The boundary focus–saddle bifurcation in planar piecewise linear systems. Application to the analysis of memristor oscillators

The focus-saddle boundary bifurcation in planar piecewise linear systems Application to the analysis of memristor oscillators Enrique Poncea, Javier Rosa, El´ısabet Velab aDepartamento de Matem´atica Aplicada II, Escuela T´ecnica Superior de Ingenier´ıa Universidad de Sevilla, Spain bDepartamento de Geometr´ıa y Topolog´ıa, Facultad de Matem´aticas Universidad de Sevilla, Spain Abstract Among the boundary equilibrium bifurcations in planar continuous piecewise linear systems with two zones separated by a straight line, the focus-saddle bifurcation corresponds with a one-parameter transition from a situation without equilibria to a configuration with two equilibria, namely a focus and a saddle point. Depending on the dynamics of the two linear systems involved, the focus can appear surrounded by a limit cycle, by a saddle-loop (homoclinic connection) or by nothing else. After introducing a criticality coefficient whose sign discriminates the different possible situations, the focus-saddle bifurcation is quantitatively characterized for the first time. The analysis requires to work in a more general framework, as is the family of planar refracting linear systems with two zones, for which a new result about existence and uniqueness of limit cycles and saddle-loops is also shown. Email addresses: [email protected] (Enrique Ponce), [email protected] (Javier Ros), [email protected] (El´ısabet Vela) This is an Accepted Manuscript of an article published by Elsevier in Nonlinear Analysis: Real World Applications, Volume 43, October 2018, available at: https://doi.org/10.1016/j.nonrwa.2018.03.011. Copyright 2018 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND The achieved results are applied to the study of oscillations in an electronic circuit involving a single memristor cell, showing rigorously the appearance of limit cycles via the focus-saddle bifurcation analyzed in the paper. Keywords: Qualitative Theory of ODEs, Piecewise linear systems, limit cycles, bifurcations 1. Introduction and statement of main results In this paper, we consider continuous piecewise linear differential systems (PWL systems, for short) with two linearity zones separated by a straight line representing the only non-smoothness manifold in the phase plane. Such systems are relevant in different application fields and have been studied by5 several authors; see [1, 2] and references therein for a thorough introduction and a revision of the state of the art in the study of these systems. Our main goal is to characterize in a precise way the focus-saddle boundary equilibrium bifurcation for such continuous PWL systems, a phenomenon leading to the creation/annihilation of two equilibrium points (a focus and10 a saddle) along with the possible appearance of a limit cycle or a saddle connection surrounding the focus, see Chapter 5 of [1] and [3, 4]. We will assume without loss of generality that the linearity regions in the phase plane are the left and right half-planes, SL={(x, y)∈R2:x < 0}, SR={(x, y)∈R2:x > 0}, separated by the straight line Σ = {(x, y)∈R2:x= 0}. We proceed by considering a specific family of systems within the more 2 general setting of discontinuous systems. Namely, we first study discontinu-15 ous PWL systems for which the two involved linear vector fields share their normal component to Σ at all the points of such discontinuity manifold. These systems are known as refracting systems and are characterized by not having any sliding segment in the discontinuity manifold Σ, see [5]. In other words, orbits arriving at Σ can be naturally extended by concatenating so-20 lutions from both sides, since all the orbits hit Σ in the so-called sewing or crossing points, see [6]. In fact, refracting systems belong to the slightly wider class of sewing systems, characterized by sharing only the sign of the normal component to the discontinuity manifold Σ, see [7]. To reduce the number of parameters in the analysis, we start by introducing a canonical form for Σ-refracting PWL systems. A general discontinuous PWL system can be written as ˙x =   ARx+bR,if x∈SR, ALx+bL,if x∈SL∪Σ, (1) where x= (x, y)∈R2is the vector of state variables, AR= (aR ij) and AL= (aL ij) are 2 ×2 constant matrices, bR= (bR 1, bR 2)T,bL= (bL 1, bL 2)T∈R2 are constant vectors, and the point denotes derivatives with respect to the time variable s. Imposing now the refracting condition on these systems, we require that for all x= (0, y)T∈Σ, with y∈R, eT 1(ARx+bR) = eT 1(ALx+bL),(2) where eT 1stands for the first unitary vector in row form. From (2) and some 3 elementary algebra, we must assume aL 12 =aR 12 and bL 1=bR 1, so that systems to be studied become ˙x = aL 11 a12 aL 21 aL 22  x+ b1 bL 2  if x∈SL∪Σ, ˙x = aR 11 a12 aR 21 aR 22  x+ b1 bR 2  if x∈SR, (3) where a12 =aL 12 =aR 12 and b1=bL 1=bR 1.25 Remark 1. Note that it is natural to assume a12 6= 0 in (3) if one wants to cope with non-elementary dynamics. Otherwise, we should have ˙x=b1for all the points in Σ, so that all the orbits would cross Σ in the same direction. Furthermore, the dynamics of the first variable would be decoupled from the second one in the whole plane, oscillations would be not possible and the30 system could not exhibit a proper two-dimensional dynamics. As done in Proposition 3.1 of [8], by assuming a12 6= 0 and applying the homeomorphism e x=h(x) given by e x= 1 0 aL 22 −a12  x− 0 b1  if x∈SL∪Σ, e x= 1 0 aR 22 −a12  x− 0 b1  if x∈SR, after dropping tildes, we transform systems (3) into the following discontin- 4 uous Li´enard canonical form for refracting systems ˙x = tL−1 dL0 x− 0 µL  if x∈SL∪Σ, ˙x = tR−1 dR0 x− 0 µR  if x∈SR, (4) where µL=a12bL 2−aL 22b1,µR=a12bR 2−aR 22b1, and t{L,R}= tr A{L,R},d{L,R}= det A{L,R}denote the traces and determinants of L- and R-matrices. Remark 2. It should be noticed that if we start from a continuous PWL vector field in (1) instead of a refracting system, then the continuity require-35 ment at Σ provides the conditions aL 22 =aR 22 and bL 2=bR 2. Thus, in such a case we obtain µL=µR, and the above canonical form turns out to be continuous. Regarding the canonical form for refracting systems (4), it is evident that our systems constitute a specific case of Filippov systems in which, excepting40 the origin, all the points in Σ belong to the so-called crossing or sewing set, not having a proper sliding set. The origin can be thought as a tangency point or a singular isolated sliding point, see [6]. When one or both of the two vector fields vanishes at the origin, such a point is a boundary equilibrium point; otherwise we can have a pseudo-equilibrium point or even a regular45 point, see [9]. Remark 3. The Li´enard canonical form for refracting systems (4) is invari- 5 ant under the following transformations: Π1: (s, x, y, tL, tR, µL, µR, dL, dR)7−→ (−s, −x, y, −tR,−tL,−µR,−µL, dR, dL) Π2: (s, x, y, tL, tR, µL, µR, dL, dR)7−→ (−s, x, −y, −tL,−tR, µL, µR, dL, dR) Π3: (s, x, y, tL, tR, µL, µR, dL, dR)7−→ (s, −x, −y, tR, tL,−µR,−µL, dR, dL) Note that as a result of transformation Π1, we should obtain the symmetrical version of the system with respect to the y-axis along with a reversal of time that maintains the rotation sense of orbits around the origin. Similarly, the transformation Π2returns the symmetrical version of the system with respect50 to the x-axis, while Π3= Π1◦Π2= Π2◦Π1gets the mirror image of the system respect to the origin of coordinates. In canonical form (4), the parameter pair (µL, µR) controls the number and position of the equilibrium points. From (4) and assuming dLdR6= 0, the candidates to be equilibrium points in the left and right zones are respectively ¯xL=µL dL , tL µL dLand ¯xR=µR dR , tR µR dR. Obviously, apart from the boundary cases, ¯xLis a real equilibrium only if µLdL<0, and so is ¯xRif µRdR>0. Otherwise, one can speak of virtual equilibrium points, since they would not be in the corresponding zone.55 Boundary equilibrium bifurcations in the case of determinants with equal sign were studied in [10]. Here, we consider the case of determinants with different sign, and for definiteness we will pay attention to the configuration dL>0, dR<0,and either µLµR>0 or µL=µR= 0,(5) 6 which has obvious dual cases that could be analyzed via Remark 3. In this situation, regarding equilibria of systems (4), we have the following general result. The proof is direct and is omitted. Proposition 1 For refracting systems (4) fulfilling hypotheses (5) the following statements hold.60 a)If µL=µR= 0, then the origin is the single equilibrium point. b)If µLµR>0, then the following cases arise. (i)If µL<0or equivalently µR<0, then the system has two equilibrium points, a saddle point in the right zone and an anti-saddle point (to be a node or a focus) in the left zone.65 (ii)If µL>0or equivalently µR>0, then the system has no equilibrium points. It is easy to rule out the existence of periodic orbits in all the cases where there are no equilibrium points of focus type, that is, when the two dynamics involved are of type node and saddle. Therefore, from the dynamical point70 of view, the most interesting case under hypotheses (5) is the focus-saddle configuration, the case to be considered in what follows. When the left dynamics is of focus type, having then a saddle dynamics in the right half plane, it is also clear that in the cases of statement (a) and (b.ii) there cannot be limit cycles. Effectively, every limit cycle must use the two half planes,75 and then it is not possible to define any return map on the y-axis by using the orbits in the right part. Regarding the possible existence of a saddle connection or limit cycles in the focus case of statement (b.i) of Proposition 1, a first useful result is 7 the following. From Proposition 3.7 in [8], which is a direct consequence of80 Bendixson’s criterion, we know that the inequality tLtR<0 is a necessary condition for that; for definiteness, we take tL>0 and tR<0. Proposition 2 Considering refracting systems (4) under hypotheses 4dL− t2 L>0and dR<0with tL>0,tR<0,µL<0,µR<0, if there exists a periodic orbit then it is a unique stable limit cycle and there are no homoclinic85 connections. Furthermore, if there exists a homoclinic connection then it is stable as seen from its interior and there are no periodic orbits. Proposition 2 will be shown in Section 3. Note that, within the realm of sewing planar piecewise linear systems, Proposition 2 gives a uniqueness statement both for limit cycles and saddle-loops in the focus-saddle case90 that cannot be obtained from the recent analysis made in [7]. Effectively, the condition µL<0 implies that we have a real focus and so a visible tangency for orbits approaching the origin from the left. Therefore, the origin is not a Σ-monodromic singularity in the terminology of the quoted paper. Thus, Proposition 2 represents a new uniqueness result that was lacking for a95 complete analysis of sewing linear systems in the focus-saddle configuration. Next result characterizes completely the existence of closed orbits in systems (4) in the focus-saddle configuration. The main ideas behind the techniques are similar to the ones needed to show Theorem 2.b.3 in [11], but here we must extend them to cope with the family of refracting systems.100 Furthermore, the discontinuous character of such a family required specific new uniqueness results for closed orbits, as given in Proposition 2 above. We introduce some auxiliary parameters to be crucial in the rest of the paper, 8 namely ωL=p4dL−t2 L 2, γL=tL 2ωL , ωR=pt2 R−4dR 2, γR=tR 2ωR ,(6) and ρ=µR µL·ωL(1 + γ2 L) ωR(1 −γ2 R),(7) along with the angles θ±=±2 arctan q1 + [γL+ρ(γR±1)]2∓[γL+ρ(γR±1)].(8) Note that from the condition dR<0 we have γR∈(−1,1) and so ρ > 0105 when µL<0, µR<0. We can state the following result. Theorem 1 Consider refracting systems (4) under hypotheses 4dL−t2 L>0 and dR<0with tL>0,tR<0,µL<0,µR<0, and γL,γR,ρ, and θ±as defined in (6)-(8). If we define the values V±=±eγLθ±q1 + [γL+ρ(γR±1)]2,(9) then the following statements hold. a) If eπγLV++V−<0, then the unstable focus is surrounded by one stable limit cycle. b) If eπγLV++V−= 0, then the unstable focus is surrounded by an homo-110 clinic orbit and there are no limit cycles. c) If eπγLV++V−>0, then the system does not have either limit cycles 9 2. Application to the analysis of memristor oscillators We consider here an electronic circuit already analyzed in [15] and [11] under different hypotheses. For the sake of completeness, we include the derivation of the model, which is very similar to the one appeared in [11],205 but with an important novelty related to the consideration of non-zero initial conditions. Following [15] memristors are two-terminal electronic passive devices for which a nonlinear relationship links charge and flux [17]. They are at the basis of future generation dynamic memories; as another important feature,210 nanoscale memristors have potential to reproduce the behavior of biological synapses. Here we apply our previous results to the analysis of an elementary oscillator endowed with one flux-controlled memristor, see Figure 3 and [18]. Note that in the shown circuit the values of Land Cfor the impedance and capacitance are positive constants, while the resistor has a negative value215 −R. We write from Kirchoff’s laws iR(τ)−iL(τ) = 0, iL(τ)−iC(τ)−iM(τ) = 0, vC(τ)−vM(τ) = 0, vR(τ) + vL(τ) + vC(τ) = 0, where v, i stand for the voltage and current, respectively, across the corresponding element of the circuit. Next, we integrate with respect to time the above equations, and it should be emphasized that we do not assume zero 16 initial conditions as in [15], to get220 qR(τ)−qL(τ) = Q1=qR(0) −qL(0),(13) qL(τ)−qC(τ)−qM(τ) = Q2=qL(0) −qC(0) −qM(0),(14) ϕR(τ) + ϕL(τ) + ϕC(τ) = Φ1=ϕR(0) + ϕL(0) + ϕC(0),(15) ϕC(τ)−ϕM(τ) = Φ2=ϕC(0) −ϕM(0),(16) where qand ϕstand respectively for the charge and flux associated to each element. The flux-charge characteristics of the memristor fMis assumed as in [18] and [15] to be the symmetric piecewise linear function fM(x) =          b(x+ 1) −afor x≤ −1, ax for −1≤x≤1, b(x−1) + afor x≥1, where it is considered a passive memristor, and in particular 0 < a < b. After recalling the constitutive equations of the bipoles, namely ϕR(τ) = −RqR(τ), ϕL(τ) = Ld dτ qL(τ), qC(τ) = Cd dτ ϕC(τ), qM(τ) = fM(ϕM(τ)), we arrive at the equations d dτ ϕC(τ) = 1 CqC(τ) = 1 C[qL(τ)−qM(τ)−Q2], d dτ qL(τ) = 1 LϕL(τ) = 1 L[−ϕC(τ)−ϕR(τ) + Φ1]. 17 −R iR LiL C iCiM Figure 3: The simple oscillator with one memristor analyzed in this section. Note that the negative value −Rconsidered for the resistor makes it the only active element in the circuit. We denote the state variables by x1=ϕC(τ) and x2=qL(τ), and using (13) and (16), to write ϕR(τ) = −RqR(τ) = −R(qL(τ) + Q1) along with225 qM(τ) = fM(ϕM(τ)) = fM(ϕC(τ)−Φ2), we have the differential system dx1 dτ =1 C[−fM(x1−Φ2) + x2−Q2],(17) dx2 dτ =1 L[−x1+R(x2+Q1) + Φ1].(18) It turns out more convenient to translate variables by introducing x= x1−Φ2, and y=x2−Q2, and write dx dτ =1 C[−fM(x) + y],(19) dy dτ =1 L[−x+R˜y+RQ1+RQ2+ Φ1−Φ2].(20) As a consequence of the conservation laws of flux and charge, we can introduce the constant h=RQ1+RQ2+ Φ1−Φ2=ϕL(0) + ϕM(0) −R(qC(0) + qM(0)), 18 and using the parameters α= 1/C,β=R/L,ξ= 1/L, we get the piecewise linear system230 dx dτ =α[−fM(x) + y],(21) dy dτ =−ξx +βy +ξh. (22) We note that until here we have reproduced exactly the model achieved in [15], but with the additional constant term involving the parameter h, because here the initial conditions are not assumed to be zero. This fact, which introduces some lack of symmetry in the model, is of special relevance in the analysis of the dynamical richness of these oscillators, as remarked in235 [16]. Also, it should be noticed that we are dealing with a PWL system with three linearity regions, but as long as the interesting dynamics involves only two adjacent regions, we can take advantage of the results of Section 1. To simplify the analysis, we rescale the time by taking as new time ˆτ=ατ and introduce the rescaled parameters ˜ h=ξh/α,˜ β=β/α =RC/L > 0,240 and ˜ ξ=ξ/α =C/L > 0, to get dx dˆτ=−fM(x) + y, (23) dy dˆτ=−˜ ξx +˜ βy +˜ h. (24) In what follows we assume the same dynamical configuration analyzed in [15], namely saddle dynamics in the external zones and anti-saddle dynamics in the central zone, what leads to the condition a˜ β < ˜ ξ < b˜ β. Of course, we also assume the necessary condition for oscillation coming from Bendixson’s245 19 criterion, which amounts to have non-constant sign for the divergence of the vector field, that is, by computing traces, a < ˜ β < b. It is straightforward to determine the number of equilibrium points and its topological type depending on the value of parameter ˜ h. Effectively, if we introduce the modified slope parameters ˜a=˜ ξ−˜ βa > 0 and ˜ b=˜ ξ−˜ βb < 0, then the conditions for equilibria allow to eliminate the value of y, to get the equivalent condition ˜ h=˜ f(x) = ˜ ξx −˜ βfM(x), where ˜ f(x) =              ˜ b(x+ 1) −˜afor x≤ −1, ˜ax for −1≤x≤1, ˜ b(x−1) + ˜afor x≥1. The following result is now direct. Proposition 3 Considering system (23)-(24) under hypotheses 0< a < b and a˜ β < ˜ ξ < b˜ β, the following statements hold.250 a) If ˜ h < −˜aor ˜ h > ˜a, then there exists only one equilibrium point, which is a saddle located at        x= 1 + ˜ h−˜a ˜ b>1, y=a+b˜ h−˜a ˜ b,        x=˜ h+ ˜a ˜ b−1<−1, y=b˜ h+ ˜a ˜ b−a, respectively. b) If |˜ h|<˜a, then there exist three equilibrium points; one in the central zone located at x=˜ h ˜a, y =a˜ h ˜a, 20 which is of anti-saddle type, while the other two are saddles and located at the external zones, with the same coordinates given in statement (a). c) When |˜ h|= ˜a, there appears a boundary equilibrium bifurcation giving rise to the generation or annihilation of two equilibria, namely a saddle255 and an anti-saddle. Regarding Proposition 3, if we impose the condition of focus dynamics in the central zone, then we see that for both ˜ h=−˜aand ˜ h= ˜awe have the focus-saddle bifurcation studied in this paper. Such a central focus configuration requires (−a+˜ β)2<4(˜ ξ−˜ βa), which is equivalent to the condition (a+˜ β)2<4˜ ξ. (25) To illustrate here the usefulness of our previous results, let us consider the focus-saddle bifurcation undergone by system (23)-(24) under condition (25) and hypotheses of Proposition 3 when ˜ h= ˜afor some specific choice of parameters. We remark that in such a case, we can neglect the left external260 zone x < −1, since for small values of |˜ h−˜a|the relevant dynamics involves only the adjacent zones to the boundary at x= 1. The first preparation task is to do the translation X=x−1, Y=y−a, to get a continuous system in the form (3) with aL 11 =−a,aR 11 =−b,a12 = 1, aL 21 =aR 21 =−˜ ξ, aL 22 =aR 22 =˜ β, along with b1= 0, bL 2=bR 2=˜ h−˜ ξ+˜ βa =˜ h−˜a. Now a265 change of variables as the one used to obtain system (4) (which turns out to be homogeneous in this case, since b1= 0) transforms our system in a continuous version of such family, with tL=˜ β−a,tR=˜ β−b,dL=˜ ξ−a˜ β, dR=˜ ξ−b˜ β, and the bifurcation parameter is µ=µL=µR=˜ h−˜a. 21 Let us assume, following [15], the values α= 1.25, β=ξ= 1, a= 0.75270 and b= 1.5. Note that the specific passivity condition 0 < a < b is fulfilled. Thus, we get ˜ β=˜ ξ= 0.8, dL= ˜a= 0.2 and dR=˜ b=−0.4, so that tL= 0.05 and tR=−0.7. Apart from the necessary condition for oscillating behaviour a < ˜ β < b, we see that both the external saddle dynamics condition a˜ β < ˜ ξ < b˜ βand the focus condition (25) hold. Next, we apply Theorem 2.275 Using (6) and (11), we get ωL≈0.446514, γL≈0.055989, ωR≈0.722842, γR≈ −0.4842, and ρ≈0.809427. Now from (8) we obtain θ+≈1.12858 and θ−≈ −0.717745. Finally, from (12) we deduce δ≈ −0.140354 <0, and then statement (a) of Theorem 2 assures that for µ < 0, that is, for ˜ h < ˜athere appears one stable limit cycle surrounding the unstable focus.280 This limit cycle (responsible for the stable oscillation in the circuit) grows linearly with |˜ h−˜a|as long as the assumption regarding that the cycle does not use the zone x < −1 is true. In fact, we could show the existence of a certain value ˜ hT>0, such that the limit cycle becomes tangent to the line x=−1. Thus, the existence of stable oscillations is guaranteed for all285 ˜ hT<˜ h < ˜a. Furthermore, the symmetry of the model allows us to assure also the existence of stable oscillations for −˜a < ˜ h < −˜ hT. We conjecture that such oscillations exist indeed for all −˜a < ˜ h < ˜a, but the proof of this conjecture is beyond the scope of this paper. If we take instead a= 0.74, keeping fixed the rest of parameters, we have290 now tL= 0.06, dL= ˜a= 0.208, ωL≈0.455082, γL≈0.0659221, and then we get ρ≈0.825955, θ+≈1.11361 and θ−≈ −0.711476. However, now δ≈0.0363403 >0 and so the bifurcation does not lead to any limit cycle. Numerically, one can estimate that the critical value for the appearance of 22 the homoclinic orbit in the bifurcation is a≈0.742014.295 3. Auxiliary results and proofs In order to show the results of Section 1, it is convenient to reduce even more the number of essential parameters by rescaling time in a different way on each half-plane along with an appropriate rescaling of variables to maintain the canonical form. Note that this change of variables preserves300 any closed orbits. Afterwards, we obtain a useful reduced canonical form for the study of the focus-saddle configuration in refracting systems. Proposition 4 If 4dL−t2 L>0and dR<0then refracting systems (4) can be written in the following reduced canonical form dx dθL = 2γL−1 γ2 L+ 1 0  x− 0 αL  if x∈SL∪Σ, dx dθR = 2γR−1 γ2 R−1 0  x− 0 αR  if x∈SR, (26) where ωL=p4dL−t2 L 2, γL=tL 2ωL , αL=µL ωL ,(27) and ωR=pt2 R−4dR 2, γR=tR 2ωR∈(−1,1), αR=µR ωR .(28) Proof. We know that the eigenvalues of the linear part at SLin (4) are σL±iωL. We make first the change X=ωLx,Y=y,θL=ωLsfor the 23 variables in the half plane SL, without altering variables and time in SR.305 Note that we do not change the coordinate y, so that periodic orbits using both half planes are preserved. Then, we get dX dθL =1 ωL dX ds =dx ds =tL ωL X−Y, dY dθL =1 ωL dY ds =1 ωL dy ds =1 ωLdL ωL X−µL=dL ω2 L X−µL ωL . Introducing the parameter γLdefined in (27), we see that tL= 2γLωLand dL= (γ2 L+ 1)ω2 L, and the left vector field is already in form of (26). Similarly to what is done in the left zone, we define ωR>0 such that310 ω2 R=t2 R/4−dRand σR=tR/2, that is, the eigenvalues of the linear part at SRin (4) are σR±ωR. In the same way as before, we make the change X=ωRx,Y=y,θR=ωRsfor the variables in the half plane SR, without altering variables and time in SL. We obtain dX dθR =1 ωR dX ds =dx ds =tR ωR X−Y, dY dθR =1 ωR dY ds =1 ωR dy ds =1 ωRdR ωR X−µR=dR ω2 R X−µR ωR . Now, introducing the parameter γRas in (28), we see that tR= 2γRωRand315 dR= (γ2 R−1)ω2 R. Finally, using the definitions of αLand αRin (27)-(28), we get the expressions given in (26). Note that under hypotheses of Proposition 4, if we assume tL>0 and tR<0 then we have γL>0 and −1< γR<0. In the canonical form (26), 24 under hypotheses αL<0, αR<0, we have a focus at the point xF, yF=αL γ2 L+ 1,2αLγL γ2 L+ 1∈SL, and a saddle at xS, yS=αR γ2 R−1,2αRγR γ2 R−1∈SR. We can compute the points (0,˜y±) where the unstable (stable) manifold of the saddle intersects Σ as follows. The eigenvectors for the saddle can be selected as (1, γR∓1)Tcorresponding to the eigenvalues γR±1. Thus, the linear invariant manifolds that emanate from the saddle intersect Σ when, for a certain value of β, we have  xS yS +β 1 γR∓1 = 0 ˜y±  . Clearly, β=−xSand then we get ˜y±=2αRγR γ2 R−1−αR(γR∓1) γ2 R−1=αR(γR±1) γ2 R−1=αR γR∓1. In the next result, we review some useful properties of the half-return maps that can be defined by using Σ as a Poincar´e section, see Fig. 4. Generically, these half-return maps cannot be explicitly computed and must320 be obtained in parametric form in terms of the respective flight times θLand θR. Their properties can be straightforward deduced from the solutions of the linear vector field involved, see Proposition 6 and 7 of [19] and Proposition 3 of [20]. 25 so that (xF, yF) = (−v, −2vγL). Thus, we work with the equivalent system ˙x = 2γL−1 γ2 L+ 1 0  x,if x∈e SL∪e Σ, ˙x = 2γR−1 γ2 R−1 0  x− 2v(γR−γL) αR+v(γ2 R−1)  ,if x∈e SR, (33) where the linearity regions in the phase plane are now the left and right half-planes e SL={(x, y)∈R2:x < v},e SR={(x, y)∈R2:x > v}, separated by the straight line e Σ = {(x, y)∈R2:x=v}. Furthermore, the saddle point becomes the point (˜xS,˜yS) = v+αR γ2 R−1,2vγL+2αRγR γ2 R−1, which after introducing the positive parameter ρ=αR αL·1 + γ2 L 1−γ2 R =µR µL·ωL(1 + γ2 L) ωR(1 −γ2 R)>0,(34) can be written in the more compact expression (˜xS,˜yS) = v(1 + ρ, 2(γL+ργR)) ,(35) 32 Y+ Y- y+ y- F S Figure 6: Sketch of the process to determine the remarkable points (0, Y±). The focus Fis already put at the origin, so that the dashed vertical lines are the discontinuity manifold e Σ at x=v, and the line x=v+xS R, where the saddle Sis located. since from (32) we have αR γ2 R−1=vρ. The linear invariant manifolds that emanate from the saddle intersect the line x=vat the points (v, y±) when, for a certain value of β, we have  ˜xS ˜yS +β 1 γR∓1 = v y±  . Taking into account (35), we get β=v−˜xS=v−v(1 + ρ) = −vρ, and so the invariant manifolds of the saddle intersect the line x=vat the points (v, y±), where y±=v[2γL+ρ(γR±1)] ,(36) see Fig. 6. We will follow the orbit with initial point (v, y+) up to determine its 33 intersection point (0, Y+) with the new y-axis. Similarly, we can compute the point (0, Y−) whose orbit eventually arrives at the point (v, y−) by integrating backwards in time from this latter point. Since the focus is at the origin, taking (v, y±) as the starting point, to compute now the value of Y±, it suffices to write the exponential matrix and to solve the equation  0 Y±  =eγLθ± cos θ±+γLsin θ±−sin θ± (γ2 L+ 1) sin θ±cos θ±−γLsin θ±   v y±  , (37) where the flight times must satisfy 0 <±θ±< π, since we must integrate370 forward (backward) in time to get Y+(Y−), see Fig. 6. From the first component of equality in (37), we get vcos θ±= (y+−vγL) sin θ±=v[γL+ρ(γR±1)] sin θ±.(38) Equivalently, we write tan θ±=1 γL+ρ(γR±1), and we obtain sin θ±=±|tanθ±| p1 + tan2θ± =±1 q1 + [γL+ρ(γR±1)]2, and then cos θ±=±γL+ρ(γR±1) q1 + [γL+ρ(γR±1)]2. To ease the angle determination with the arctan function, we use the 34 trigonometric identity tan θ± 2=sin θ± 1 + cos θ± =±1 q1 + [γL+ρ(γR±1)]2±[γL+ρ(γR±1)] . After multiplying numerator and denominator in the argument by the conjugate, we get θ±=±2 arctan q1 + [γL+ρ(γR±1)]2∓[γL+ρ(γR±1)],(39) as given in (8). Noting that cos θ±−γLsin θ±=ρ(γR±1) sin θ±, we get from the second component of equality of (37) that Y±=eγLθ±(γ2 L+ 1)v+ρ(γR±1)y±sin θ±=±veγLθ±q1 + [γL+ρ(γR±1)]2, where θ±is given in (39) and we have used the above expresions for y±and sin θ±, leading to (γ2 L+ 1)v+ρ(γR±1)y±=v1 + [γL+ρ(γR±1)]2. Therefore, the common expression for both ordinates is Y±=vV±, with V±as given in (9). Summarizing, the unstable invariant manifold of the saddle after intersecting the vertical x=vat the point (v, y+) passes through the point (0, Y+)375 and will come back again to the new y-axis at the point (0,−eπγLY+). Analogously, the linear stable invariant manifold of the saddle is reached when we consider the orbit starting at the point (0, Y−), see Fig. 6. 35 Assume first that −eπγLY+> Y−, which after reordering and removing the common factor vleads to eπγLV++V−<0. Then we have a trapping region380 containing the unstable focus, so that by Poincar´e-Bendixson’s theorem we must have at least a periodic orbit surrounding the focus. The uniqueness and stability of the corresponding limit cycle comes from Proposition 2 and statement (a) follows. To show statement (b) it suffices to see that if the condition of the385 statement holds, then the two linear invariant manifolds of the saddle are connected forming a saddle-loop or homoclinic connection. Therefore, from Proposition 2 there cannot be limit cycles. In the case of statement (c), since −eπγLY+< Y−, we conclude that for the original system (26) we have PL(˜y+)<˜y−, see Fig. 4. Then, regarding390 the proof of Proposition 2, if there exists an intersection between the graphs of PLand P−1 R, then the concavity of both graphs implies the existence of at least two intersection points below the point y∗. This is impossible, as shown in Proposition 2, so that we conclude that there are no limit cycles. The theorem is so completely shown.395 Finally, we give the proof of Theorem 2. Proof of Theorem 2. All the assertions about the equilibrium points and their stability come easily from Proposition 1, taking into account the hypothesis 4dL−t2 L>0. Applying Proposition 4, we arrive at a system (26) with αL=µ/ωL,400 αR=µ/ωR. Note that even we start from a continuous vector field in (4), after the transformation involved in the quoted proposition, we pass to a 36 refracting system. This is the reason why we needed to consider the more general framework of discontinuous refracting systems. Note also that from (34) the parameter ρbecomes ρ=ωL(1 + γ2 L) ωR(1 −γ2 R)>0, as given in (11), not depending on µ.405 If µis negative, then the appearance of a limit cycle, an homoclinic connection or nothing else, is a direct consequence of Theorem 1, considering the sign of the expression eπγLV++V−. Taking into account that eπγLV+− V−>0, we can write sign(eπγLV++V−) = sign(e2πγLV2 +−V2 −) = sign e2γLπ−V2 − V2 +, from where the expression of the coefficient δfollows. To complete the proof, we need to show the linear evolution of the limit cycle or the saddle-loop with respect to |µ|. It suffices to note that in systems (10), when µ6= 0, the system can be rewritten with µ=±1 by the homothety change X=|µ|x,Y=|µ|y. Therefore, if µ < 0 and there exists410 a periodic orbit or an homoclinic connection, then there exists also the same distinguished closed orbit for µ=−1 and both orbits are homothetic. So, the size of the such orbit evolves linearity with |µ|, and the theorem follows. Acknowledgements Authors are partially supported by the Spanish Ministerio de Econom´ıa,415 Industria y Competitividad, in the frame of projects MTM2014-56272-C2-1-P 37 and MTM2015-65608-P, and by the Consejer´ıa de Econom´ıa y Conocimiento de la Junta de Andaluc´ıa under grant P12-FQM-1658. 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