for a given (r, s, α), the same is true for the pair (1/u, 1/v). NON-EXISTENCE AND UNIQUENESS OF LIMIT CYCLES 7 By doing the changes u → 1/U and v → 1/V , we can get Y0 and Z0 from y0 and z0 , respectively. Now we compute the four equations ei (y0 , z0 , u, v) (i = 1, 2, 3, 4) from (12). By removing the factor (u − 1)/2 in the equation e1 = 0, we obtain the new first equation ee1 = 0, where (19) ee1 = 2 + u + 9u2 + (−u + 3u2 )y0 + (−u + u2 )z0 . The second equation, after multiplying it by α2 (r − 1)(s − 1)(r − s) becomes ee2 = 0, where ee2 =(r − s + r2 (s − 1) − (r − 1)s2 + (−r + s)v − r2 (s − 1)v r (20) + (r − 1)s2 v s )α2 + ((−r + s)v − r(s − 1)v r + (r − 1)sv s )y0 α + ((r − s)v + (s − 1)v r + (1 − r)v s )z0 . To obtain a simplified third equation we define the polynomial ee3 = 3(r − s)α(e3 + (13 − 9u)αe e2 /((−1 + r)(r − s)(−1 + s)α)) that the equation becomes e1 /6 + 2(r + s)e ee3 = 0, where ee3 = − α(13(r − s) + 5(r − s)u + 6(r − s)u2 + 3(−r2 + s2 )α (21) + 3(r2 v r − s2 v s )α) + ((s − r)u − 3rv r + 3sv s )y0 α + ((−r + s)uα + (r − s)u2 α + 3v r − 3v s )z0 . Finally to simplify the fourth equation we have ee4 =(r − s)(3e4 + (5 − 3u)e e2 /((r − 1)(r − s)(s − 1))) e1 + 3rse =10(r − s) + 8(r − s)u + 6(r − s)u2 + (3r(r − s)s − 3rs(rv r − sv s ))α2 (22) + (4(r − s)u − 3rs(v r − v s )α)y0 + (4(r − s)u + (−r + s)u2 + 3sv r − 3rv s )z0 . We obtain that the above equations eei (i = 1, 2, 3, 4) are linear in y0 and z0 . Furthermore ee1 is a quadratic polynomial with respect to u and independent on v, ee2 is the polynomial with respect to v and independent on u, while ee3 and ee4 are both the polynomials with respect to u and v without mixed terms of the form um v n . From the two equations ee1 = ee2 = 0 we have F1 F2 (23) y0 = , z0 = , F3 F3 where Fi (i = 1, 2, 3) are polynomials in the variables u and v. Substituting y0 and z0 into ee3 = ee4 = 0 we obtain the polynomial expressions of Ej (u, v) for j = 3, 4 in function of the variables u and v. Since we do not directly these expression in the proof, we do not list them here. Note that since we are assuming that α(r − 1)(s − 1)(r − s) 6= 0, u = e−t and v = e−αT with t > 0 and T > 0, we have u ∈ (0, 1) and v ∈ (0, 1) ∪ (1, +∞). We claim that the polynomial system E3 (u, v) = E4 (u, v) = 0 in the open square W = {(u, v) : u ∈ (0, 1), v ∈ (0, 1) ∪ (1, +∞)} with F3 (u, v) 6= 0 has at most one solution satisfying Proposition 3. If our claim holds than the conjecture will have a positive answer. 8 TING CHEN, LIHONG HUANG AND JAUME LLIBRE We have proved this claim for the following values of (r, s) = (2, 3), (2, 4), (2, 10), (3, 4), (2, −3), (2, −4), (3, −2). Here we provide all the details for the case (r, s) = (2, 3) and (r, s) = (2, −3), all the other cases follow in a similar way. The proof of Theorem 1: The polynomials E3 (u, v) and E4 (u, v) for (r, s) = (2, 3) are E3 =(1 − v)v(−5u + 11u2 + 6u4 + 2v 2 + uv 2 + 9u2 v 2 ) − u(u − v)v(12 + 3u − 5u2 + 3v − 14uv + 15u2 v − 5v 2 + 15uv 2 )α + (u − 1)u(2u2 − 5v + u2 v + 11v 2 + 9u2 v 2 + 6v 4 )α2 , E4 =6(u − 1)u(1 + u2 )(v − 1)v + v(6u + 11u3 − 5u4 − 18uv − 33u3 v + 15u4 v + 6v 2 + 3uv 2 + 27u2 v 2 − 2v 3 − uv 3 − 9u2 v 3 )α − u(−6u2 + 2u3 − 6v + 18uv − 3u2 v + u3 v − 27u2 v 2 + 9u3 v 2 − 11v 3 + 33uv 3 + 5v 4 − 15uv 4 )α2 + 6(u − 1)u(v − 1)v(1 + v 2 )α3 , and the numerators and denominator of y0 and z0 in this case are F1 =(2 + u + 9u2 )(1 − v)v − (u − 1)u(2 + v + 9v 2 )α2 , F2 = − α(2 + u + 9u2 )v(−1 + 3v) + u(−1 + 3u)(2 + v + 9v 2 )α2 , F3 =uv((−1 + 3u)(v − 1) + (u − 1)(−1 + 3v)α). By computing the resultant polynomial of the elimination of α from the equations E3 = 0 and E4 = 0, we get a long expression in the form 4(u − 1)u2 (u − v)(v − 1)v 2 p(u, v)q(u, v), where p(u, v) is p(u, v) = − (1 − 3v)2 v + u(−1 + v − 7v 2 + 9v 3 ) + 9u3 (−1 + v − 7v 2 + 9v 3 ) + u2 (6 + v(−7 + 48v − 63v 2 )), and q(u, v) is given in Appendix 1. If u = v we have E3 =(v − 1)v 2 (−1 + 3v)(5 + 2v + 5v 2 )(−1 + α)(1 + α), E4 =6(v − 1)2 v 2 (1 + v 2 )(1 + α)(1 + α2 ). The equations E3 = E4 = 0 have only one common solution α = −1, but the denominator F3 = 0 when u = v and α = −1. Then we need to analyze the solutions of p(u, v) and q(u, v) with (u, v) in the open rectangle W . Proposition 5. The number of solutions of the equations p(u, v) = 0 or q(u, v) = 0 in W 0 = {(u, v) : u ∈ (0, 1), v ∈ (0, +∞)} appears in Table 1. Proof. The following computations are based in the root determination of polyno- mials in only one variable. We can select any value of v in (0, +∞) and get the roots u ∈ (0, 1) of the corresponding polynomials p(u, v) and q(u, v). It is easy get that the roots of these polynomials are continuous complex functions. By varying v, the number of real roots of the polynomials p(u, v) and q(u, v) with u ∈ (0, 1) can change only at values of v where the curve p(u, v) = 0 or q(u, v) = 0 has a hor- izontal tangent, a branching point, or one solution escapes through the boundary of the rectangle W 0 . NON-EXISTENCE AND UNIQUENESS OF LIMIT CYCLES 9 2.0 1.5 v 1.0 p1 • 0.5 p0 p3 • • 0.0 • p2 0.0 0.2 0.4 0.6 0.8 1.0 u Figure 3. The zero level set of p(u, v)q(u, v) = 0 in the rectangle W 0 . Points p0 , p1 , p2 are in the boundary of the open rectangle. Table 1. Number of solutions of p(u, v)q(u, v) = 0 in the open rectangle W 0 . Range Solutions of p(u, v) Solutions of q(u, v) Periodic orbits 0 < v < 1/3 0 2 0 v = 1/3 2 2 0 1/3 < v < 1 0 2 0 v=1 1 1 0 1 1, which is in contradiction with v > 1 (0 < v < 1) if α < 0 (α > 0). Then we discard all branches for counting periodic solutions, i.e. systems (1)–(3) has no limit cycles when r = 2 and s = 3. Therefore we have concluded the proof of Theorem 1.  NON-EXISTENCE AND UNIQUENESS OF LIMIT CYCLES 11 e3 (u, v) and E The proof of Theorem 2: The polynomials E e4 (u, v) for (r, s) = (2, −3) are e3 =(1 − v)(−5u + 11u2 + 6u4 + 8v − 6uv + 58u2 v + 12u4 v + 6v 2 − 12uv 2 E + 60u2 v 2 + 18u4 v 2 + 4v 3 − 18uv 3 + 62u2 v 3 + 24u4 v 3 + 2v 4 + uv 4 + 9u2 v 4 ) − u(−18 + 18u − 9u2 + 15u3 − 10v − 6uv − 9u2 v + 15u3 v − 15v 2 + 9uv 2 − 9u2 v 2 + 15u3 v 2 − 15v 3 + 9uv 3 − 9u2 v 3 + 15u3 v 3 − 45v 4 + 39uv 4 − 24u2 v 4 + 40u3 v 4 + 3v 5 − 9uv 5 ) + (u − 1)u(v − 1)(−9 + 9u2 − 30v + 18u2 v − 21v 2 + 27u2 v 2 − 12v 3 + 16u2 v 3 + 2v 4 )α2 , e4 =6(u − 1)u(1 + u2 )(v − 1)(1 + 2v + 3v 2 + 4v 3 ) + (−18u − 33u3 + 15u4 + 16v E − 10uv + 72u2 v − 33u3 v + 15u4 v + 6v 2 − 15uv 2 + 27u2 v 2 − 33u3 v 2 + 15u4 v 2 + 6v 3 − 15uv 3 + 27u2 v 3 − 33u3 v 3 + 15u4 v 3 + 6v 4 − 45uv 4 + 27u2 v 4 − 88u3 v 4 + 40u4 v 4 + 6v 5 + 3uv 5 + 27u2 v 5 )α + u(1 − v)(2 − 6u − 27u2 + 9u3 − 12v + 36uv − 54u2 v + 18u3 v − 21v 2 + 63uv 2 − 81u2 v 2 + 27u3 v 2 − 30v 3 + 90uv 3 − 48u2 v 3 + 16u3 v 3 − 9v 4 + 27uv 4 )α2 + 6(1 − u)u(v − 1)(1 + 6v + 6v 2 + 6v 3 + v 4 )α3 , and the numerators and denominator of y0 and z0 in this case are F1 =(1 − v)((2 + u + 9u2 )(1 + 2v + 3v 2 + 4v 3 ) + (u − 1)u(9 + 18v + 27v 2 + 16v 3 )α2 ), F2 = − α(2 + u + 9u2 )(3 + 3v + 3v 2 + 3v 3 + 8v 4 ) + u(3u − 1)(v − 1)(9 + 18v + 27v 2 + 16v 3 )α2 , F3 =u(3u − 1)(v − 1)(1 + 2v + 3v 2 + 4v 3 ) + (u − 1)u(3 + 3v + 3v 2 + 3v 3 + 8v 4 )α. The calculation of the resultant of the polynomials E e3 and Ee4 with respect to the parameter α yields 4(u − 1)u2 (−1 + v)2 P (u, v)Q(u, v) where P (u, v) is (27) P (u, v) = − (3 + 3v + 3v 2 + 3v 3 + 8v 4 )2 + (9 + 18v + 27v 2 + 26v 3 + 15v 4 + 6v 5 − 3v 6 + 38v 7 + 64v 8 )u + (−63 − 126v − 189v 2 − 192v 3 − 165v 4 − 102v 5 − 39v 6 − 276v 7 − 448v 8 )u2 + 9(9 + 18v + 27v 2 + 26v 3 + 15v 4 + 6v 5 − 3v 6 + 38v 7 + 64v 8 )u3 and Q(u, v) is given in Appendix 1. Similarly to the proof of Theorem 1 we consider the solutions of P (u, v) and Q(u, v) with (u, v) in the open rectangle W. Proposition 6. The number of solutions of the equation P (u, v) = 0 or Q(u, v) = 0 in W 0 = {(u, v) : u ∈ (0, 1), v ∈ (0, +∞)} is as displayed in Table 2. If r = 2 and s = −3 the number of limit cycles of system (1)–(3) with exactly 2 points in the plane P+ is as indicated in last column of Table 2. 12 TING CHEN, LIHONG HUANG AND JAUME LLIBRE 2.0 1.5 P2 1.0 • P1 • 0.5 P5 • 0.0 • P0 P3 • P4 • 0.0 0.2 0.4 0.6 0.8 1.0 Figure 4. The zero level set of P (u, v)Q(u, v) = 0 in the rectangle W 0 . Table 2. Number of solutions of P (u, v)Q(u, v) = 0 in the open rectangle W . Range Solutions of P (u, v) Solutions of Q(u, v) Periodic orbits 0 < v ≤ v5 1 3 1 v5 < v < 1 1 3 0 v=1 1 2 0 1 1 (0 < v < 1) if α < 0 (α > 0). To prove the second statement of Proposition 6 we need to show the inequalities −6 − y0 > 0. We start by looking for possible values of (u, v) in the branch from P0 to P1 , where such inequality is not true any longer, that is H(u, v, α) = − (2 + 9u)(v − 1)(1 + 2v + 3v 2 + 4v 3 ) − 6u(3 + 3v + 3v 2 + 3v 3 (29) + 8v 4 )α + u(v − 1)(9 + 18v + 27v 2 + 16v 3 )α2 . 14 TING CHEN, LIHONG HUANG AND JAUME LLIBRE Computing the resultants of the polynomials E e3 and H, E e4 and H, respectively, then we have the following polynomials (30) e3 , H, α] = 2u2 (−1 + v)2 P (u, v)M1 (u, v), h1 (u, v) =Resultant[E e4 , H, α] = 2u2 (−1 + v)2 (1 + 2v + 3v 2 + 4v 3 )P (u, v)M2 (u, v), h2 (u, v) =Resultant[E where the expression of P (u, v) is as displayed in (27) and M1 (u, v) = − 2(v − 1)4 (3 + 4v + 3v 2 )2 + u(414 + 4824v + 13824v 2 + 21748v 3 + 22685v 4 + 14616v 5 + 6106v 6 + 636v 7 − 153v 8 ) + 4u2 (72 + 432v + 1071v 2 + 1880v 3 + 2110v 4 + 1488v 5 + 623v 6 + 24v 7 ) − 6u3 (18 + 120v + 297v 2 + 520v 3 + 580v 4 + 392v 5 + 165v 6 + 8v 7 ) − 4u4 (1 + 2v + 3v 2 + 4v 3 )(9 + 18v + 27v 2 + 16v 3 ) + u5 (1 + 2v + 3v 2 + 4v 3 )(9 + 18v + 27v 2 + 16v 3 ), M2 (u, v) =4(v − 1)(−2 + 12v + 21v 2 + 30v 3 + 9v 4 )2 + 12u(−48 + 624v + 2070v 2 + 2690v 3 + 1789v 4 + 1143v 5 + 1476v 6 + 2460v 7 + 1845v 8 + 351v 9 ) + 3u2 (−3168 − 1440v + 19728v 2 + 31856v 3 + 27109v 4 − 2817v 5 − 5970v 6 + 762v 7 + 16209v 8 + 4131v 9 ) + 6u4 (−162 − 486v − 1377v 2 − 1515v 3 + 840v 4 + 5112v 5 + 6671v 6 + 4597v 7 + 720v 8 ) − 12u5 (v − 1)(9 + 18v + 27v 2 + 16v 3 )2 + u6 (v − 1)(9 + 18v + 27v 2 + 16v 3 )2 . It is easy to get that P (u, v) 6= 0 on the branch P0 -P1 , see Figure 4. By com- puting the resultant we eliminate v between polynomials M1 and M2 , then we obtain a polynomial in u whose roots in the interval W 0 are 0.00483442, 0.0163653, 0.0995178, 0.194119. Going back with these values to M1 = 0 and M2 = 0, we obtain the points (0.004834, 0.221817), (0.016365, 0.087319), (0.099517, −1.342034) and (0.194119, −0.860593). Clearly, only the second point P5 = (0.016365, 0.087319) is in W 0 being in fact on the branch P0 -P1 . The conclusion is that, by continuity, the sign of H(u, v, α) changes along the branch P0 -P1 . We select v = 0.05 and have the values of u ≈ 0.010455 and α ≈ 2.96596 for the corresponding point on the branch P0 -P1 , so that the value of −6 − y0 is approximately 8.55439. Next we select v = 0.1 and compute the values of u ≈ 0.018116 and α ≈ 2.943606 for the corresponding point, then we obtain the value of −6 − y0 ≈ −1.45632. Then we choose v = 0.3 and obtain the values of u ≈ 0.035656 and α ≈ 3.00209 for the corresponding point, and the value of −6 − y0 is approximately −8.26739. Hence we conclude from Proposition 3 that all the points of the branch P0 -P5 correspond to periodic orbits of systems (1)–(3).  We consider the branch from P0 to P5 . By continuity the value of α reaches a maximum positive value and a minimum positive value. These values correspond to αN2 and αN1 , respectively, which can be calculated by standard procedures with as much accuracy as possible. Hence the proof of Theorem 2 is complete.  NON-EXISTENCE AND UNIQUENESS OF LIMIT CYCLES 15 4. Appendix 1 The polynomials q(u, v): q(u, v) =2u7 (9 + u + 2u2 )2 + u5 (855 − 1626u + 1156u2 − 1592u3 + 225u4 − 178u5 + 8u6 )v + u3 (−2700 + 420u − 1986u2 + 1707u3 + 1510u4 + 1796u5 + 682u6 + 225u7 + 74u8 )v 2 + 2u2 (−1350 + 5565u − 4791u2 + 11352u3 − 10267u4 + 5941u5 − 7290u6 + 898u7 − 796u8 + 18u9 )v 3 + 2u2 (210 − 4791u − 43u2 − 6811u3 + 5518u4 − 862u5 + 5941u6 + 755u7 + 578u8 + 81u9 )v 4 − u(−855 + 1986u − 22704u2 + 13622u3 − 32762u4 + 31872u5 − 11036u6 + 20534u7 − 1707u8 + 1626u9 )v 5 + u(−1626 + 1707u − 20534u2 + 11036u3 − 31872u4 + 32762u5 − 13622u6 + 22704u7 − 1986u8 + 855u9 )v 6 + 2(81 + 578u + 755u2 + 5941u3 − 862u4 + 5518u5 − 6811u6 − 43u7 − 4791u8 + 210u9 )v 7 − 2(−18 + 796u − 898u2 + 7290u3 − 5941u4 + 10267u5 − 11352u6 + 4791u7 − 5565u8 + 1350u9 )v 8 + (74 + 225u + 682u2 + 1796u3 + 1510u4 + 1707u5 − 1986u6 + 420u7 − 2700u8 )v 9 + (8 − 178u + 225u2 − 1592u3 + 1156u4 − 1626u5 + 855u6 )v 10 + 2(2 + u + 9u2 )2 v 11 Q(u, v) = − 2(v − 1)v 3 (4 + 3v + 2v 2 + v 3 )(16 + 27v + 18v 2 + 9v 3 )2 − uv 2 (12352 + 73648v + 136805v 2 + 171365v 3 + 138730v 4 + 68846v 5 + 16704v 6 − 7980v 7 − 8490v 8 − 2430v 9 + 27v 10 + 423v 11 ) − u2 v(−16640 − 129584v − 332928v 2 − 578585v 3 − 731935v 4 − 677770v 5 − 458242v 6 − 206364v 7 − 36240v 8 + 16410v 9 + 17010v 10 + 12501v 11 + 2367v 12 ) − u3 v(12432 + 210790v + 812054v 2 + 1522535v 3 + 1975495v 4 + 1684186v 5 + 976910v 6 + 302672v 7 − 26860v 8 − 49120v 9 − 11268v 10 + 2025v 11 + 4149v 12 ) − 2u4 (−1352 − 42764v − 264806v 2 − 679324v 3 − 1124475v 4 − 1362011v 5 − 1231697v 6 − 831833v 7 − 414017v 8 − 117325v 9 − 13037v 10 + 15561v 11 + 16464v 12 + 2616v 13 ) + u5 (2587 − 65653v − 571412v 2 − 1665416v 3 − 2831825v 4 − 3434849v 5 − 2891784v 6 − 1905256v 7 − 1171223v 8 − 864575v 9 − 697988v 10 − 375888v 11 − 82707v 12 − 4011v 13 ) + u6 (1 + v)(8173 + 92982v + 492568v 2 + 968910v 3 + 1253915v 4 + 1288284v 5 + 1150336v 6 + 1288284v 7 + 1253915v 8 + 968910v 9 + 492568v 10 + 92982v 11 + 8173v 12 ) + u7 (−4011 − 82707v − 375888v 2 − 697988v 3 − 864575v 4 − 1171223v 5 − 1905256v 6 − 2891784v 7 − 3434849v 8 − 2831825v 9 − 1665416v 10 − 571412v 11 − 65653v 12 + 2587v 13 ) + 2u8 (−2616 − 16464v − 15561v 2 + 13037v 3 16 TING CHEN, LIHONG HUANG AND JAUME LLIBRE + 117325v 4 + 414017v 5 + 831833v 6 + 1231697v 7 + 1362011v 8 + 1124475v 9 + 679324v 10 + 264806v 11 + 42764v 12 + 1352v 13 ) + u9 (−4149 − 2025v + 11268v 2 + 49120v 3 + 26860v 4 − 302672v 5 − 976910v 6 − 1684186v 7 − 1975495v 8 − 1522535v 9 − 812054v 10 − 210790v 11 − 12432v 12 ) + u10 (−2367 − 12501v − 17010v 2 − 16410v 3 + 36240v 4 + 206364v 5 + 458242v 6 + 677770v 7 + 731935v 8 + 578585v 9 + 332928v 10 + 129584v 11 + 16640v 12 ) + u11 (−423 − 27v + 2430v 2 + 8490v 3 + 7980v 4 − 16704v 5 − 68846v 6 − 138730v 7 − 171365v 8 − 136805v 9 − 73648v 10 − 12352v 11 ) + 2u12 (v − 1)(1 + 2v + 3v 2 + 4v 3 )(9 + 18v + 27v 2 + 16v 3 )2 . 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Bifurcation Chaos 19 (2009) 2391–2399. 1 College of Mathematics and Econometrics, Hunan University, Changsha, 410082, PR China 2 School of Mathematics and Statistics, Changsha University of Science and Tech- nology, Changsha, 410082, PR China 3 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain Email address: chenting0715@126.com (T.Chen), Email address: lhhuang@csust.edu.cn (L.Huang), Email address: jllibre@mat.uab.cat (J.Llibre).