PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS WITH TWO ZONES BY A STRAIGHT LINE

. This paper provides the classiﬁcation of the phase portraits in the Poincar´e 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.


Introduction and statement of the main results
The Liénard second order differential equation 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 equivalent planar differential system (2) dx dt = y − F (x), 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 system 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 directions. On one hand, piecewise linear differential systems are the most natural extensions to the nonlinear differential systems in order to capture nonlinear phenomena 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, especially 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 investigate the global dynamics of the planar piecewise linear continuous slow-fast differential systems of the form where f (x) is a continuous piecewise linear function given by with ε > 0 sufficiently small and k 1 k 2 = 0.
System (3) is the most simplicity piecewise linear continuous slow-fast differential system and has been studied in the papers [7,33,34]. From the results of [33], we can conclude that if k 1 = −k 2 , then system (3) has neither maximal nor faux maximal canard orbits exist. In 2016 Roberts [34] proved that if k 1 > 0 and k 2 < 0, then system (3) can have an unstable canard limit cycle. He also investigated the mechanism of canard explosive. If k 1 = k 2 , 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 D 2 , Σ is a closed set and the component of 1}. We denote by S and R the number of separatrices and canonical regions, respectively.
We say that two phase portraits of X 1 and X 2 of systems (3) are topologically equivalent if there exists a homeomorphism h : , and h mapping orbits of X 1 into orbits of X 2 either preserving the orientation, or reversing the orientation of all orbits.
We denote Theorem 1. The phase portrait on the Poincaré disc of piecewise linear continuous differential system (3) with k 1 > 0, k 2 > 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. Theorem 2. The phase portrait on the Poincaré disc of piecewise linear continuous differential system (3) with k 1 < 0, k 2 > 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.
Note that statements (I) and (IV) of Corollary 3 can be obtained from Proposition 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.

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 S 2 be the set of points s= (s 1 , s 2 , s 3 ) ∈ R 3 such that s 2 1 + s 2 2 + s 2 3 = 1. We will call this set the Poincaré sphere. Given a polynomial differential system of degree d = max{deg(P ), deg(Q)} in R 2 identified with the plane x 3 = 1 of R 3 , it can be extended analytically to the Poincaré sphere by projecting each point (x 1 , x 2 , 1) ∈ R 3 onto the Poincaré sphere using a straight line through (x 1 , x 2 , 1) PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS 5 Figure 2. Topological phase portraits of system (3) with k 1 < 0, k 2 > 0 in Theorem 2. and the origin of R 3 . In this way we obtain a new differential system formed by two copies of (6): one on the northern hemisphere S − = {(s 1 , s 2 , s 3 ) ∈ S 2 , s 3 > 0} and another on the southern hemisphere S + = {(s 1 , s 2 , s 3 ) ∈ S 2 , s 3 < 0} . Note that the equator S 1 = {(s 1 , s 2 , s 3 ) ∈ S 2 , s 3 = 0} corresponds to the infinity of R 2 .
The local charts needed for doing the calculations on the Poincaré sphere are where s = (s 1 , s 2 , s 3 ).
The expression for the corresponding differential system on S 2 in the local chart U 1 is given by with v > 0.
The expression for the corresponding differential system on S 2 in the local chart U 2 is given by with v > 0.
The expression for the corresponding differential system on S 2 in the local chart V j is the same than in the chart U j multiplied by (−1) d , for j = 1, 2.
The expression for the corresponding differential system on S 2 in the local charts U 3 and V 3 are just We note that to study the differential system (6), it is enough to study its Poincaré compactification restricted to the northern hemisphere plus S 1 . To draw the phase portraits we will consider the orthogonal projection π(s 1 , s 2 , s 3 ) = (s 1 , s 2 ) of the northern hemisphere onto the closed unit disc centered at the origin of coordinates in the plane x 3 = 0, called the Poincar disc.
Finite singular points of (6) are the singular points of its compactification which are in S 2 \ S 1 , and they can be studied using U 3 . Infinite singular points of (6) are the singular points of the corresponding differential system on the Poincaré disc lying on S 1 . Note that for studying the infinite singular points it suffices to look the ones at U 1 | v=0 , V 1 | v=0 , and the origins of U 2 and V 2 .
We note that the coordinates (u, v) means different things in every local charts, but in the local charts U i and V i for i = 1, 2 the infinite points have always the coordinate v = 0.
(III) If ∆ 1 > 0, then system (11) has two infinite singular points which is a saddle for k 1 > 0 and a stable node for , 0 , which is an unstable node for k 1 > 0, and a saddle for Proof. The singular points should satisfy the equation u 2 − k 1 u + ε = 0. The statement (I) is obvious.
We first consider the case According to the Theorem 2.19 of the book [8], we can deduce that E is a saddlenode 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.

SHIMIN LI AND JAUME LLIBRE
The phase portraits of E described in Figure 3.2 can be analyzed similarly for the case k 1 > 0.
(III) The Jacobian matrix for singular point The Jacobian matrix for singular point It is worth to note that system (16) is similar to (11), and just substitute k 1 of system (11) for k 2 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.
which is a saddle for k 2 > 0 and a stable node for , 0 , which is an unstable node for k 2 > 0 and a saddle for

PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS 9
with u 0, and with u 0.
Since (0, 0) is neither a singular point of system (17) nor a singular point of system (18), the origins of U 2 and V 2 are not infinite singular points.
2.5. Chart U 3 . 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.
is an unstable node, see 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 where (20) F (u) = µ 1 u u > 0, We introduce the following parameters which will simplify our analysis 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 D 1 > 0 and D 2 > 0.

PHASE PORTRAITS OF PIECEWISE LINEAR CONTINUOUS DIFFERENTIAL SYSTEMS 11
(iv) If γ 1 + γ 2 > 0 and sign(A) = sign(µ 2 ), then the equilibrium point is asymptotically stable, and it is surrounded by a unique unstable limit cycle.
is global attractor and no limit cycle exist.
is a global repeller and no limit cycle exist.
is unstable, and it is surrounded by a unique stable limit cycle.
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.

Global phase portraits
In this section we classify the global phase portraits of system (3) with k 1 k 2 = 0.
3.1. Global phase portraits of (3) with k 1 > 0 and k 2 > 0. First we consider the case a > 0. (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 pointẼ + 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 saddlenode by statement (II) of Proposition 4,Ẽ + is a saddle, andẼ − 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, andẼ − 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. Proof. Recall that a > 0, k 1 < 0 and k 2 > 0. According to statement (a) of Proposition 1 and Lemma 1 of [23], if k 1 k 2 < 0 then system has at most one limit cycle.
(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, andẼ − 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, andẼ − is an unstable node by statement (III) of Proposition 5. Hence the global phase portrait is topologically equivalent to Figure  2.17.
(I) For the case ∆ 1 < 0, ∆ 2 < 0 and k 1 +k 2 0, we have µ 1 µ 2 < 0, D 1 > 0, D 2 > 0, γ 1 + γ 2 0 and sign(A) = sign(µ 2 ). By statement (v) of Lemma 7 the finite singular point (a, k 2 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 (II) For the case ∆ 1 < 0, ∆ 2 < 0 and k 1 + k 2 < 0, we have µ 1 µ 2 < 0, D 1 > 0, D 2 > 0, γ 1 + γ 2 > 0 and sign(A) = sign(µ 2 ). Form statement (iv) of Lemma 7 the finite singular point (a, k 2 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)  (III) For the case ∆ 1 = 0 and ∆ 2 < 0 the finite singular point (a, k 2 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, k 2 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, k 2 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, k 2 a) , and we know that when a limit cycle exists is unique, and consequently unstable. So the global phase portrait is topologically equivalent to (VII) For the case ∆ 1 > 0 and ∆ 2 = 0 the finite singular point (a, k 2 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, andẼ 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, k 2 a) is a stable node by (III.2) of Proposition 6, and the infinite singular pointẼ + is a saddle, andẼ − is an unstable node by statement (III) of Proposition 5. Consequently the global phase portrait is topologically equivalent to Figure 2 (IX) For the case ∆ 1 = 0 and ∆ 2 > 0 the finite singular point (a, k 2 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, andẼ − 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, k 2 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, andẼ − is an unstable node by statement (III) of Proposition 5. So the global phase portrait is topologically equivalent to Figure 2.10.

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 V i , i = 1, 2, 3 are those for the charts U i 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 V i , i = 1, 2, 3 are different from the expressions of U i multiplied by (−1) d−1 , see for instance (11) and (16).
Second, for the smooth differential systems if s ∈ S 1 is an infinite singular point, then −s ∈ S 1 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.