This is a preprint of: “Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line”, Shimin Li, Jaume Llibre, J. Differential Equations, vol. 266, 8094–8109, 2019. DOI: [10.1016/j.jde.2018.12.024] PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS WITH TWO ZONES SEPARATED BY A STRAIGHT LINE SHIMIN LI1 AND JAUME LLIBRE2 Abstract. This paper provides the classification of the phase portraits in the Poincaré disc of all piecewise linear continuous slow–fast differential systems with two zones separated by a straight line. The sufficient and necessary conditions for existence and uniqueness of limit cycles are also given. 1. Introduction and statement of the main results The Liénard second order differential equation (1) ẍ + f (x)ẋ + g(x) = 0, where the dot denotes differentiation with respect to the time t and f, g ∈ C 1 was first introduced by Liénard in [20]. This differential equation has many important applications in several branches of science, such as biology, chemistry, mechanics, electronics and so on. The Liénard second order differential equation (1) can be written as the equiv- alent planar differential system dx = y − F (x), dt (2) dy = g(x), dt Z x where F (x) = f (t)dt. 0 When F (x) and g(x) are polynomial functions in the variable x, the Liénard differential system (2) is called the generalized polynomial Liénard differential sys- tem has been studied extensively, see [14, 38, 41] for center conditions, [13, 16, 21, 22, 27, 37] for the number of limit cycles, [1, 40] for the amplitude of limit cycles, [15, 26] for integrability conditions, [38, 39] for isochronous conditions, and [2, 3, 17] for global phase portraits and bifurcation diagrams. The investigation of differential system (2) are mainly extended into two di- rections. On one hand, piecewise linear differential systems are the most natural 2010 Mathematics Subject Classification. 34C37, 34C07, 37G15. Key words and phrases. Phase portraits, Piecewise linear slow–fast differential system, Limit cycle. 1 2 SHIMIN LI AND JAUME LLIBRE extensions to the nonlinear differential systems in order to capture nonlinear phe- nomena observed in real world applications. There are several papers studying the piecewise linear continuous Liénard systems, see [23, 24, 25, 32] and references therein. On the other hand, differential systems with multiple time scales, espe- cially for slow–fast systems, regularly appear in a great variety of areas. Discussions regarding Liénard type slow–fast systems have dominated research in recent years, see [4, 9, 18, 19, 30, 31, 36] for smooth cases, and [5, 6, 12, 35] for piecewise smooth cases. Motivated by the above two lines of research, the aim of this paper is to in- vestigate the global dynamics of the planar piecewise linear continuous slow–fast differential systems of the form dx = y − f (x), dt (3) dy = ε(a − x), dt where f (x) is a continuous piecewise linear function given by ( k1 x if x > 0, (4) f (x) = k2 x if x 6 0, with ε > 0 sufficiently small and k1 k2 6= 0. System (3) is the most simplicity piecewise linear continuous slow–fast differen- tial system and has been studied in the papers [7, 33, 34]. From the results of [33], we can conclude that if k1 = −k2 , then system (3) has neither maximal nor faux maximal canard orbits exist. In 2016 Roberts [34] proved that if k1 > 0 and k2 < 0, then system (3) can have an unstable canard limit cycle. He also investigated the mechanism of canard explosive. If k1 = k2 , then system (3) becomes a smooth linear slow–fast system. In 2006 Dumortier [7] studied the behaviour near infinity on the Poincaré disc. For the piecewise linear continuous differential systems (3) it is known (see for instance the Chapter 1 of [8]) that the separatrices include all the infinite singular points, 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 disc D2 , Σ is a closed set and the component 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, respectively. We say that two phase portraits of X1 and X2 of systems (3) are topologically equivalent if there exists a homeomorphism h : D2 → D2 such that h(∂D2 ) = ∂D2 , h(D2 ∩ {s1 = 0}) = D2 ∩ {s1 = 0}, either h(D2 ∩ {s1 > 0}) = D2 ∩ {s1 > 0} and h(D2 ∩ {s1 < 0}) = D2 ∩ {s1 < 0}, or h(D2 ∩ {s1 > 0}) = D2 ∩ {s1 < 0} and h(D2 ∩ {s1 < 0}) = D2 ∩ {s1 > 0}, and h mapping orbits of X1 into orbits of X2 either preserving the orientation, or reversing the orientation of all orbits. We denote (5) ∆1 = k12 − 4ε, ∆2 = k22 − 4ε. PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS 3 According to the sign of a, ∆1 , ∆2 , we classified all the global phase portraits of system (3) with k1 k2 6= 0. 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. Our main results are the following two theorems. Theorem 1. The phase portrait on the Poincaré disc of piecewise linear continuous differential system (3) with k1 > 0, k2 > 0 and ε > 0 sufficiently small, or the phase portrait with the sense of all orbits reversed, is topologically equivalent to one of the 15 phase portraits described in Figure 1. Figure 1. Topological phase portraits of system (3) with k1 > 0, k2 > 0 in Theorem 1. The dashed line represents the straight line of x = 0. 4 SHIMIN LI AND JAUME LLIBRE Theorem 2. The phase portrait on the Poincaré disc of piecewise linear continuous differential system (3) with k1 < 0, k2 > 0 and ε sufficiently small, or the phase portrait with the sense of all orbits reversed, is topologically equivalent to one of the 18 phase portraits described in Figure 2. From Figures 1 and 2, we know that system (3) has at most one limit cycle, this result was conjectured by Lum and Chua [28, 29] in 1990, and proved in 1998 by Freire et al [11]. Later on a new and shorter proof was given by Llibre et al in [23]. The following corollary outlines the conditions for the existence and stability of limit cycle. The result on the stability is new. Corollary 3. Piecewise linear continuous slow–fast differential system (3) with k1 < 0, k2 > 0 has a unique limit cycle if and only if one of the following six conditions hold: (I) If a > 0, ∆1 < 0, ∆2 < 0 and k1 + k2 > 0, then the limit cycle is stable, see Figure2.1. (II) If a > 0, ∆1 < 0, ∆2 = 0, then the limit cycle is stable, see Figure2.5. (III) If a > 0, ∆1 < 0, ∆2 > 0, then the limit cycle is stable, see Figure2.8. (IV) If a < 0, ∆1 < 0, ∆2 < 0 and k1 + k2 < 0, then the limit cycle is unstable, see Figure2.1 for reversing all the orientation of the orbits. (V) If a < 0, ∆1 = 0, ∆2 < 0, then the limit cycle is unstable, see Figure2.5 for reversing all the orientation of the orbits. (VI) If a < 0, ∆1 > 0, ∆2 < 0, then the limit cycle is unstable, see Figure2.8 for reversing all the orientation of the orbits. Note that statements (I) and (IV) of Corollary 3 can be obtained from Proposi- tion 4 and Theorem 5 of the paper [10]. This paper is organized as follows. In section 2, we first introduce the Poincaré compacification, and then analyze the local phase portraits of the finite and infinite singular points. Applying the results from section 2 we prove our main theorems in section 3. Finally we discuss the differences of phase portraits between smooth and piecewise smooth differential systems in section 4. 2. Singular points In order to analyze the global behaviour of trajectories it is possible to use Poincaré compactification, see for example the chapter 5 of the book [8]. 2.1. Poincaré compactification. Let S2 be the set of points s= (s1 , s2 , s3 ) ∈ R3 such that s21 + s22 + s23 = 1. We will call this set the Poincaré sphere. Given a polynomial differential system x˙1 = P (x1 , x2 ), (6) x˙2 = Q(x1 , x2 ), of degree d = max{deg(P ), deg(Q)} in R2 identified with the plane x3 = 1 of R3 , it can be extended analytically to the Poincaré sphere by projecting each point (x1 , x2 , 1) ∈ R3 onto the Poincaré sphere using a straight line through (x1 , x2 , 1) PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS 5 Figure 2. Topological phase portraits of system (3) with k1 < 0, k2 > 0 in Theorem 2. 6 SHIMIN LI AND JAUME LLIBRE and the origin of R3 . In this way we obtain a new differential system formed by two copies of (6): one on the northern hemisphere S− = {(s1 , s2 , s3 ) ∈ S2 , s3 > 0} and another on the southern hemisphere S+ = {(s1 , s2 , s3 ) ∈ S2 , s3 < 0} . Note that the equator S1 = {(s1 , s2 , s3 ) ∈ S2 , s3 = 0} corresponds to the infinity of R2 . The local charts needed for doing the calculations on the Poincaré sphere are (7) Ui = {s ∈ S2 : si > 0}, Vi = {s ∈ S2 : si < 0}, where s = (s1 , s2 , s3 ). The expression for the corresponding differential system on S2 in the local chart U1 is given by        1 u 1 u 1 u (8) u̇ = v d − uP , +Q , , v̇ = −v d+1 P , ; v v v v v v with v > 0. The expression for the corresponding differential system on S2 in the local chart U2 is given by        u 1 u 1 u 1 (9) u̇ = v d − uP , − uQ , , v̇ = −v d+1 Q , ; v v v v v v with v > 0. The expression for the corresponding differential system on S2 in the local chart Vj is the same than in the chart Uj multiplied by (−1)d , for j = 1, 2. The expression for the corresponding differential system on S2 in the local charts U3 and V3 are just (10) u̇ = P (u, v), v̇ = Q(u, v). We note that to study the differential system (6), it is enough to study its Poincaré compactification restricted to the northern hemisphere plus S1 . To draw the phase portraits we will consider the orthogonal projection π(s1 , s2 , s3 ) = (s1 , s2 ) of the northern hemisphere onto the closed unit disc centered at the origin of coor- dinates in the plane x3 = 0, called the Poincar disc. Finite singular points of (6) are the singular points of its compactification which are in S2 \ S1 , and they can be studied using U3 . Infinite singular points of (6) are the singular points of the corresponding differential system on the Poincaré disc lying on S1 . Note that for studying the infinite singular points it suffices to look the ones at U1 |v=0 , V1 |v=0 , and the origins of U2 and V2 . We note that the coordinates (u, v) means different things in every local charts, but in the local charts Ui and Vi for i = 1, 2 the infinite points have always the coordinate v = 0. 1 u 2.2. Chart U1 . Let x = , y = , v > 0, then system (3) becomes v v du = εav − ε − u2 + k1 u, dt (11) dv = v(k1 − u). dt PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS 7 Let v = 0, the infinite singular points of system (3) in chart U1 should satisfy u2 − k1 u + ε = 0. Recall that ε > 0 and ∆1 = k12 − 4ε, we have Proposition 4. For system (11) the following statements hold. (I) If ∆1 < 0, then system (11) has no infinite singular point.   k1 (II) If ∆1 = 0, then system (11) has one infinite singular point E = ,0 , 2 and it is a saddle-node, see Figure 3.  √  k1 + ∆1 (III) If ∆1 > 0, then system (11) has two infinite singular points E+ = ,0 , 2 which  is √ a saddle  for k1 > 0 and a stable node for k1 < 0. E− = k1 − ∆1 , 0 , which is an unstable node for k1 > 0, and a saddle for 2 k1 < 0. Figure 3. The local phase portrait of the saddle-node singular point E of system (11). Proof. The singular points should satisfy the equation u2 − k1 u + ε = 0. The statement (I) is obvious.   k12 k1 (II) Since ε = , the Jordan matrix of singular point E = , 0 is 4 2   k12 a  0 (12)  4 . k1  0 2 k1 k1 We first consider the case k1 < 0, we do the change U = − + u − av, V = 2 2 v, T = −t, then system (11) becomes dU k1 a = U2 + U V, dT 2 (13) dV k1 k1 a = V (− k1 + U + V ). dT 2 2 According to the Theorem 2.19 of the book [8], we can deduce that E is a saddle- node singular point whose local phase portrait is described in Figure 3.1. Note that we have reverse the time, so the orientation of the orbits is the converse. 8 SHIMIN LI AND JAUME LLIBRE The phase portraits of E described in Figure 3.2 can be analyzed similarly for the case k1 > 0.  √  k 1 + ∆1 (III) The Jacobian matrix for singular point E+ = , 0 is 2  √  − ∆1 εa√ (14)  k1 − ∆1  . 0 2 If k1 > 0, then E+ is a saddle. If k1 < 0, then E+ is a stable node.  √  k1 − ∆1 The Jacobian matrix for singular point E− = , 0 is 2  √  ∆1 εa√ (15)  k 1 + ∆1  . 0 2 If k1 > 0, then E− is an unstable node. If k1 < 0, then E− is a saddle.  1 u 2.3. Chart V1 . Let x = , y = , v < 0, then system (3) becomes v v du = εav − ε − u2 + k2 u, dt (16) dv = v(k2 − u). dt It is worth to note that system (16) is similar to (11), and just substitute k1 of system (11) for k2 of system (16). The proof of the next proposition uses the same arguments than the proof of Proposition 4. Proposition 5. For system (16) the following statements hold. (I) If ∆2 < 0, then system (16) has no infinite singular point.   k2 (II) If ∆2 = 0, then system (16) has one infinite singular point Ẽ = ,0 , 2 which is a saddle-node, see Figure 3.  √  k2 + ∆2 (III) If ∆2 > 0, then system (16) has two infinite singular points Ẽ+ = ,0 , 2 which  is √ a saddle  for k 2 > 0 and a stable node for k 2 < 0. Ẽ − = k2 − ∆2 , 0 , which is an unstable node for k2 > 0 and a saddle for 2 k2 < 0. u 1 2.4. Charts U2 and V2 . Let x = , y = , v > 0, then system (3) becomes v v du = 1 − εauv + εu2 − k1 u, dt (17) dv = εv(u − av). dt PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS 9 with u > 0, and du = 1 − εauv + εu2 − k1 u, dt (18) dv = εv(u − av), dt with u 6 0. Since (0, 0) is neither a singular point of system (17) nor a singular point of system (18), the origins of U2 and V2 are not infinite singular points. 2.5. Chart U3 . It is obvious that system (3) has one finite singular point (a, f (a)). The local phase portrait of the singular point (a, f (a)) is characterized in the result. Proposition 6. For system (3) with k2 > 0 and ε > 0 sufficiently small the following statements hold. (I) If a > 0 then the unique singular point is (a, k1 a), which is unstable (stable) when k1 < 0 (k1 > 0). Furthermore, (I.1) If ∆1 < 0, then (a, k1 a) is a focus. (I.2) If ∆1 > 0, then (a, k1 a) is a node. (II) If a = 0 then the unique singular point is (0, 0). Furthermore, (II.1) If ∆1 < 0, ∆2 < 0 and k1 < 0, k1 + k2 < 0, then (0, 0) is a global unstable focus. (II.2) If ∆1 < 0, ∆2 < 0 and k1 < 0, k1 + k2 = 0, then (0, 0) is a global center. (II.3) If ∆1 < 0, ∆2 < 0 and k1 < 0, k1 + k2 > 0, then (0, 0) is a global stable focus. (II.4) If k1 < 0 and ∆1 > 0, ∆2 < 0, then (0, 0) is an unstable node, see Figure 4.1. (II.5) If k1 < 0 and ∆1 < 0, ∆2 > 0, then (0, 0) is a stable node, see Figure 4.2. (II.6) If k1 < 0 and ∆1 > 0, ∆2 > 0, then (0, 0) is formed by an elliptic sector and a hyperbolic sector, see Figure 4.3. (II.7) If k1 > 0 and ∆1 < 0, ∆2 < 0, then (0, 0) is a stable focus. (II.8) If k1 > 0, and either ∆1 > 0 or ∆2 > 0, then (0, 0) is a stable node, see Figure 4.4. (III) If a < 0 then the unique singular point (a, k2 a) is stable. Furthermore, (III.1) If ∆2 < 0, then (a, k2 a) is a focus. (III.2) If ∆2 > 0, then (a, k2 a) is a node. We first introduce the results of Freire, Ponce and Torres [10], which is important for the proof of Proposition 6. Consider the following piecewise linear system du = F (u) − v, dτ (19) dv = u − Bv − A, dτ 10 SHIMIN LI AND JAUME LLIBRE Figure 4. The local phase portrait of the singular point (a, f (a)) of system (3) when a = 0. where ( µ1 u u > 0, (20) F (u) = µ2 u u 6 0. We introduce the following parameters which will simplify our analysis (µi − B)2 µi − B (21) Di = 1 − Bµi − , γi = √ , 4 2 Di for i = 1, 2. We have the following result, for a proof see Proposition 4 and Theorem 5 of [10]. Lemma 7. The following statements hold for system (19) under the assumptions D1 > 0 and D2 > 0. Assume A = 0. (i) If γ1 + γ2 > 0 then (0, 0) is a global unstable nonlinear focus. (ii) If γ1 + γ2 = 0 then (0, 0) is a global nonlinear center. (iii) If γ1 + γ2 < 0 then (0, 0) is a global stable nonlinear focus. Assume A 6= 0 and µ1 µ2 < 0. PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS 11 (iv) If γ1 + γ2 > 0 and sign(A) = sign(µ2 ), then the equilibrium point (A, F (A)) is asymptotically stable, and it is surrounded by a unique unstable limit cycle. (v) If γ1 + γ2 6 0 and sign(A) = sign(µ2 ), then the equilibrium point (A, F (A)) is global attractor and no limit cycle exist. (vi) If γ1 + γ2 > 0 and sign(A) = sign(µ1 ), then the equilibrium point (A, F (A)) is a global repeller and no limit cycle exist. (vii) If γ1 + γ2 < 0 and sign(A) = sign(µ1 ), then the equilibrium point (A, F (A)) is unstable, and it is surrounded by a unique stable limit cycle. Proof of Proposition 6. (I) The characteristic polynomial of system (3) at its equi- librium point is λ2 + k1 λ + ε. Recall that ε > 0, then statements (I) follows easily. √ √ (II) Let u = εx, v = −y and τ = εt, then system (3) can be written in the √ k1 k2 canonical form (19) with A = εa, B = 0 and µ1 = − √ , µ2 = − √ . It is easy to ε ε compute that 4ε − k12 k1 D1 = , γ1 = − p , 4ε 4ε − k12 (22) 4ε − k22 k2 D2 = , γ2 = − p . 4ε 4ε − k22 Recall that a = 0, k2 > 0 and ε > 0 sufficiently small. If ∆1 < 0 and ∆2 < 0, then D1 > 0 and D2 > 0. First we consider the case ∆1 < 0, ∆2 < 0. If k1 < 0, k2 > 0 and k1 + k2 < 0, then γ1 + γ2 > 0. From statement (i) of Lemma 7, we can conclude that (0, 0) is an unstable focus. This is the proof of statement (II.1) of proposition 6. If k1 < 0, k2 < 0 and k1 + k2 = 0, then γ1 + γ2 = 0. From statement (ii) of Lemma 7, (0, 0) is a global center. This is the proof of statement (II.2) of Proposition 6. If k1 < 0, k2 > 0, k1 + k2 > 0 or k1 > 0, k2 > 0, then γ1 + γ2 < 0. From statement (iii) of Lemma 7, (0, 0) is a stable focus. This is the proof of statement (II.3) and (II.7) of Proposition 6. In the following we discuss the other cases. Since the singular point (0, 0) is located in the switching line x = 0 for a = 0, the type and stability of (0, 0) depends on both differential systems, the one in x > 0 and the other in x < 0. (II.4) For the case k1 < 0, k2 > 0 and ∆1 > 0, ∆2 < 0, then (0, 0) is an unstable node in the right zone and a stable focus in the left zone, thus (0, 0) is an unstable focus, see Figure 4.1. (II.5) For the case k1 < 0, k2 > 0 and ∆1 < 0, ∆2 > 0, then (0, 0) is an unstable focus in the right zone and a stable node in the left zone, thus (0, 0) is a stable node, see Figure 4.2. (II.6) For the case k1 < 0, k2 > 0 and ∆1 > 0, ∆2 > 0, thus (0, 0) is formed by an elliptic sector and a hyperbolic sector, see Figure 4.3. (II.8) For the case k1 > 0, k2 > 0 and either ∆1 > 0 or ∆2 > 0, then (0, 0) is a stable node, see Figure 4.4. 12 SHIMIN LI AND JAUME LLIBRE (III) The characteristic polynomial of systems (3) is λ2 + k2 λ + ε. Note that ε > 0 and k2 > 0, then statements (III) follows easily.  3. Global phase portraits In this section we classify the global phase portraits of system (3) with k1 k2 6= 0. 3.1. Global phase portraits of (3) with k1 > 0 and k2 > 0. First we consider the case a > 0. Proposition 8. The phase portrait of system (3) with k1 > 0, k2 > 0 and a > 0 is topologically equivalent to (I) Figure 1.1 if ∆1 < 0 and ∆2 < 0; (II) Figure 1.2 if ∆1 = 0 and ∆2 < 0; (III) Figure 1.3 if ∆1 > 0 and ∆2 < 0; (IV) Figure 1.4 if ∆1 < 0 and ∆2 = 0; (V) Figure 1.5 if ∆1 = 0 and ∆2 = 0; (VI) Figure 1.6 if ∆1 > 0 and ∆2 = 0; (VII) Figure 1.7 if ∆1 < 0 and ∆2 > 0; (VIII) Figure 1.8 if ∆1 = 0 and ∆2 > 0; (XI) Figure 1.9 if ∆1 > 0 and ∆2 > 0. Proof. Recall that if a > 0, k1 > 0 and k2 > 0, then system (3) has no limit cycles according to Proposition 2 of [10]. (I) For the case ∆1 < 0 and ∆2 < 0 the finite singular point (a, k1 a) is a stable focus by statement (I.1) of Proposition 6, and by statement (I) of Propositions 4 and 5 the system has no infinite singular points. Therefore the global phase portrait is topologically equivalent to Figure 1.1. (II) For the case ∆1 = 0 and ∆2 < 0 the finite singular point (a, k1 a) is a stable node by statement (I.2) of Proposition 6. The infinite singular point E is a saddle- node by statement (II) of Proposition 4. Consequently the global phase portrait is topologically equivalent to Figure 1.2. (III) For the case ∆1 > 0 and ∆2 < 0 the finite singular point (a, k1 a) is a stable node by statement (I.2) of Proposition 6. The infinite singular points E+ is a saddle and E− is an unstable node by statement (III) of Proposition 4. So the global phase portrait is topologically equivalent to Figure 1.3. (IV) For the case ∆1 < 0 and ∆2 = 0 the finite singular point (a, k1 a) is a stable focus by statement (I.1) of Proposition 6. The infinite singular point Ẽ is a saddle-node by statement (II) of Proposition 5. Hence the global phase portrait is topologically equivalent to Figure 1.4. (V) For the case ∆1 = 0 and ∆2 = 0 the finite singular point (a, k1 a) is a stable node by statement (I.2) of Proposition 6. The infinite singular points E and Ẽ are PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS 13 two saddle-nodes by statement (II) of Propositions 4 and 5. Then the global phase portrait is topologically equivalent to Figure 1.5. (VI) For the case ∆1 > 0 and ∆2 = 0 the finite singular point (a, k1 a) is a stable node by statement (I.2) of Proposition 6. The infinite singular points E+ is a saddle and E− is an unstable node by statement (III) of Proposition 4, and the infinite singular point Ẽ is a saddle-nodes by statement (II) of Proposition 5. Therefore the global phase portrait is topologically equivalent to Figure 1.6. (VII) For the case ∆1 < 0 and ∆2 > 0 the finite singular point (a, k1 a) is a stable focus by statement (I.1) of Proposition 6. The infinite singular point Ẽ+ is a saddle, and the other infinite singular point Ẽ− is an unstable node by statement (III) of Proposition 5. So the global phase portrait is topologically equivalent to Figure 1.7. (VIII) For the case ∆1 = 0 and ∆2 > 0 the finite singular point (a, k1 a) is a stable node by statement (I.2) of Proposition 6. The infinite singular point E is a saddle-node by statement (II) of Proposition 4. The infinite singular point Ẽ+ is a saddle, and the other infinite singular point Ẽ− is an unstable node by statement (III) of Proposition 5. Hence the global phase portrait is topologically equivalent to Figure 1.8. (IX) For the case ∆1 > 0 and ∆2 > 0 the finite singular point (a, k1 a) is a stable node by statement (I.2) of Proposition 6. The infinite singular point E+ is a saddle and E− is an unstable node by statement (III) of Proposition 4. The infinite singular point Ẽ+ is a saddle and Ẽ− is an unstable node by statement (III) of Proposition 5. Consequently the global phase portrait is topologically equivalent to Figure 1.9.  Second we consider the case a = 0. Proposition 9. The phase portraits of system (3) with k1 > 0, k2 > 0 and a = 0 is topologically equivalent to (I) Figure 1.10 if ∆1 < 0 and ∆2 < 0; (II) Figure 1.11 if ∆1 = 0 and ∆2 < 0; (III) Figure 1.12 if ∆1 > 0 and ∆2 < 0; (IV) Figure 1.11 if ∆1 < 0 and ∆2 = 0; (V) Figure 1.13 if ∆1 = 0 and ∆2 = 0; (VI) Figure 1.14 if ∆1 > 0 and ∆2 = 0; (VII) Figure 1.12 if ∆1 < 0 and ∆2 > 0; (VIII) Figure 1.14 if ∆1 = 0 and ∆2 > 0; (IX) Figure 1.15 if ∆1 > 0 and ∆2 > 0. Proof. Recall that k1 > 0 and k2 > 0 and a = 0. (I) For the case ∆1 < 0 and ∆2 < 0 the finite singular point (0, 0) is a stable focus by statement (II.7) of Proposition 6. There is no infinite singular points by 14 SHIMIN LI AND JAUME LLIBRE statement (I) of Propositions 4 and 5. So the global phase portrait is topologically equivalent to Figure 1.10. (II) For the case ∆1 = 0 and ∆2 < 0 the finite singular point (0, 0) is a stable node by statement (II.8) of Proposition 6. The infinite singular point E is a saddle- node by statement (II) of Proposition 4. Therefore the global phase portrait is topologically equivalent to Figure 1.11. (III) For the case ∆1 > 0 and ∆2 < 0 the finite singular point (0, 0) is a stable node by statement (II.8) of Proposition 6. The infinite singular point E+ is a saddle and E− is an unstable node by statement (III) of Proposition 4. Hence the global phase portrait is topologically equivalent to Figure 1.12. (IV) For the case ∆1 < 0 and ∆2 = 0 the finite singular point (0, 0) is a stable node by statement (II.8) of Proposition 6. The infinite singular point Ẽ is a saddle- node by statement (II) of Proposition 5. Consequently the global phase portrait is topologically equivalent to Figure 1.11 by the homeomorphism h : (x, y) → (−X, −Y ). (V) For the case ∆1 = 0 and ∆2 = 0 the finite singular point (0, 0) is a stable node by statement (II.8) of Proposition 6. The infinite singular points E and Ẽ are saddle-nodes by statement (II) of Propositions 4 and 5, respectively. Then the global phase portrait is topologically equivalent to Figure 1.13. (VI) For the case ∆1 > 0 and ∆2 = 0 the finite singular point (0, 0) is a stable node by statement (II.8) of Proposition 6. The infinite singular point E+ is a saddle, E− is an unstable node by statement (III) of Proposition 4, and Ẽ is a saddle-node by statement (II) of Proposition 5. So the global phase portrait is topologically equivalent to Figure 1.14. (VII) For the case ∆1 < 0 and ∆2 > 0 the finite singular point (0, 0) is a stable node by statement (II.8) of Proposition 6. The infinite singular point Ẽ+ is a saddle, and E− is an unstable node by statement (III) of Proposition 5. Therefore the global phase portrait is topologically equivalent to Figure 1.12. (VIII) For the case ∆1 = 0 and ∆2 > 0 the finite singular point (0, 0) is a stable node by statement (II.8) of Proposition 6. The infinite singular point E is a saddle- node by statement (II) of Proposition 4, Ẽ+ is a saddle, and Ẽ− is an unstable node by statement (III) of Proposition 5. Consequently the global phase portrait is topologically equivalent to Figure 1.14. (IX) For the case ∆1 > 0 and ∆2 > 0 the finite singular point (0, 0) is a stable node by statement (II.8) of Proposition 6. The infinite singular point E+ is a saddle, E− is an unstable node by statement (III) of Proposition 4, Ẽ+ is a saddle, and Ẽ− is an unstable node by statement (III) of Proposition 5. Hence the global phase portrait is topologically equivalent to Figure 1.15.  Finally, we consider the case a < 0. Proposition 10. The phase portraits of system (3) with k1 > 0, k2 > 0 and a < 0 is topologically equivalent to (I) Figure 1.1 if ∆1 < 0 and ∆2 < 0; PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS 15 (II) Figure 1.4 if ∆1 = 0 and ∆2 < 0; (III) Figure 1.7 if ∆1 > 0 and ∆2 < 0; (IV) Figure 1.2 if ∆1 < 0 and ∆2 = 0; (V) Figure 1.5 if ∆1 = 0 and ∆2 = 0; (VI) Figure 1.8 if ∆1 > 0 and ∆2 = 0; (VII) Figure 1.3 if ∆1 < 0 and ∆2 > 0; (VIII) Figure 1.8 if ∆1 = 0 and ∆2 > 0; (XI) Figure 1.9 if ∆1 > 0 and ∆2 > 0. Proof. The proof of Proposition 10 is similar to the one of Proposition 8, and we omit it here.  3.2. Global phase portraits of (3) with k1 < 0 and k2 > 0. First we consider the case a > 0. Proposition 11. The phase portraits of system (3) with k1 < 0, k2 > 0 and a > 0 is topologically equivalent to (I) Figure 2.1 if ∆1 < 0, ∆2 < 0 and k1 + k2 > 0; (II) Figure 2.2 if ∆1 < 0, ∆2 < 0 and k1 + k2 6 0; (III) Figure 2.3 if ∆1 = 0 and ∆2 < 0; (IV) Figure 2.4 if ∆1 > 0 and ∆2 < 0; (V) Figure 2.5 if ∆1 < 0 and ∆2 = 0; (VI) Figure 2.6 if ∆1 = 0 and ∆2 = 0; (VII) Figure 2.7 if ∆1 > 0 and ∆2 = 0; (VIII) Figure 2.8 if ∆1 < 0 and ∆2 > 0; (IX) Figure 2.9 if ∆1 = 0 and ∆2 > 0; (X) Figure 2.10 if ∆1 > 0 and ∆2 > 0. Proof. Recall that a > 0, k1 < 0 and k2 > 0. According to statement (a) of Proposition 1 and Lemma 1 of [23], if k1 k2 < 0 then system has at most one limit cycle. (I) For the case ∆1 < 0, ∆2 < 0 and k1 + k2 > 0, we have µ1 µ2 < 0, D1 > 0, D2 > 0, γ1 + γ2 < 0 and sign(A) = sign(µ1 ). By statement (vii) of Lemma 7 the finite singular point (a, k1 a) is an unstable focus and there is a unique sta- ble limit cycle surround it. It is obvious that system (3) has no infinite singular points by statement (I) of Propositions 4 and 5. Then the global phase portrait is topologically equivalent to Figure 2.1. (II) For the case ∆1 < 0, ∆2 < 0 and k1 + k2 6 0, we have µ1 µ2 < 0, D1 > 0, D2 > 0, γ1 + γ2 > 0 and sign(A) = sign(µ1 ). According to statement (vi) of Lemma 7, the finite singular point (a, k1 a) is an unstable focus and there is 16 SHIMIN LI AND JAUME LLIBRE no limit cycles. It is obvious that system (3) has no infinite singular points by statement (I) of Propositions 4 and 5. So the global phase portrait is topologically equivalent to Figure 2.2. (III) For the case ∆1 = 0 and ∆2 < 0, then the finite singular point (a, k1 a) is an unstable node by (I.2) of Proposition 6, and the infinite singular point E is a saddle-node by statement (II) of Proposition 4. Hence the global phase portrait is topologically equivalent to Figure 2.3. (IV) For the case ∆1 > 0 and ∆2 < 0, then the finite singular point (a, k1 a) is an unstable node by (I.2) of Proposition 6, and the infinite singular point E+ is a stable node, and E− is a saddle by statement (III) of Proposition 4. Consequently the global phase portrait is topologically equivalent to Figure 2.4. (V) For the case ∆1 < 0 and ∆2 = 0, then the finite singular point (a, k1 a) is an unstable focus by (I.1) of Proposition 6, and the infinite singular point Ẽ is a saddle-node by statement (II) of Proposition 5. Therefore the global phase portrait is topologically equivalent to Figure 2.5. (VI) For the case ∆1 = 0 and ∆2 = 0, then the finite singular point (a, k1 a) is an unstable node by (I.2) of Proposition 6, and the infinite singular points E and Ẽ are saddle-nodes by statement (II) of Propositions 4 and 5. So the global phase portrait is topologically equivalent to Figure 2.6. (VII) For the case ∆1 > 0 and ∆2 = 0, then the finite singular point (a, k1 a) is an unstable node by (I.2) of Proposition 6, and the infinite singular pointsE+ is a stable node, E− is a saddle by statement (III) of Proposition 4, and Ẽ is a saddle-node by statement (II) of Proposition 5. Therefore the global phase portrait is topologically equivalent to Figure 2.7. (VIII) For the case ∆1 < 0 and ∆2 > 0, then the finite singular point (a, k1 a) is an unstable focus by (I.1) of Proposition 6, and the infinite singular point Ẽ+ is a saddle, and Ẽ− is an unstable node by statement (III) of Proposition 5. Hence the global phase portrait is topologically equivalent to Figure 2.8. (IX) For the case ∆1 = 0 and ∆2 > 0, then the finite singular point (a, k1 a) is an unstable node by (I.2) of Proposition 6, and the infinite singular point E is a saddle-node by statement (II) of Proposition 4, Ẽ+ is a saddle, and Ẽ− is an unstable node by statement (III) of Proposition 5. Consequently the global phase portrait is topologically equivalent to Figure 2.9. (X) For the case ∆1 > 0 and ∆2 > 0, then the finite singular point (a, k1 a) is an unstable node by (I.2) of Proposition 6, and the infinite singular point E+ is a stable node, E− is a saddle by statement (III) of Proposition 4, Ẽ+ is a saddle, and Ẽ− is an unstable node by statement (III) of Proposition 5. Hence the global phase portrait is topologically equivalent to Figure 2.10.  Second we consider the case a = 0. Proposition 12. The phase portraits of system (3) with k1 < 0, k2 > 0 and a = 0 is topologically equivalent to (I) Figure 2.11 if ∆1 < 0, ∆2 < 0 and k1 + k2 6= 0; PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS 17 (II) Figure 2.12 if ∆1 < 0, ∆2 < 0 and k1 + k2 = 0; (III) Figure 2.13 if ∆1 = 0 and ∆2 < 0; (IV) Figure 1.14 if ∆1 > 0 and ∆2 < 0; (V) Figure 2.13 if ∆1 < 0 and ∆2 = 0; (VI) Figure 2.15 if ∆1 = 0 and ∆2 = 0; (VII) Figure 2.16 if ∆1 > 0 and ∆2 = 0; (VIII) Figure 2.14 if ∆1 < 0 and ∆2 > 0; (IX) Figure 2.16 if ∆1 = 0 and ∆2 > 0; (X) Figure 2.17 if ∆1 > 0 and ∆2 > 0. Proof. Recall that a = 0, k1 < 0 and k2 > 0. (I) Assume ∆1 < 0, ∆2 < 0 and k1 + k2 6= 0. If k1 + k2 > 0, then the finite singular point (0, 0) is a global stable focus by statement (II.1) of Proposition 6. If k1 + k2 < 0, then the finite singular point (0, 0) is a global unstable focus by statement (II.3) of Proposition 6. It is obvious that system (3) has no infinite singular point by statement (I) of Propositions 4 and 5. Then the global phase portrait is topologically equivalent to Figure 2.11. (II) Assume ∆1 < 0, ∆2 < 0 and k1 + k2 = 0. The finite singular point (0, 0) is a global center by statement (II.2) of Proposition 6. It is obvious that system (3) has no infinite singular points by statement (I) of Proposition 4 and 5. Hence the global phase portrait is topologically equivalent to Figure 2.12. (III) For the case ∆1 = 0 and ∆2 < 0 the finite singular point (0, 0) is an unstable node by statement (II.4) of Proposition 6. The infinite singular point E is a saddle-node by statement (II) of Proposition 4. Consequently the global phase portrait is topologically equivalent to Figure 2.13. (IV) For the case ∆1 > 0 and ∆2 < 0 the finite singular point (0, 0) is an unstable node by statement (II.4) of Proposition 6. The infinite singular point E+ is a stable node, and E− is a saddle by statement (III) of Proposition 4. So the global phase portrait is topologically equivalent to Figure 2.14. (V) For the case ∆1 < 0 and ∆2 = 0 the finite singular point (0, 0) is a stable node by statement (II.5) of Proposition 6. The infinite singular point Ẽ is a saddle- node by statement (II) of Proposition 5. Therefore the global phase portrait is topologically equivalent to Figure 2.13. (VI) For the case ∆1 = 0 and ∆2 = 0 the finite singular point (0, 0) is formed by an elliptic sector and a hyperbolic sector by statement (II.6) of Proposition 6. The infinite singular points E and Ẽ are saddle-node by statement (II) of Propositions 4 and 5. Consequently the global phase portrait is topologically equivalent to Figure 2.15. (VII) For the case ∆1 > 0 and ∆2 = 0 the finite singular point (0, 0) is formed by an elliptic sector and a hyperbolic sector by statement (II.6) of Proposition 6. The infinite singular point E+ is a stable node, E− is a saddle by statement (III) 18 SHIMIN LI AND JAUME LLIBRE of Proposition 4, and Ẽ is a saddle-node by statement (II) of Proposition 5. Hence the global phase portrait is topologically equivalent to Figure 2.16. (VIII) For the case ∆1 < 0 and ∆2 > 0 the finite singular point (0, 0) is a stable node by statement (II.5) of Proposition 6. The infinite singular point Ẽ+ is a saddle, and Ẽ− is an unstable node by statement (III) of Proposition 5. Therefore the global phase portrait is topologically equivalent to Figure 2.14. (IX) For the case ∆1 = 0 and ∆2 > 0 the finite singular point (0, 0) is formed by an elliptic sector and a hyperbolic sector by statement (II.6) of Proposition 6. The infinite singular point E is a saddle-node by statement (II) of Proposition 4, Ẽ+ is a saddle, and Ẽ− is an unstable node by statement (III) of Proposition 5. So the global phase portrait is topologically equivalent to Figure 2.16. (XI) For the case ∆1 > 0 and ∆2 > 0 the finite singular point (0, 0) is formed by an elliptic sector and a hyperbolic sector by statement (II.6) of Proposition 6. The infinite singular point E+ is a stable node, E− is a saddle by statement (III) of Proposition 4, Ẽ+ is a saddle, and Ẽ− is an unstable node by statement (III) of Proposition 5. Hence the global phase portrait is topologically equivalent to Figure 2.17.  Finally we consider the case a < 0. Proposition 13. The phase portraits of system (3) with k1 > 0, k2 > 0 and a < 0 is topologically equivalent to (I) Figure 2.2 if ∆1 < 0, ∆2 < 0 and k1 + k2 > 0; (II) Figure 2.1 if ∆1 < 0, ∆2 < 0 and k1 + k2 < 0; (III) Figure 2.5 if ∆1 = 0 and ∆2 < 0; (IV) Figure 2.8 if ∆1 > 0 and ∆2 < 0; (V) Figure 2.18 if ∆1 < 0 and ∆2 = 0; (VI) Figure 2.6 if ∆1 = 0 and ∆2 = 0; (VII) Figure 2.9 if ∆1 > 0 and ∆2 = 0; (VIII) Figure 2.4 if ∆1 < 0 and ∆2 > 0; (IX) Figure 2.7 if ∆1 = 0 and ∆2 > 0; (X) Figure 2.10 if ∆1 > 0 and ∆2 > 0. Proof. Recall that a < 0, k1 < 0 and k2 > 0. (I) For the case ∆1 < 0, ∆2 < 0 and k1 +k2 > 0, we have µ1 µ2 < 0, D1 > 0, D2 > 0, γ1 + γ2 6 0 and sign(A) = sign(µ2 ). By statement (v) of Lemma 7 the finite singular point (a, k2 a) is a stable focus and there is no limit cycles. It is obvious that system (3) has no infinite singular points by statement (I) of Propositions 4 and 5. Therefore the global phase portrait is topologically equivalent to Figure 2.2. (II) For the case ∆1 < 0, ∆2 < 0 and k1 + k2 < 0, we have µ1 µ2 < 0, D1 > 0, D2 > 0, γ1 + γ2 > 0 and sign(A) = sign(µ2 ). Form statement (iv) of Lemma 7 the finite singular point (a, k2 a) is a stable focus and there is a unique PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS 19 unstable limit cycle surround it. It is obvious that system (3) has no infinite singu- lar points by statement (I) of Propositions 4 and 5. Then the global phase portrait is topologically equivalent to Figure 2.1. (III) For the case ∆1 = 0 and ∆2 < 0 the finite singular point (a, k2 a) is a stable focus by (III.1) of Proposition 6. The infinite singular point E is a saddle-node by statement (II) of Proposition 4. According to the Poincaré-Bendixson Theorem, there is at least one unstable limit cycle surround the singular point (a, k2 a), and we know that when a limit cycle exists it is unique. Hence the global phase portrait is topologically equivalent to Figure 2.5. (IV) For the case ∆1 > 0 and ∆2 < 0 the finite singular point (a, k2 a) is a stable focus by (III.1) of Proposition 6. The infinite singular point E+ is a stable node, E− is a saddle by statement (III) of Proposition 4. By the Poincaré-Bendixson Theorem there is a limit cycle surround the singular point (a, k2 a) , and we know that when a limit cycle exists is unique, and consequently unstable. So the global phase portrait is topologically equivalent to Figure 2.8. (V) For the case ∆1 < 0 and ∆2 = 0 the finite singular point (a, k2 a) is a stable node by (III.2) of Proposition 6. The infinite singular point Ẽ is a saddle-node by statement (II) of Proposition 5. So the global phase portrait is topologically equivalent to Figure 2.18. (VI) For the case ∆1 = 0 and ∆2 = 0 the finite singular point (a, k2 a) is a stable node by (III.2) of Proposition 6. The infinite singular points E and Ẽ are saddle-nodes by statement (II) of Propositions 4 and 5, respectively. Then the global phase portrait is topologically equivalent to Figure 2.6. (VII) For the case ∆1 > 0 and ∆2 = 0 the finite singular point (a, k2 a) is a stable node by (III.2) of Proposition 6. The infinite singular point E+ is a stable node, E− is a saddle by statement (III) of Proposition 4, and Ẽ is a saddle-node by statement (II) of Proposition 5. Therefore the global phase portrait is topologically equivalent to Figure 2.9. (VIII) For the case ∆1 < 0 and ∆2 > 0 the finite singular point (a, k2 a) is a stable node by (III.2) of Proposition 6, and the infinite singular point Ẽ+ is a saddle, and Ẽ− is an unstable node by statement (III) of Proposition 5. Consequently the global phase portrait is topologically equivalent to Figure 2.4. (IX) For the case ∆1 = 0 and ∆2 > 0 the finite singular point (a, k2 a) is a stable node by (III.2) of Proposition 6. The infinite singular points E is a saddle-node by statement (II) of Proposition 4, Ẽ+ is a saddle, and Ẽ− is an unstable node by statement (III) of Proposition 5. Hence the global phase portrait is topologically equivalent to Figure 2.7. (X) For the case ∆1 > 0 and ∆2 > 0 the finite singular point (a, k2 a) is an unstable node by (III.2) of Proposition 6. The infinite singular point E+ is a stable node, E− is a saddle by statement (III) of Proposition 4, Ẽ+ is a saddle, and Ẽ− is an unstable node by statement (III) of Proposition 5. So the global phase portrait is topologically equivalent to Figure 2.10.  20 SHIMIN LI AND JAUME LLIBRE 4. Remarks In this paper we classify the global phase portraits for the class of piecewise linear continuous differential system (3) separated by the straight line x = 0. We can find three main differences between the smooth and the piecewise smooth differential systems when we analyze the global phase portraits. First, for the smooth differential systems the expressions for the charts Vi , i = 1, 2, 3 are those for the charts Ui multiplied by (−1)d−1 , where d is the degree of the polynomial differential systems. This symmetry property in general does not hold for piecewise smooth differential systems because the expressions for charts Vi , i = 1, 2, 3 are different from the expressions of Ui multiplied by (−1)d−1 , see for instance (11) and (16). Second, for the smooth differential systems if s ∈ S1 is an infinite singular point, then −s ∈ S1 is another infinite singular point. Thus the number of infinite singular points is even and the local behaviour of one is that of the other multiplied by (−1)d−1 . But for the piecewise smooth differential systems in general the singular points at infinity are not diametrally opposite, see Figure 1.2 for example. Finally, when a finite singular point of the piecewise smooth differential system is located at the separating line x = 0, the analysis of its local phase portrait is more complex than in the smooth differential systems. 5. Acknowledgements The first author is partially supported by the Natural Science Foundation of Guangdong Province (2017A030313010), National Natural Science Foundations of China (No. 11401111), China Scholarship Council (No. 201608440447) and Science and Technology Program of Guangzhou (No. 201707010426). The second author is partially supported by the MINECO grants MTM2013- 40998-P and MTM2016-77278-P (FEDER), the AGAUR grant 2014 SGR568, and the projecte MDM-2014-0445 (BGSMath). 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