This is a preprint of: “Phase portraits of planar piecewise linear refracting systems: Focus-Saddle case”, Shimin Li, Jaume Llibre, Nonlinear Anal. Real World Appl., vol. 56, 103153:1–11, 2020. PHASE PORTRAITS OF PLANAR PIECEWISE LINEAR REFRACTING SYSTEMS: FOCUS-SADDLE CASE SHIMIN LI1 AND JAUME LLIBRE2 Abstract. This paper deal with planar piecewise linear refracting systems with a straight line of separation. Using the Poincaré compactification, we provide the classification of the phase portraits in the Poincaré disc of piecewise linear refracting systems with focus-saddle dynamics. 1. Introduction Global phase portraits are an invaluable tool in studying the long dynamical behaviour of differential systems. They reveals information such as whether an attractor, a repellor or a limit cycle is present for a given parameter value. Hence the global phase portraits analysis is the one of most important problems in the qualitative theory of differential systems. The possibilities of topological distinct phase portraits for a general polynomial differential system are huge, it is expected that the quadratic polynomial differential systems have more than 2000 topological distinct phase portraits. As far as we know, most of known results about global phase portraits are mainly deal with smooth differential systems, see for instance [1, 2, 6, 7, 8, 9, 11, 15, 18, 25]. Many real-world systems involve a discontinuity or sudden change, such as fric- tion in mechanical systems [10] and switching in electrical circuits [29]. Smooth differential systems generally do not provide ideal mathematical models for such situations. It becomes necessary to incorporate a non-smooth component into the model. Often this yields a piecewise smooth differential systems [3, 4]. In order to state precisely our results we introduce first some notations and definitions. Let us denote by Ωr the sets of C r planar vector fields (r > 1), The set of planar piecewise smooth vector fields given by ( X(x, y), f (x, y) < 0, (1.1) Z(x, y) = Y (x, y), f (x, y) > 0, where X, Y ∈ Ωr , and the smooth function f have 0 ∈ R as a regular value (i.e. ∇f (p) 6= 0, for any p ∈ f −1 (0)). We say that p ∈ R2 is a visible (resp. invisible) equilibrium of X if X(p) = 0 and f (p) < 0 (resp. f (p) > 0). Similarly, p is a visible (resp. invisible) equilibrium of Y if Y (p) = 0 and f (p) > 0 (resp. f (p) < 0). 2010 Mathematics Subject Classification. 34C37, 34C07, 37G15. Key words and phrases. Phase portraits; Refracting systems; Limit cycle. 1 2 S. LI AND J. LLIBRE Σc Σa Σe 1.1 crossing region 1.2 attracting region 1.3 escaping region Figure 1. Definition of the vector field on Σ. Definition 1.1. The discontinuous set Σ = f −1 (0) can be divided into three open regions: (i) Crossing region Σc = {p ∈ Σ|Xf (p)Y f (p) > 0}, see Fig.1.1; (ii) Attracting region Σa = {p ∈ Σ|Xf (p) > 0, Y f (p) < 0}, see Fig.1.2; (iii) Escaping region Σe = {p ∈ Σ|Xf (p) < 0, Y f (p) > 0}, see Fig.1.3. where Xf (p) = h∇f (p), X(p)i. The boundary of the above regions are called Σ−tangential points, that is Σt = {p ∈ Σ|Xf (p)Y f (p) = 0}. The simplest tangency is the fold point, which is defined as follows. Definition 1.2. p ∈ Σ is a fold point of X if Xf (p) = 0 and X 2 f (p) 6= 0. Here X 2 f = X(Xf ). The fold is visible if X 2 f (p) > 0 and it is invisible if X 2 f (p) < 0. Analogously, a fold point p ∈ Σ of Y satisfies Y f (p) = 0 and Y 2 f (p) 6= 0, and it is visible if Y 2 f (p) < 0 and invisible if Y 2 f (p) > 0. We are interested in a special kind of discontinuous differential systems which are known as refracting systems [5, 26]. The precise definition is given as follows. Definition 1.3. If Xf (p) = Y f (p) for any p ∈ Σ, then systems (1.1) are known as refracting systems. The most simplest piecewise smooth differential systems are planar piecewise lin- ear systems with a straight line of separation. In 2012, Freire, Ponce and Torres[14] deduced planar piecewise linear systems into Liénard canonical forms   −     T −1 x 0      D− 0 − − , x < 0, ẋ y a (1.2) =  +     ẏ   T −1 x −b  − + , x > 0, D+ 0 y a where T ± and D± are denote the traces and determinants of the left (right) sub- systems, respectively. We call systems (1.2) with x < 0 (resp. x > 0) the left (resp. right) subsystems for convenience. If b = 0, a− = a+ , then systems (1.2) become continuous systems. In 1990, Lum and Chua[23] conjectured that planar piecewise linear continuous systems (1.2) have at most one limit cycle, and this conjecture was proved by Freire, etc. [13]. PHASE PORTRAITS OF PLANAR REFRACTING SYSTEMS 3 Recently, Li and Llibre [19] provided the global phase portraits in the poincaré dics of continuous systems (1.2). For the discontinuous systems (1.2), most of the known results are concerned with the lower bounds of the number of limit cycles, see [21, 22] and the references therein. According to the equilibrium of left and right subsystems (1.2), we can classify systems (1.2) into F F, F S, F N, SS, SN, N N cases, where F, S, N denote focus/center, saddle and node respectively. If b = 0, then systems (1.2) become refracting systems by Definition 1.3. Piece- wise linear refracting systems (1.2) have been studied in several papers [14, 16, 17, 26, 27, 28], all of these results show that piecewise linear refracting systems (1.2) have at most one limit cycle. In the present paper, we investigate the global phase portraits of planar piecewise linear refracting systems of the focus-saddle type. Without loss of generality, we assume that the left subsystems of (1.2) have a focus and the right subsystems of (1.2) have a saddle. We introduce some auxiliary parameters as follows: p 4D− − (T − )2 T− ωL = , γL = , 2 2ωL p (T + )2 − 4D+ T+ ωR = , γR = . 2 2ωR x̄ t̄ x̄ Do the change of variables x = , y = ȳ, t = for x < 0 and x = ,y = ωL ωL ωR t̄ ȳ, t = for x > 0, after dropping tildes, the canonical forms (1.2) with b = 0 can ωR be written as       2γL −1 x 0      γ2 + 1 0 − , x < 0, ẋ L y α L (1.3) =       ẏ   2γR −1 x 0  2 − , x > 0, γR −1 0 y αR where γL 6= 0 and |γR | < 1. We say that two phase portraits of Z1 and Z2 of systems (1.1) are topologically equivalent if there exists a homeomorphism h : D2 → D2 such that it takes orbits of Z1 onto orbits of Z2 either preserving the orientation, or reversing the orientation of all orbits; also the discontinuous sets are preserved by the homeomorphism h. Without loss of generality we assume that γL > 0, otherwise doing the change of variables X = x, Y = −y, T = −t, we change γL < 0 into the former one. Our main result is the following one: Theorem 1.4. The phase portrait on the Poincaré disc of refracting system (1.3) with γL > 0, |γR | < 1, is topologically equivalent to one of the 18 phase portraits described in Figure 2. The rest of the paper is organized as follows. In section 2 we give a preliminary introduction for poincaré compactification, which is a crucial tool to investigate the 4 S. LI AND J. LLIBRE 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 Figure 2. Global phase portraits of refracting systems (1.3) global phase portraits of our systems. In section 3 we study the number of limit cycles of systems (1.3). The proof of Theorem 1.4 will be given in section 4. PHASE PORTRAITS OF PLANAR REFRACTING SYSTEMS 5 2. Singular points 2.1. Poincaré compactification. For a given polynomial differential system dx dy (2.1) = P (x, y), = Q(x, y), dt dt of degree d = max{deg(P ), deg(Q)}. Let X = (P, Q) be the vector field associated to system (2.1). We call S2 = {s = (s1 , s2 , s3 ) ∈ R3 : s21 + s22 + s23 = 1} the Poincaré sphere. The Poincaré compactified vector field p(X ) corresponding to X is an analytic vector field induced on S2 as follows. First we take R2 as the plane in R3 defined by (x, y, 1) ∈ R3 , and then project each point (x, y, 1) in two points of the Poincaré sphere S2 using the straight line through (x, y, 1) and the origin (0, 0, 0). It is obvious that the equator S1 = {s ∈ S2 , s3 = 0} 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 S− = {s ∈ S2 : s3 > 0}, and the other in the open southern hemisphere S+ = {s ∈ S2 : s3 < 0}. 0 This vector field X on S2 \ S1 can be extended to a vector field p(X ) defined in the 0 whole S2 by multiplying X by sd3 . For studying the Poincaré sphere we use the following six local charts (2.2) Ui = {s ∈ S2 : si > 0}, Vi = {s ∈ S2 : si < 0}, i = 1, 2, 3, with the corresponding diffeomorphisms ϕk : Uk → R2 and ψk : Vk → R2 defined by ϕk (s) = −ψk (s) = (sm /sk , sn /sk ) = (u, v) for m < n and m, n 6= k. Note that the coordinates (u, v) play a different role in each local chart. The expression of p(X ) in the local chart U1 is        du u 1 u 1 dv u 1 (2.3) = v d − uP , − uQ , , = −v d+1 Q , . dt v v v v dt v v The expression of p(X ) in the local chart U2 is        du d 1 u 1 u dv 1 u (2.4) = v − uP , +Q , , = −v d+1 P , . dt v v v v dt v v In the charts Vi for i = 1, 2 the expressions of p(X ) are the same in Ui but multiplied by (−1)d−1 . The expression of p(X ) in the local charts U3 ≡ V3 are just du dv (2.5) = P (u, v), = Q(u, v). dt dt For studying the phase portrait of a polynomial differential system (2.1), we just need to study its Poincaré compactification p(X ) restricted to the closed northern hemisphere. We do the orthogonal projection π(s1 , s2 , s3 ) = (s1 , s2 ) of the closed northern hemisphere onto the Poincaré disc D2 = {s21 + s22 6 1, s3 = 0} for drawing the phase portrait. It is obvious that the finite equilibria of system (2.1) are the equilibria in the interior of D2 , and they can be studied using U3 . The infinite equilibria of system 6 S. LI AND J. LLIBRE (2.1) are the equilibria of p(X ) in the boundary of D2 . Note that for studying the infinite equilibria it suffices to look for the ones at the local charts U1 |v=0 and V1 |v=0 , and at the origin of the local charts U2 and V2 . For more details on the poincaré compactification, see Chapter 5 of [12]. 1 u 2.2. Chart U1 . Let x = , y = , v > 0, then systems (1.3) become v v  du   = u2 − 2γR u − αR v + γR 2 − 1, dt (2.6)   dv = v(u − 2γ ). R dt Since the equilibrium of systems (2.6) with v = 0 correspondence with the infinite equilibrium of systems (1.3) in chart U1 , we have the following results directly. Proposition 2.1. For systems (1.3) with |γR | < 1, then systems (1.3) have two infinite equilibrium: E1 = (γR − 1, 0) is a stable node, and E2 = (γR + 1, 0) is an unstable node. 1 u 2.3. Chart V1 . Let x = , y = , v < 0, then systems (1.3) become v v  du   = u2 − 2γL u − αL v + γL2 + 1, dt (2.7)   dv = v(u − 2γ ). L dt It is obvious that systems (1.3) have no infinite equilibrium in chart V1 because systems (2.7) have no equilibrium when v = 0. u 1 2.4. Charts U2 and V2 . Let x = , y = , then systems (1.3) become v v  du   2 = (1 − γR )u2 + αR uv + 2γR u − 1, dt (2.8)   dv = v(α v + u − γ 2 u). R R dt with uv > 0, and  du   = −(1 + γL2 )u2 + αL uv + 2γL u − 1, dt (2.9)   dv = v(α v − u − γ 2 u). L L dt with uv 6 0. Since (0, 0) is neither a singular point of system (2.8) nor a singular point of system (2.9), the origins of U2 and V2 are not infinite equilibrium. PHASE PORTRAITS OF PLANAR REFRACTING SYSTEMS 7 F ig.3.1 F ig.3.2 F ig.3.3 F ig.3.4 Figure 3. Phase portraits for boundary equilibrium of systems (1.3). 2.5. Chart U3 . Let     αL 2αL γL αR 2αR γR (2.10) PL = , , PR = 2 − 1 γ2 − 1 . , γL2 + 1 γL2 + 1 γR R Proposition 2.2. For systems (1.3) with γL > 0 and |γR | < 1 the following state- ments hold. (I) If αL < 0 and αR < 0, then systems (1.3) have two equilibrium: PL is an unstable focus; PR is a saddle. (II) If αL < 0 and αR = 0, then systems (1.3) have two equilibrium: PL is a unstable focus; O = (0, 0) is a boundary equilibrium which is known as visible fold-saddle, see Fig.3.1. (III) If αL < 0 and αR > 0, then systems (1.3) have one equilibrium PL which is an unstable focus. (IV) If αL > 0 and αR > 0, then systems (1.3) have no equilibrium. (V) If αL > 0 and αR = 0, then systems (1.3) have a boundary equilibrium O = (0, 0) which is known as invisible fold-saddle, see Fig.3.2. (VI) If αL > 0 and αR < 0, then systems (1.3) have one equilibrium PR , which is a saddle. (VII) If αL = 0 and αR < 0 then systems (1.3) have two equilibrium: PR is a saddle; O = (0, 0) is a boundary equilibrium, see Fig.3.3. (IV) If αL = 0 and αR = 0, then systems (1.3) have a boundary equilibrium O = (0, 0), see Fig.3.2. (IX) If αL = 0 and αR > 0, then systems (1.3) have a boundary equilibrium O = (0, 0), see Fig.3.4. Proof. The qualitative analysis of the visible equilibrium are directly, so we omit it here. In the following we just need to analyze the phase portraits of the boundary equilibrium O(0, 0). (II) For the case αL < 0, αR = 0. In the region x > 0, the right subsystems of (1.3) is governed by the saddle (0, 0). It has two invariant straight lines y = (γR −1)x and y = (γR + 1)x. While in the region x < 0, the dynamics of left subsystems of (1.3) are governed by a visible focus PL . From the above analysis, the phase portraits of the boundary equilibrium (0, 0) is Fig.3.1. 8 S. LI AND J. LLIBRE (V) For the case αL > 0, αR = 0. In the region x > 0, the dynamics of the origin is the same as case (II). While in the region x < 0, the dynamics of left subsystems of (1.3) are governed by an invisible focus PL . From the above analysis, the phase portraits of the boundary equilibrium (0, 0) is Fig.3.2. (VII) For the case αL = 0, αR < 0. In the region x < 0, the dynamics of left subsystems of (1.3) are governed by the focus O(0, 0). While in the region x > 0, the right subsystems of (1.3) have a visible saddle. Hence the phase portraits of the boundary equilibrium (0, 0) is Fig.3.3. (IV) For the case αL = 0, αR = 0. The phase portraits of the boundary equilib- rium (0, 0) is the same as Fig.3.2. (IX) For the case αL = 0, αR > 0. In the region x < 0, the dynamics of left subsystems of (1.3) is the same as case (VII). In the region x > 0, the right subsystems of (1.3) have an invisible saddle. Hence the phase portraits of the boundary equilibrium (0, 0) is Fig.3.4.  3. limit cycles This section devote to study the limit cycles, which is very important for the investigation of global phase portraits of systems (1.3). According to the proposition 3.7 of [14] and note that b = 0, then a necessary condition for the existence of limit cycles is γL γR < 0. In order to have limit cycles it is obvious that the saddle PR should visible, that is αR < 0. In the paper [26], the authors obtain the uniqueness of limit cycles of systems (1.3) when the finite equilibrium are visible. Theorem 3.1. Consider refracting systems (1.3) with γL > 0, γR < 0 and αL < 0, αR < 0. If we define the values q (3.1) γL θ ± 2 V± = ±e 1 + [γL + ρ(γR ± 1)] , with µR ωL (1 + γL2 ) (3.2) ρ= 2 ), µL ωR (1 − γR and q  2 (3.3) θ± = ±2 arctan 1 + [γL + ρ(γR ± 1)] ∓ [γL + ρ(γR ± 1)] . Then the following statements hold. (a) If eπγL V+ + V− < 0, then the unstable focus PL is surrounded by one stable limit cycle. (b) If eπγL V+ + V− = 0, then the unstable focus PL is surrounded by a homo- clinic orbit and there are no limit cycle. (a) If eπγL V+ + V− > 0, then the the system does not have either limit cycles or homoclinic connections. PHASE PORTRAITS OF PLANAR REFRACTING SYSTEMS 9 For the remain case αL > 0, αR < 0, we have the following result. Theorem 3.2. Assume that γL > 0, γR < 0 and αL > 0, αR < 0, then refracting systems (1.3) have at most one limit cycle. Proof. If αL > 0 then the left subsystems of (1.3) have an invisible focus, and an equilibrium on Σ when αL = 0. Do the change of variables X = 2γL x − y, Y = x, if x < 0, X = 2γR x − y, Y = x, if x > 0, then the refracting systems (1.3) become       2γL −(γL2 + 1) X αL      + if Y < 0, Ẋ 1 0 Y 0 (3.4) =  2     Ẏ   2γR −(γR − 1) X αR  + if Y > 0. 1 0 Y 0 It is easy to check that (0, 0) is the unique Σ−monodromic singularity of systems (3.4). According with Theorem 1.1 of [24], systems (3.4) have at most one limit cycle because γR γL < 0.  4. Proof of Theorem 1.4 We divide the proof of Theorem 1.4 into two cases. 4.1. γR < 0. Proposition 4.1. The phase portrait of refracting systems (1.3) with γL > 0 and γR < 0 is topologically equivalent to (I)the ones of Fig.2.1, Fig.2.2 or Fig.2.3 if αL < 0, αR < 0 ; (II)Fig.2.4 if αL < 0 and αR = 0; III)Fig.2.5 if αL < 0 and αR > 0; (IV) the ones of Fig.2.6, Fig.2.7 or Fig.2.8 if αL = 0 and αR < 0; (V) Fig.2.9 if αL = 0 and αR = 0; (VI) Fig.2.10 if αL = 0 and αR > 0; (VII) the ones of Fig.2.11, Fig.2.12, Fig.2.13, Fig2.14, Fig.2.15 or Fig2.16 if αL > 0 and αR < 0; (VIII) Fig.2.17 if αL > 0 and αR = 0; (XI) Fig.2.18 if αL > 0 and αR > 0; Proof. Proposition 2.1 shows that systems (1.3) have two infinite equilibrium: E1 is a stable node, and E2 is an unstable node. From Theorems 3.1 and 3.2 we know that systems (1.3) have at most one limit cycle. (I) For the case αL < 0 and αR < 0. According to the statement (I) of Proposi- tion 2.2, systems (1.3) have two finite equilibrium, PL is an unstable focus and PR is a saddle. If eπγL V+ + V− > 0, then systems (1.3) do not have either limit cycle or homo- clinic connections by Theorem 3.1. We know that the two stable separatrices of the 10 S. LI AND J. LLIBRE saddle PR cannot go together to the stable node E1 at infinity, because one of the unstable separatrix of the saddle PR would not has its ω−limit set. So one of the stable separatrices of the saddle PR goes to the unstable focus PL and the other goes to the unstable node E2 at infinity. Finally the two unstable separatrices of PR have their ω−limit sets at the unstable node E2 at infinity. Thus the phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.1. If eπγL V+ + V− < 0, then systems (1.3) have a stable limit cycle which surround the focus PL . Thus the stable separatrices of the saddle PR goes to the unstable node E2 at infinity, one of the unstable separatrices of the saddle PR encircle the limit cycle and the other one goes to the stable node E1 at infinity. Hence the phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.2. If eπγL V+ + V− = 0, then the unstable focus PL is surrounded by a homoclinic orbit and there are no limit cycles. We know that the one of the stable separatrix of the saddle PR and unstable separatrix must connect by continuity. The phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.3. (II) For the case αL < 0, αR = 0. Systems (1.3) have two finite equilibrium: PL is an unstable focus, and O(0, 0) is a boundary equilibrium, see Fig.3.1. It is obvious that systems (1.3) can not have limit cycles because the boundary equilibrium O(0, 0) has two invariant straight lines y = (γR ± 1)x in the region x > 0. Therefore the phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.4. (III) For the case αL < 0, αR > 0. Systems (1.3) have one finite equilibrium PL which is an unstable focus. Since αR > 0 systems (1.3) can not have limit cycles. Therefore the phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.5. (IV) For the case αL = 0, αR < 0. Systems (1.3) have two finite equilibrium: PR is a saddle, and O(0, 0) is a boundary equilibrium, see Fig.3.3. If systems (1.3) have no limit cycles, then the two unstable separatrices of saddle PR must go to the stable node E1 at the infinity since O(0, 0) is an unstable focus. The stable separatrices of the saddle PR have their α limit set at the unstable node E2 and boundary equilibrium O(0, 0) respectively. Hence the phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.6. If systems (1.3) have a unique limit cycle, its must stable and surround the boundary equilibrium O(0, 0). It is obvious that both of the stable separatrices of saddle PR have their α limit sets at the unstable node E2 . And one of the unstable separatrix of the saddle PR goes to the stable node E1 at infinity, the other one encircle the stable limit cycle. Thus the phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.7. From the above analysis, we know that the one of the stable separatrix of the saddle PR and unstable separatrix must connect by continuity. Note that limit cycle and homoclinic connection can not coexist. Therefore the phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.8. (V) For the case αL = 0, αR = 0. Systems (1.3) have a unique equilibrium O(0, 0) which is a boundary equilibrium, see Fig.3.2. It is obvious that systems (1.3) have PHASE PORTRAITS OF PLANAR REFRACTING SYSTEMS 11 no limit cycles. Hence the phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.9. (VI) For the case αL = 0, αR > 0. Systems (1.3) have a unique boundary equi- librium O(0, 0), see Fig.3.4. Note that systems (1.3) have no limit cycles because αR > 0. Thus the phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.10. (VII) For the case αL > 0, αR < 0. Systems (1.3) have a unique saddle PR . The origin O(0, 0) is a invisible two-fold singularity. If O(0, 0) is stable and systems (1.3) have no limit cycles, then both of the stable separatrices of the saddle PR have their α limit sets at the unstable node E2 at infinity, and the unstable separatricies of the saddle PR go to O(0, 0) and infinity stable node E1 , respectively. Therefore the phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.11. If O(0, 0) is stable and systems (1.3) have a unique limit cycle, then both of the unstable separatrices of the saddle PR have their ω limit sets at the unstable node E1 at infinity, and the stable separatricies of the saddle PR have their α limit sets at the infinity unstable node E2 and the unstable limit cycle, respectively. Therefore the phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.13. The phase portrait of Figure 2.12 can be obtained by the continuity of Figure 2.11 and Figure 2.13. If O(0, 0) is unstable and systems (1.3) have no limit cycles, then both of the unstable separatrices of the saddle PR have their ω limit sets at the stable node E1 at infinity, and the stable separatricies of the saddle PR go to O(0, 0) and infinity unstable node E2 , respectively. Therefore the phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.14. If O(0, 0) is unstable and systems (1.3) have a unique limit cycle, then both of the stable separatrices of the saddle PR have their α limit sets at the unstable node E2 at infinity. One of the unstable separatrix of the saddle PR goes to the infinity stable node E1 , and the other ones encircle the stable limit cycle. So the phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.16. The phase portrait of Figure 2.15 can be obtained by the continuity of Figure 2.14 and Figure 2.16. (VIII) For the case αL > 0, αR = 0. Systems (1.3) have a unique boundary equi- librium O(0, 0), see Figure 3.2. The phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.17. (IX) For the case αL > 0, αR > 0. Systems (1.3) have no finite equilibrium. The phase portrait of systems (1.3) is topologically equivalent to the one of Figure 2.18.  4.2. γR > 0. Proposition 4.2. The phase portrait of refracting systems (2.1) with γL > 0 and γR > 0 is topologically equivalent to 12 S. LI AND J. LLIBRE (I) Fig.2.1 if αL < 0 and αR < 0; (II) Fig.2.4 if αL < 0 and αR = 0; (III) Fig.2.5 if αL < 0 and αR > 0; (IV) Fig.2.6 if αL = 0 and αR < 0; (V) Fig.2.9 if αL = 0 and αR = 0; (VI) Fig.2.10 if αL = 0 and αR > 0; (VII) the ones of Fig.2.11 or Fig.2.14 if αL > 0 and αR < 0; (VIII) Fig.2.17 if αL > 0 and αR = 0; (IX) Fig.2.18 if αL > 0 and αR > 0. Proof. Recall that systems (1.3) have no limit cycles and homoclinic connections since γL γR > 0, the proof of Proposition 4.2 similar with Proposition 4.1.  5. Acknowledgements The first author is partially supported by the Natural Science Foundation of Guangdong Province (2017A030313010) and Science and Technology Program of Guangzhou (No. 201707010426). 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