This is a preprint of: “Phase portraits of continuous piecewise linear Liénard differential systems with three zones”, Shimin Li, Jaume Llibre, Chaos Solitons Fractals, vol. 120, 149–157, 2019. DOI: [10.1016/j.chaos.2018.12.037] PHASE PORTRAITS OF CONTINUOUS PIECEWISE LINEAR LIÉNARD DIFFERENTIAL SYSTEMS WITH THREE ZONES SHIMIN LI1 AND JAUME LLIBRE2 Abstract. This paper deals with a class of planar continuous piecewise linear Liénard differential systems with three zones separated by two vertical lines without symmetry. We provide the topological classification of the phase por- traits in the Poincaré disc for systems having a unique singular point located in the middle zone. 1. Introduction and main results In recent years there is a growing interest in the analysis of planar continuous piecewise linear Liénard differential systems of the form: dx dy (1) = y − f (x), = a − g(x), dt dt where   k1 (x − 1) + k2  if x > 1,   (2) f (x) = k2 x if −1 6 x 6 1,     k3 (x + 1) − k2 if x 6 −1, and    l1 (x − 1) + l2 if x > 1,   (3) g(x) = l2 x if −1 6 x 6 1,     l3 (x + 1) − l2 if x 6 −1. Systems (1) satisfying (2) and (3) have been studied extensively, see for instance [7, 9, 10, 14, 15, 17, 21]. For the symmetrical cases, that is a = 0, k1 = k3 and l1 = l3 , see [9] for the bifurcation sets and existence of the limit cycles, see [10] for the amplitude and period of the limit cycles, see [17] for the global phase portraits and bifurcation diagrams. While for the non-symmetrical case, the analysis of systems (1) satisfying (2) and (3) become more complicated. In [14] it is considered the existence and uniqueness of limit cycles for the case k1 > 0, k1 < 0, k3 > 0. In [15] the authors studied the uniqueness and non-uniqueness of limit cycles for the more general cases, they showed that systems (1) satisfying (2) and (3) can have at leat two limit cycles for the cases either k1 > 0, k2 < 0, k3 < 0, or k1 < 0, k2 < 2010 Mathematics Subject Classification. 34A36, 34C07, 37G05. Key words and phrases. Limit cycle; Piecewise linear differential system; Phase portrait; Liénard system. 1 http://www.gsd.uab.cat 2 SHIMIN LI AND JAUME LLIBRE 0, k3 > 0. In [7] the authors found that systems (1) satisfying (2) and (3) can exist jump bifurcations when l2 = 0. In [21] it is investigated the boundary equilibrium bifurcations of systems (1) satisfying (2) and (3). It is worth to note that if either k1 = k2 , l1 = l2 , or k2 = k3 , l2 = l3 , then systems (1) satisfying (2) and (3) become continuous piecewise linear Liénard differential systems with two zones. In 1990 Lum and Chua [18, 19] conjectured that continuous piecewise linear differential systems with two zones separated by a straight line have at most one limit cycle. In 1998 Freire et al. [8] gave a positive answer to this conjecture. Note that this conjecture cannot be extended to discontinuous piecewise linear differential systems see [1, 12], and to continuous piecewise linear differential systems with non-regular separation line see [3, 16]. In this paper we study the global phase portraits of systems (1) satisfying (2) and (3). Since the number of different phase portraits is high, in the present paper we only consider the special case g(x) = x, that is dx dy (4) = y − f (x), = a − x, dt dt with −1 < a < 1, k1 k2 k3 6= 0 and f (x) given in (2). Without loss of generality we assume that k2 < 0. Otherwise we can do the change X = x, Y = −y, T = −t, and then obtain k2 < 0. For polynomial differential systems it is well known that the separatrices in the Poincaré disc are all the infinite orbits, all the finite singular points, the separatrices of the hyperbolic sectors of the finite and infinite singular points, and the limit cycles. If Σ denote the set of all separatrices in the Poincaré disk D2 , Σ is a closed set and the components of D2 \Σ are called the canonical regions. See subsection 2.1 for the definition of the Poincaré compactification, there it is defined the Poincaré disc D2 = {(s1 , s2 ) : s21 +s22 6 1}. We denote by S and R the number of separatrices and canonical regions of a given phase portrait in the Poincaré disc, respectively. We say that two phase portraits P1 and P2 of systems (4) are topologically equivalent if there exists a homeomorphism h : D2 → D2 such that h(∂D2 ) = ∂D2 , h(int(D2 )) = int(D2 ), and h maps orbits of P1 into orbits of P2 either preserving all the orientations, or reversing all the orientations of the orbits. Here ∂D2 and int(D2 ) denote the boundary and interior of D2 , respectively. It it convenient to introduce the following crucial parameters. (5) ∆i = ki2 − 4, i = 1, 2, 3. In section 3 we will see that the type and stability of the finite and infinite singu- lar points of systems (4) satisfying (2) are determined by the signs of ∆i and ki respectively. We classify all the topological equivalent phase portraits in the Poincaré disc for systems (4) satisfying (2) according to the sign of ki . Our main results are the following three theorems. Theorem 1. The phase portrait on the Poincaré disc of a continuous piecewise Liénard differential systems (4) satisfying (2) with −1 < a < 1 and k1 < 0, k2 < 0, k3 < 0, is topologically equivalent to one of the 6 phase portraits described in Figure 1. PHASE PORTRAITS OF CONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 3 1.1 S=2, R=1 1.2 S=4, R=1 1.3 S=6, R=1 1.4 S=7, R=2 1.5 S=9, R=2 1.6 S=11, R=2 Figure 1. Topological phase portraits of systems (4) satisfying (2) with k1 < 0, k2 < 0 and k3 < 0. The right dashed line represents the straight line of x = 1, and the left dashed curve represents the straight line of x = −1. Theorem 2. The phase portrait on the Poincaré disc of a continuous piecewise Liénard differential systems (4) satisfying (2) with −1 < a < 1 and k1 > 0, k2 < 0, k3 > 0, is topologically equivalent to one of the 6 phase portraits described in Figure 2. 2.1 S=3, R=2 2.2 S=5, R=2 2.3 S=7, R=2 2.4 S=8, R=2 2.5 S=10, R=2 2.6 S=12, R=2 Figure 2. Topological phase portraits of systems (4) satisfying (2) with k1 > 0, k2 < 0 and k3 > 0. Theorem 3. The phase portrait on the Poincaré disc of a continuous piecewise Liénard differential systems (4) satisfying (2) with −1 < a < 1 and either k1 > 0, k2 < 0, k3 < 0, or k1 < 0, k2 < 0, k3 > 0, is topologically equivalent to one of the 20 phase portraits described in Figure 3. 4 SHIMIN LI AND JAUME LLIBRE 3.1 S=2, R=1 3.2 S=3, R=2 3.3 S=5, R=2 3.4 S=7, R=2 3.5 S=4, R=1 3.6 S=6, R=1 3.7 S=12, R=3 3.8 S=11, R=3 3.9 S=10, R=2 3.10 S=11, R=2 3.11 S=10, R=3 3.12 S=9, R=3 3.13 S=9, R=2 3.14 S=10, R=3 3.15 S=8, R=2 3.16 S=9, R=2 3.17 S=8, R=2 3.18 S=7, R=2 3.19 S=6, R=2 3.20 S=7, R=2 Figure 3. Topological phase portraits of systems (4) satisfying (2) with k1 > 0, k2 < 0 and k3 < 0. x t Doing the change of variables X = √ , Y = y, T = √ , systems (4) satisfying ε ε (2) become continuous piecewise linear slow fast Liénard differential systems of the form dX dY (6) = Y − F (X), = ε(A − X), dT dT √ √ where A = a/ ε and F (X) = f ( εX). PHASE PORTRAITS OF CONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 5 In [13] the authors studied systems (6) with k2 = −1 and k1 = k3 a fixed constant greater than 1, but close to 1. They proved that systems (6) have at most one limit cycle using first integrals, and also shown that an usual relaxation oscillation occurs for such systems. In [4] it is investigated systems (6) with k1 = k3 > 0 and k2 < 0. They found that this kind of differential systems admit limit cycles that share a lot of similarity with the van der Pol canards. In [5] the authors analyzed systems (6) that display fold singularities, primary and secondary canards. This paper is organized as follows. In section 2 we introduce the Poincaré com- pactification, which is important for studying the infinite singular points. Section 3 is devoted to investigate the existence and uniqueness of limit cycles. In section 4 we prove our main results by careful qualitative analysis. The conclusions on this work are given in section 5. 2. Singular points We use Poincaré compactification to analyze the behaviour of finite and infinite singular points, see more details in chapter 5 of [6]. 2.1. Poincaré compactification. For a given polynomial differential systems dx1 dx2 (7) = P (x1 , x2 ), = Q(x1 , x2 ), dt dt of degree d=max{deg(P ), deg(Q)} in R2 . Let X = (P, Q) the vector field associate systems (7). First we consider R2 as the plane in R3 defined by (s1 , s2 , s3 ) = (x1 , x2 , 1). We call the unit sphere S2 = {(s1 , s2 , s3 ) : s21 + s22 + s23 = 1} the Poincaré sphere. The equator S1 = {(s1 , s2 , s3 ) ∈ S2 , s3 = 0} divide S2 into two parts: the northern hemisphere S− = {(s1 , s2 , s3 ) ∈ S2 : s3 > 0}, and the southern hemisphere S+ = {(s1 , s2 , s3 ) ∈ S2 : s3 < 0}. Now we project each point (x1 , x2 , 1) ∈ R3 onto the Poincaré sphere using a straight line through (x1 , x2 , 1) and the origin, it is obvious that the equator S1 corresponds to the infinity of R2 . So we have two copies of the vector field X on the poincaré sphere S2 , one in the open northern hemisphere and the other in the open southern hemisphere. This vector field on S2 \ S1 can be extended to a vector field p(X ) defined in the whole S2 multiplying it by sd3 . For studying the Poincaré sphere we consider the local charts (8) Ui = {s ∈ S2 : si > 0}, Vi = {s ∈ S2 : si < 0}, where s = (s1 , s2 , s3 ). The expression of p(X ) in the local chart U1 is given by        du d 1 u 1 u dv d+1 1 u (9) = v − uP , +Q , , = −v P , ; dt v v v v dt v v with v > 0. The expression of p(X ) in the local chart V1 is also given by (9) with v < 0. 6 SHIMIN LI AND JAUME LLIBRE The expression of p(X ) in the local chart U2 is given by        du u 1 u 1 dv u 1 (10) = v d − uP , − uQ , , = −v d+1 Q , ; dt v v v v dt v v with v > 0. The expression of p(X ) in the local chart V2 is also given by (10) with v < 0. The expression of p(X ) in the local charts U3 and V3 are just du dv (11) = P (u, v), = Q(u, v). dt dt It is clear that to study the phase portrait of the differential systems (7), it suffices to study its Poincaré compactification p(X ) restricted to the northern hemisphere. For drawing the phase portraits we do the orthogonal projection π(s1 , s2 , s3 ) = (s1 , s2 ) of the closed northern hemisphere onto the closed unit disc centered at the origin of coordinates in the plane x3 = 0. This closed disc is called the Poincaré disc D2 . The finite singular points of systems (7) are the singular points in the interior of D2 , and they can be studied using U3 . The infinite singular points of systems (7) are the singular points of p(X ) in the boundary of D2 . Note that for studying the infinite singular points it suffices to look the ones at the local charts U1 |v=0 , V1 |v=0 , and at the origin of the local chart U2 . 1 u 2.2. Chart U1 . Let x = , y = , v > 0, then systems (4) satisfying (2) become v v du 2 dv (12) = av − 1 − u + k1 u − k1 uv + k2 uv, = v(k1 − k1 v + k2 v − u), dt dt with 0 < v 6 1, and du dv (13) = av − 1 − u2 + k2 u, = v(k2 − u), dt dt with v > 1. In order to study the infinite singular points of systems (4) satisfying (2) in chart U1 , we impose that v = 0. It is obvious that systems (13) have no singular points because v > 1. Let v = 0, then the singular points of systems (12) should satisfy u2 − k1 u + 1 = 0. Recall that ∆1 = k12 − 4, then we have: Proposition 4. For systems (12) the following statements hold. (I) If ∆1 < 0, then systems (12) have no infinite singular points.   k1 (II) If ∆1 = 0, then systems (12) have one infinite singular point E1 = ,0 , 2 and it is a saddle-node, see Figure 4. + (III)  If ∆1 > √ 0, then  systems (12) have two infinite singular points: E1 = k1 + ∆1 , 0 , which is a saddle for k1 > 0, and a stable node for k1 < 0; 2 √  k 1 − ∆1 E1− = , 0 , which is an unstable node for k1 > 0, and a saddle 2 for k1 < 0. PHASE PORTRAITS OF CONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 7 4.1 4.2 Figure 4. The local phase portrait of the saddle-node singular point E1 of systems (12). Proof. (I) Since the infinite singular points must satisfy the equation u2 − k1 u + 1 = 0, statement (I) is obvious. 2  (II) Since ∆1 = k1 − 4 = 0, the Jacobian matrix at the singular point E1 = k1 , 0 is 2   k1 (k2 − k1 )  0 a +  (14)  2 . k1 0 2 We first consider the case k1 = −2, we do the change U = 1 + u − (2 + k2 )v, V = v, T = −t, then systems (12) become dU dV (15) = U 2 − aU V, = V (1 + U − aV ). dT dT According with Theorem 2.19 of [6] we can deduce that E1 is the saddle-node singular point described in Fig. 4.1. Note that we have reverse the time, so the direction is the converse. The phase portraits of E1 described in Fig. 4.2 can be analyzed similarly for the case k1 = 2.  √  + k 1 + ∆1 (III) The Jacobian matrix at the singular point E1 = , 0 is 2  √  √ k 1 + ∆1  − ∆ 1 a − (k 1 − k 2 )  (16)  √ 2 . k1 − ∆1 0 2 If k1 > 0, then E1 is a saddle. If k1 < 0, then E1+ is a stable node. +  √  − k 1 − ∆1 The Jacobian matrix at the singular point E1 = , 0 is 2  √  √ k 1 − ∆1  ∆ 1 a − (k1 − k 2 )  (17)  √ 2 . k 1 + ∆1 0 2 If k1 > 0, then E1 is an unstable node. If k1 < 0, then E1− is a saddle. −  8 SHIMIN LI AND JAUME LLIBRE 1 u 2.3. Chart V1 . Let x = , y = , v < 0, then systems (4) satisfying (2) become v v du dv (18) = av − 1 − u2 + k3 u − k2 uv + k3 uv, = v(k3 − k2 v + k3 v − u), dt dt with −1 6 v < 0, and du dv (19) = av − 1 − u2 + k2 u, = v(k2 − u), dt dt with v 6 −1. Note that systems (19) have no infinite singular points because v 6 −1. For systems (18) we have Proposition 5. For systems (18) the following statements hold. (I) If ∆3 < 0, then systems (18) have no infinite singular points.   k3 (II) If ∆3 = 0, then systems (18) have one infinite singular point E3 = ,0 , 2 and it is a saddle-node. + (III)  If ∆3 > √ 0, then  systems (18) have two infinite singular points: E3 = k3 + ∆3 , 0 , which is a saddle for k3 > 0, and a stable node for k3 < 0; 2 √  − k 3 − ∆3 E3 = , 0 , which is an unstable node for k3 > 0, and a saddle 2 for k3 < 0. Proof. The proof of this proposition is similar to the proof of Proposition 5, we omit it here.  u 1 2.4. Chart U2 . Let x = , y = , v > 0, then systems (4) satisfying (2) become v v du dv (20) = 1 − εauv + εu2 − k1 u, = εv(u − av), dt dt u with > 1, and v du dv (21) = 1 − εauv + εu2 − k2 u, = εv(u − av), dt dt u with −1 6 6 1, and v du dv (22) = 1 − εauv + εu2 − k3 u − (k3 − k2 )v, = εv(u − av), dt dt u with 6 −1. v Since (0, 0) is not singular point of systems (20), (21) and (22), the origin of the local chart U2 is not an infinite singular points. Consequently the origin of V2 is not an infinite singular point. PHASE PORTRAITS OF CONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 9 2.5. Chart U3 . It is obvious that systems (4) satisfying (2) have a unique finite singular point E2 = (a, k2 a). The local phase portrait of E2 is described in the next result. Proposition 6. Consider systems (4) satisfying (2) with k2 < 0 and −1 < a < 1, then the following statements hold: (I) If ∆2 < 0, then the finite singular point E2 is an unstable focus. (II) If ∆2 > 0, then E2 is an unstable node. Note that focus and node are topologically equivalent, thus the topological type of the finite singular point E2 is independent of ∆2 . The proof of this result is straightforward and it is omitted. 3. Limit cycles This section is devoted to study the number of limit cycles for systems (4) sat- isfying (2). First we introduce some definitions. A period annulus is a region in the plane completely filled by periodic orbits. A differential system has the radial angular monotonicity property (RAM property, for simplicity) if the differential system rotates monotonically along rays as the radius increases. A closed orbit surrounding the origin is called star like with respect to the origin if any rays through the origin intersect with the closed orbit in a unique point. The following lemma provides an useful criteria on the existence and uniqueness of a limit cycle for continuous differential systems, see [14]. Lemma 7. [Massera’s Method] Consider continuous piecewise Liénard differential systems of the form dX dY (23) = F (X) − Y, = G(X), dT dT with XG(X) > 0 for X 6= 0, and that F (0) = 0, so that the only singular point is at the origin. Assume that systems (23) have the RAM property and that they have no period annuli. If the systems have a periodic orbit then it is star-like with respect to the origin and it is a limit cycle which is unique and stable. Using the results of [15] and Lemma 7, we can obtain the existence and unique- ness of limit cycles of systems (4) satisfying (2). Proposition 8. For continuous piecewise Liénard differential systems (4) satisfy- ing (2) with −1 < a < 1 and k2 < 0, the following statements hold. (I) A necessary condition for the existence of a limit cycle is that ki , i = 1, 2, 3 have not the same sign. (II) If k1 > 0, k3 > 0, then the singular point E2 is surrounded by a unique stable limit cycle. (III) If k1 < 0, k3 > 0, then the singular point E2 is surrounded by at most one limit cycle, which is unstable if it exist. (IV) If k1 > 0, k3 < 0, then the singular point E2 is surrounded by at most one limit cycle, which is stable if it exist. 10 SHIMIN LI AND JAUME LLIBRE Proof. Doing the change of variables X = x − a, Y = y − k2 a, T = −t, then systems (4) satisfying (2) become dX dY (24) = F (X) − Y, = X, dT dT with    k1 (X + a − 1) + k2 (1 − a) if X > 1 − a,   (25) F (X) = k2 X if −(1 + a) 6 X 6 1 − a,     k3 (X + 1 + a) − k2 (1 + a) if X 6 −(1 + a). We note that statements (I) and (II) of Proposition 8 follow as particular cases r = l = 1 of statements (a) and (b) of Theorem 6 of [15] respectively. Since we have reverse the time, the stability of limit cycle has changed. (III) According to statement (d) of Theorem 6 of [15], we just need to prove statement (III) of Proposition 8 for k1 − k2 < 0. To show the nonnegative rotation property, we will compute the slope of the vector field along half-rays of the form Y = λX. It is obvious that dY X (26) mλ (X) = = . dX Y =λX F (X) − λX In order to analyze the monotone character of this slope along the half-rays Y = λX, we need to compute its derivative with respect to X, i.e. dmλ (X) F (X) − XF 0 (X) (27) = . dX (F (X) − λX)2 dmλ (X) (k2 − k2 )X If −(1 + a) 6 X 6 1 − a, then F (X) = k2 X. Thus = = dX (F (X) − λX)2 0. If X > 1 − a, then F (X) = k1 (X + a − 1) + k2 (1 − a). It is obvious that dmλ (X) (k2 − k1 )(1 − a) = > 0, because k1 − k2 < 0 and 1 − a > 0. dX (F (X) − λX)2 If X 6 −(1 + a), then F (X) = k3 (X + 1 + a) − k2 (1 + a). It follows that dmλ (X) (k3 − k2 )(1 + a) = > 0, because k3 > 0, k2 < 0 and 1 + a > 0. dX (F (X) − λX)2 According to Massera’s Method, if k1 < 0, k3 > 0 and k1 − k2 < 0, then the origin is surrounded by at most one limit cycle, and if it exists is stable. For the case (IV) doing the change of variables X̄ = −X and Ȳ = −Y , we have k̄2 = k2 > 0, k̄1 = k3 < 0 and k̄3 = k1 > 0. Then we can guarantee the uniqueness of limit cycles by statement (III).  Note that if either k2 = k3 , or k1 = k2 , then systems (4) satisfying (2) become continuous piecewise linear Liénard differential systems with two zones. Then ac- cording to Theorem 5 of [11] we have Proposition 9. Consider continuous piecewise Liénard differential systems (4) satisfying (2) with −1 < a < 1 and k2 < 0. Assume that ∆1 < 0, ∆3 < 0, then the following statements hold. PHASE PORTRAITS OF CONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 11 k1 k3 (I) If k1 > 0, k2 = k3 < 0 and √ +√ > 0, then the singular point −∆1 −∆3 E2 is surrounded by a unique stable limit cycle. k1 k3 (II) If k1 > 0, k2 = k3 < 0 and √ +√ 6 0, then the singular point −∆1 −∆3 E2 is globally asymptotically unstable and no limit cycle exist. k1 k3 (III) If k1 = k2 > 0, k3 < 0 and √ +√ > 0, then the singular point −∆1 −∆3 E2 is surrounded by a unique unstable limit cycle. k1 k2 (IV) If k1 = k2 > 0, k3 < 0 and √ +√ < 0, then the singular point −∆1 −∆2 E2 is globally asymptotically stable and no limit cycle exist. Proof. First we consider the case k1 > 0, k2 = k3 < 0. Let X = x − 1, Y = y − k2 , T = −t, then systems (4) satisfying (2) become dX dY (28) = F (X) − Y, = X − A, dT dT with A = a − 1 < 0 and ( k1 X if X > 0, (29) F (X) = k2 X if X 6 0. According with statements (a) and (b) of Theorem 5 of [11], we can prove statements (I) and (II), respectively. Similar to the proofs of statements (I) and (II), we can deduce statements (III) and (IV) from statements (c) and (d) of Theorem 5 of [11].  4. Proof of the results This section is devoted to prove our main results. 4.1. Proof of Theorem 1. The proof of Theorem 1 directly follows from the following result. Proposition 10. The phase portraits of systems (4) satisfying (2) with k1 < 0, k2 < 0, k3 < 0 and −1 < a < 1 are topologically equivalent to (I) Fig.1.1 if ∆1 < 0, ∆3 < 0; (II) Fig.1.2 if either ∆1 = 0, ∆3 < 0, or ∆1 < 0, ∆3 = 0; (III) Fig.1.3 if either ∆1 > 0, ∆3 < 0, or ∆1 < 0, ∆3 > 0; (IV) Fig.1.4 if ∆1 = 0, ∆3 = 0; (V) Fig.1.5 if either ∆1 = 0, ∆3 > 0, or ∆1 > 0, ∆3 = 0; (VI) Fig.1.6 if ∆1 > 0, ∆3 > 0. Proof. Recall that the finite singular point E2 is an unstable focus or node by Proposition 6, and there is no limit cycle by statement (I) of Proposition 8. (I) For the case ∆1 < 0, ∆3 < 0, it is obvious that systems (4) satisfying (2) have no infinite singular points by statement (I) of Propositions 4 and 5. Therefore 12 SHIMIN LI AND JAUME LLIBRE the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.1.1. (II) For the case ∆1 = 0, ∆3 < 0, the infinite singular point E1 is a saddle-node by statement (II) of Proposition 4. Hence the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.1.2. For the case ∆1 < 0, ∆3 = 0, the infinite singular point E3 is a saddle-node by statement (II) of Proposition 5. So the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.1.2. (III) For the case ∆1 > 0, ∆3 < 0, the infinite singular points E1+ is a stable node, and E1− is a saddle by statement (III) of Proposition 4. Then the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.1.3. For the case ∆1 < 0, ∆3 > 0, the infinite singular points E3+ is a stable node, and E3− is a saddle by statement (III) of Proposition 5. Therefore the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.1.3. (IV) For the case ∆1 = 0, ∆3 = 0, the infinite singular points E1 and E3 are saddle-nodes by statement (II) of Propositions 4 and 5. Hence the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.1.4. (V) For the case ∆1 > 0, ∆3 = 0, the infinite singular points E1+ is a stable node, and E1− is a saddle by statement (III) of Proposition 4, and the infinite singular point E3 is a saddle-node by statement (II) of Propositions 5. So the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.1.5. For the case ∆1 = 0, ∆3 > 0, the infinite singular points E3+ is a stable node, and E3− is a saddle by statement (III) of Proposition 5, and the infinite singular point E1 is a saddle-node by statement (II) of Propositions 4. Then the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.1.5. (VI) For the case ∆1 > 0, ∆3 > 0, the infinite singular points E1+ is a stable node, and E1− is a saddle by statement (III) of Proposition 4. The infinite singular points E3+ is a stable node, and E3− is a saddle by statement (III) of Proposition 5. So the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.1.6.  4.2. Proof of Theorem 2. The proof of Theorem 2 directly follows from the following result. Proposition 11. The phase portraits of systems (4) satisfying (2) with k1 > 0, k2 < 0, k3 > 0 and −1 < a < 1 are topologically equivalent to (I) Fig.2.1 if ∆1 < 0, ∆3 < 0; (II) Fig.2.2 if either ∆1 = 0, ∆3 < 0, or ∆1 < 0, ∆3 = 0; (III) Fig.2.3 if either ∆1 > 0, ∆3 < 0, or ∆1 < 0, ∆3 > 0; (IV) Fig.2.4 if ∆1 = 0, ∆3 = 0; (V) Fig.2.5 if either ∆1 = 0, ∆3 > 0, or ∆1 > 0, ∆3 = 0; (VI) Fig.2.6 if ∆1 > 0, ∆3 > 0. PHASE PORTRAITS OF CONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 13 Proof. Recall that the finite singular point E2 is an unstable focus or node by Propo- sition 6, and there is a unique stable limit cycle by statement (II) of Proposition 8. (I) For the case ∆1 < 0, ∆3 < 0, it is obvious that systems (4) satisfying (2) have no infinite singular points by statement (I) of Propositions 4 and 5. Then the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.2.1. (II) For the case ∆1 = 0, ∆3 < 0, the infinite singular point E1 is a saddle-node by statement (II) of Proposition 4. Therefore the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.2.2. For the case ∆1 < 0, ∆3 = 0, the infinite singular point E3 is a saddle-node by statement (II) of Proposition 5. Hence the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.2.2. (III) For the case ∆1 > 0, ∆3 < 0, the infinite singular points E1+ is a saddle, and E1− is an unsaddle node by statement (III) of Proposition 4. So the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.2.3. For the case ∆1 < 0, ∆3 > 0, the infinite singular points E3+ is a saddle, and E3− is a stable node by statement (III) of Proposition 5. Then the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.2.3. (IV) For the case ∆1 = 0, ∆3 = 0, the infinite singular points E1 and E3 are saddle-nodes by statement (II) of Propositions 4 and 5. So the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.2.4. (V) For the case ∆1 > 0, ∆3 = 0, the infinite singular points E1+ is a saddle, and E1− is an unstable node by statement (III) of Proposition 4, and the infinite singular point E3 is a saddle-node by statement (II) of Propositions 5. Therefore the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.2.5. For the case ∆1 = 0, ∆3 > 0, the infinite singular points E3+ is a saddle, and E3− is an unstable node by statement (III) of Proposition 5, and the infinite singular point E1 is a saddle-node by statement (II) of Propositions 4. Hence the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.2.5. (VI) For the case ∆1 > 0, ∆3 > 0, the infinite singular points E1+ is a saddle, and E1− is an unstable node by statement (III) of Proposition 4. The infinite singular points E3+ is a saddle, and E3− is an unstable node by statement (III) of Proposition 5. Then the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.2.6.  4.3. Proof of Theorem 3. The proof of Theorem 3 for the case k1 > 0, k2 < 0, k3 < 0 directly follows by the following result. For the case k1 < 0, k2 < 0, k3 > 0, we can do the change of variables X = −x and Y = −y. Proposition 12. The phase portraits of systems (4) satisfying (2) with k1 > 0, k2 < 0, k3 < 0 and −1 < a < 1 are topologically equivalent to (I) Fig.3.1 and Fig.3.2 if ∆1 < 0, ∆3 < 0; 14 SHIMIN LI AND JAUME LLIBRE (II) Fig.3.3 if ∆1 = 0, ∆3 < 0; (III) Fig.3.4 if ∆1 > 0, ∆3 < 0; (IV) Fig.3.5 if ∆1 < 0, ∆3 > 0; (V) Fig. 3.6 if ∆1 < 0, ∆3 = 0; (VI) Fig.3.7, Fig.3.8, Fig. 3.9 and Fig. 3.10 if ∆1 > 0, ∆3 > 0; (VII) Fig.3.11, Fig.3.12 and Fig.3.13 if ∆1 > 0, ∆3 = 0; (VIII) Fig.3.14, Fig.3.15 and Fig.3.16 if ∆1 = 0, ∆3 > 0; (IX) Fig.3.17, Fig.3.18, Fig.3.19 and Fig.3.20 if ∆1 = 0, ∆3 = 0. In order to prove Proposition 12 we need the following result, for a proof see Theorem XVII of [20]. Theorem 13. Consider a heteroclinic loop with two singular points which are either two hyperbolic saddles, or one saddle and one saddle-node, with eigenvalues µi 6 0 6 λi for i = 1, 2 such that µ1 µ2 6= 0 or λ1 λ2 6= 0. Define K = |µ1 µ2 |/(λ1 λ2 ), if K < 1 then the heteroclinic loop is stable, and if K > 1 it is unstable. Proof of Proposition 12. Recall that the finite singular point E2 is an unstable focus or node by Proposition 6, and there is at most one limit cycle by statement (IV) of Proposition 8. (I) For the case ∆1 < 0 and ∆3 < 0, it is obvious that systems (4) satisfying (2) has no infinite singular points by statement (I) of Propositions 4 and 5. In the following we will prove that systems (4) satisfying (2) can have no limit cycles and then they are topologically equivalent to Fig.3.1, or they have one limit cycle and then they are topologically equivalent to Fig.3.2. We consider the special case k2 = k3 < 0. From statements (II) of Proposition 9, k1 k3 if √ +√ 6 0, then there is no limit cycles surrounding the singular point −∆1 −∆3 E2 . Therefore the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.3.1. k1 k3 If √ +√ > 0, then the singular point E2 is surrounded by a stable limit −∆1 −∆3 cycle. Then the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.3.2. (II) For the case ∆1 = 0, ∆3 < 0, the infinite singular point E1 is a saddle-node by statement (II) of Proposition 4. Hence the global phase portrait is topologically equivalent to Fig.3.3. (III) For the case ∆1 > 0, ∆3 < 0, the infinite singular points E1+ is a saddle, and E1− is an unstable node by statement (III) of Proposition 4. So the global phase portrait is topologically equivalent to Fig.3.4. (IV) For the case ∆1 < 0, ∆3 > 0, the infinite singular points E3+ is a stable node, and E3− is a saddle by statement (III) of Proposition 5. Then the global phase portrait is topologically equivalent to Fig.3.5. (V) For the case ∆1 < 0, ∆3 = 0, the infinite singular point E3 is a saddle- node by statement (II) of Proposition 5. Therefore the global phase portrait is topologically equivalent to Fig.3.6. PHASE PORTRAITS OF CONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 15 (VI) For the case ∆1 > 0, ∆3 > 0, the infinite singular points E1+ is a saddle, and E1− is an unstable node by statement (III) of Proposition 4. The infinite singular points E3+ is a stable node, and E3− is saddle node by statement (III) of Proposition 5. First we consider the subcase ∆2 < 0. The finite singular point E2 is an unstable focus. Systems (4) satisfying (2) in the middle zone −1 6 x 6 1 have the first integral   2a + (k22 − 2)x − k2 y 2k22 ArcTan √  √ H(x, y) = exp (k2 x − y) −∆2 |x − a|2k2 −∆2 (30)   a2 + a(k22 − 2)x + x2 − k2 (a+x )y + y 2 × . (a − x)2 Note that H(a, k2 a) = +∞, so the first integral (30) is a decreasing function in a neighborhood of E2 . Since ∆1 > 0 the finite singular point (a, k1 (a − 1) + k2 ) is a virtual node with the stable invariant straight line √ k 1 + ∆1 (31) ΓR : y = (x − a) + k1 (a − 1) + k2 , 2 which intersects the switching line x = 1 at the point (1, y1 ), with √ (a − 1)k1 + ∆1 (32) y1 = + k2 . 2 Since ∆3 > 0 the finite singular point (a, k3 (a + 1) − k2 ) is a virtual node with the unstable invariant straight line √ k 3 − ∆3 (33) ΓL : y = (x − a) + k3 (a + 1) − k2 , 2 which intersects the witching line x = −1 at the point (−1, y2 ), with √ (a + 1)k3 + ∆3 (34) y2 = − k2 . 2 We define the auxiliary function (35) H̄(a) = H(1, y1 ) − H(−1, y2 ), a ∈ (−1, 1). In the following we prove that H̄(a) given by (35) is an increasing function. 0 Computing the derivative of H̄(a) with respect to a we have that H̄ (a) is equal to  √  √ √  √ 4 − k1 k2 + ∆1 k2 k2 −∆2 2k2 ArcTan 2 √ √  −2k2 −∆2 (1 − a)2 (k1 − k2 )(k1 − ∆1 ) −k1 + ∆1 −∆2 exp 1−a  √  √ √  √ −4 + k2 k3 + ∆3 k2 k −∆ 2k 2 ArcTan √ √  . −2k2 −∆2 (1 + a)2 (k3 − k2 )(k3 + ∆3 ) 2 2 2 + exp k3 + ∆3 −∆2 1+a 0 It is obvious that H̄ (a) > 0. Note that lim H̄(a) = −∞ and lim H̄(a) = +∞, a→−1 a→1 we can conclude that H̄(a) has a unique zero a∗ in the interval a ∈ (−1, 1). 16 SHIMIN LI AND JAUME LLIBRE If a < a∗ then the separatrix ΓR of the saddle E1+ is over the separatrix ΓL of the saddle E3− . According to Poincaré-Bendixson Theorem there is a stable limit cycle surrounding the finite singular point E2 . Hence the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.3.7. If a = a∗ then there is a heteroclinic loop connecting E1+ and E3− . In order to study the stability of this loop, we compute the eigenvalues of systems (12) and (18) at the singular point E1+ and E3− , respectively. Then we obtain √ √ ∆1 (k3 + ∆3 ) (36) K= √ √ . ∆3 (k1 − ∆1 ) If K > 1, then the loop is unstable by Theorem 13. According to Poincaré- Bendixson Theorem, there is a stable limit cycle surrounding the finite singular point E2 . Therefore the global phase portrait of systems (4) satisfying (2) is topo- logically equivalent to Fig.3.8. If K < 1, then the loop is stable by Theorem 13. Hence the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.3.9. If a > a∗ , then the separatrix ΓR of the saddle E1+ is under the separatrix ΓL of the saddle E3− . So the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.3.10. Second we consider the subcase ∆2 = 0. Note that k2 = −2, then the finite singular point E2 = (a, −2a) is an unstable node with the invariant straight line ΓC = −x − a, which intersects the switching line x = −1 at the point (−1, ȳL ) with ȳL = 1 − a, see Figure 5. It is obvious that yL > ȳL , thus the phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.3.10. Figure 5. The invariant straight lines of systems (4) with k1 > 0, k2 < 0, k3 < 0 and ∆1 > 0, ∆2 > 0, ∆3 > 0. PHASE PORTRAITS OF CONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 17 Finally we consider the subcase ∆2 > 0. The finite singular √ point E2 is an k 2 + ∆2 unstable node with the invariant straight line ΓC = (x − a) + k2 a, 2 which intersects √ the switching line x = −1 at the point (−1, ȳL ) with ȳL = k2 + ∆2 − (1 + a) + k2 a. It is easy to deduce that yL > ȳL when k2 < −2. 2 Thus the phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.3.10. (VII) For the case ∆1 > 0, ∆3 = 0, the infinite singular points E1+ is a saddle, E1− is an unstable node by statement (III) of Proposition 4, and the infinite singular point E3 is a saddle-node by statement (II) of Propositions 5. Note that k3 = −2, then K = +∞. Similarly with the proof of statement (VI) we have: If a < a∗ , ∆2 < 0, then the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.3.11. If a = a∗ , ∆2 < 0, then the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.3.12. If either a > a∗ , ∆2 < 0, or ∆2 > 0, then the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.3.13. (VIII) For the case ∆1 = 0, ∆3 > 0, the infinite singular point E1 is a saddle- node by statement (II) of Propositions 4, the infinite singular points E3+ is a stable node, and E3− is a saddle by statement (III) of Proposition 5. Note that k1 = 2, then K = 0. Similarly with the proof of statement (V) we have: If a < a∗ , ∆2 < 0, then the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.3.14. If a = a∗ , ∆2 < 0, then the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.3.15. If either a > a∗ , ∆2 < 0, or ∆2 > 0, then the global phase portrait of systems (4) satisfying (2) is topologically equivalent to Fig.3.16. (IX) For the case ∆1 = 0, ∆3 = 0, the infinite singular points E1 and E3 are two saddle-nodes by statement (II) of Propositions 4 and 5. The proof of statement (IX) can be deduced from the proof of statement (V) we omit it here.  5. Conclusion In this paper we classify the topological phase portraits in the Poincaré disc of the continuous piecewise Liénard differential systems (4) satisfying (2) with a unique singular point E2 located at the middle zone. From the proofs of the three theorems which are given in section 4, we can obtain the conditions for the existence and uniqueness of limit cycles for systems (4) satisfying (2). It is worth to point out that the amplitude of limit cycles is also very important, see for instance [2, 4, 22]. For the case k1 > 0, k1 < 0, k3 > 0, we obtain that systems (4) satisfying (2) have a unique stable limit cycle by Theorem 2. In fact we can proved that if ∆2 > 0, then the limit cycle is contained in three zones. While if ∆2 < 0 and 1 − a > 0 (resp. a + 1 > 0) sufficiently small, then the limit cycle is contained in the middle 18 SHIMIN LI AND JAUME LLIBRE and the right (resp. left) zones. These three cases have the same phase portrait according to the definition of topological equivalence given in section 1. 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