Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region

Communications in Nonlinear Science and Numerical Simulation 123 (2023) 107257 Contents lists available at ScienceDirect Communications in Nonlinear Science and Numerical Simulation journal homepage: www.elsevier.com/locate/cnsns Research paper Uniqueness and stability of limit cycles in planar piecewise linear differential systems without sliding region Victoriano Carmona a,∗, Fernando Fernández-Sánchez b, Douglas D. Novaes c aDpto. Matemática Aplicada II & IMUS, Universidad de Sevilla, Escuela Politécnica Superior. Calle Virgen de África 7, 41011 Sevilla, Spain bDpto. Matemática Aplicada II & IMUS, Universidad de Sevilla, Escuela Técnica Superior de Ingeniería. Camino de los Descubrimientos s/n, 41092 Sevilla, Spain cDepartamento de Matemática, Instituto de Matemática, Estatística e Computação Científica (IMECC), Universidade Estadual de Campinas (UNICAMP), Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz, 13083–859, Campinas, SP, Brazil article info Article history: Received 8 December 2022 Received in revised form 27 February 2023 Accepted 5 April 2023 Available online 17 April 2023 MSC: 34A26 34A36 34C05 34C25 Keywords: Piecewise planar linear differential systems Sewing systems Limit cycles Optimal uniform upper bound Poincaré half-maps Integral characterization abstract In this paper, we consider the family of planar piecewise linear differential systems with two zones separated by a straight line without sliding regions, that is, differential systems whose flow transversally crosses the switching line except for at most one point. In the research literature, many papers deal with the problem of determining the maximum number of limit cycles that these differential systems can have. This problem has been usually approached via large case-by-case analyses which distinguish the many different possibilities for the spectra of the matrices of the differential systems. Here, by using a novel integral characterization of Poincaré half-maps, we prove, without unnecessary distinctions of matrix spectra, that the optimal uniform upper bound for the number of limit cycles of these differential systems is one. In addition, it is proven that this limit cycle, if it exists, is hyperbolic and its stability is determined by a simple condition in terms of the parameters of the system. As a byproduct of our analysis, a condition for the existence of the limit cycle is also derived. ©2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction In this paper, we consider planar discontinuous piecewise linear differential systems with two zones separated by a straight line. Without loss of generality, these differential systems can be written as ˙ x={ALx+bL,if x1<0, ARx+bR,if x1>0,(1) where x=(x1,x2)T∈R2,AL=(aL ij)2×2,AR=(aR ij)2×2,bL=(bL 1,bL 2)T∈R2, and bR=(bR 1,bR 2)T∈R2. Notice that Σ= {(x1,x2)∈R2:x1=0}the switching line of system (1). We assume Filippov’s convention (see [1]) for the definition of the trajectories of (1). ∗Corresponding author. E-mail addresses: [email protected] (V. Carmona), [email protected] (F. Fernández-Sánchez), [email protected] (D.D. Novaes). https://doi.org/10.1016/j.cnsns.2023.107257 1007-5704/©2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons. org/licenses/by-nc-nd/4.0/). V. Carmona, F. Fernández-Sánchez and D.D. Novaes Communications in Nonlinear Science and Numerical Simulation 123 (2023) 107257 Moreover, we are interested in the family of such differential systems which do not have any sliding regions, that is, for each point in the switching line, except for at most one, the corresponding trajectory crosses it transversally. Such systems are sometimes called sewing systems. This can be analytically expressed by (aL 12 x2+bL 1)(aR 12 x2+bR 1)⩾0 for every x2∈R(2) and (aL 12 x2+bL 1)(aR 12 x2+bR 1)=0 at most at one value of x2∈R.(3) Notice that, in this paper, the concept of ‘‘sliding region’’ also includes the sometimes called in other papers ‘‘escaping region’’. We focus on determining an optimal uniform upper bound for the number of limit cycles of this family of differential systems. In the study of planar piecewise smooth differential systems, a limit cycle is usually defined as a non-trivial closed crossing orbit which is isolated from other closed orbits. In 1991, Lum and Chua [2], assuming the continuity of the differential system (1) conjectured that they have at most one limit cycle. This conjecture was proven in 1998 by Freire et al. [3]. Their proof was performed by distinguishing every possible configuration depending on the spectra of the matrices of the differential system. Recently, Carmona et al. [4] provided a new simple proof for Lum–Chua’s conjecture using a novel integral characterization of Poincaré halfmaps (see [5]). In addition, this limit cycle, if it exists, has been proven to be hyperbolic and its stability was explicitly determined by an easy condition in terms of the parameters of the system. This novel characterization has proven to be an effective method to avoid the case-by-case study performed in the former proof and has also been used by the same authors in [6] to show the existence of a uniform upper bound for the maximum number of limit cycles of general planar piecewise linear differential systems with two zones separated by a straight line. Dropping the continuity assumption, Freire et al. [7] studied the limit cycles of the differential system (1) assuming that each linear differential system has a real or virtual equilibrium of focus type (the concepts of real,boundary or virtual, referred to an equilibrium of a system corresponding to one of the zones of linearity, just describe if it is located, respectively, inside, on the boundary or outside this zone). In this paper, the authors qualitatively described the different phase portraits taking into account the number of real focus equilibria, namely zero, one, or two. For the cases of zero and one real focus equilibrium, they obtained results on the existence and uniqueness of limit cycles. For the case of two real focus equilibria, based on extensive numerical simulations, they conjectured that such differential systems have at most one limit cycle. The case of two virtual foci had already been considered by Llibre et al. [8], who obtained the uniqueness of limit cycles via a generalized criterium for Liénard differential equations allowing discontinuities. The uniqueness of limit cycles for the node–node and saddle–saddle cases have been addressed by Huan and Yang in [9,10], respectively. Medrado and Torregrosa [11] provided the uniqueness of limit cycles assuming the existence of a monodromic singularity in the switching line. Their proof also distinguishes some configurations depending on the spectra of matrices of the differential system. Li and Llibre [12] gave a proof of the uniqueness of limit cycles for the focus–saddle case. Recently, Li et al. [13] provided the uniqueness of limit cycles for the focus–node and focus–focus scenarios, which were the remaining non-degenerate cases (that is, det(AL) det(AR)= 0). By means of a different approach, Tao Li et al. [14] also provided the uniqueness of limit cycles for the non-degenerate cases. These last two papers provided a positive answer for the conjecture stated in [7] for the case of two real focus equilibria. However, taking into account all the previous results, a positive answer to the uniqueness of limit cycles of planar piecewise linear differential systems without sliding region was only obtained for the non-degenerate cases. As far as we are concerned in all previous case-by-case studies, the degenerate cases have not been exhausted. It is worth noting all the effort and time needed, the large number of papers devoted to different cases, and the specific techniques developed in each one of them to reach the result for the non-degenerate cases. In this paper we provide the uniqueness of limit cycles for piecewise linear differential systems without sliding region. Moreover, this result is proven in a unified way (with a single approach and a common technique) that does not distinguish cases depending on the spectrum of the matrices. This unified way allows us to establish the hyperbolicity of the limit cycle and also to determine its stability by a simple condition in terms of the parameters of system (1). These results are collected in the statement of the main theorem of this paper. Theorem A. Consider the planar piecewise linear differential system (1). Let TLand TRbe the traces of the matrices ALand AR, respectively. Denote aL=aL 12bL 2−aL 22bL 1, aR=aR 12bR 2−aR 22bR 1, and ξ=aRTL−aLTR. If the differential system (1) does not have sliding region (that is, conditions (2) and (3) hold), then it has at most one limit cycle. This limit cycle, if it exists, is hyperbolic and ξ= 0. Moreover, it is asymptotically stable (resp. unstable) provided ξ < 0(resp. ξ > 0). The proof of Theorem A is a direct consequence of Propositions 1,12 and 13, as follows: Proposition 1 recalls the normal form (5) for piecewise linear differential systems given by (1); then, Proposition 12 provides the uniqueness of hyperbolic limit cycles and characterizes its stability for piecewise linear differential systems without sliding region; lastly, Proposition 13 concludes that such systems do not admit non-hyperbolic limit cycles. The main basis of this paper lies in an original integral characterization, presented in [5], of the Poincaré half-maps for planar linear differential systems associated to a straight line. This novel characterization is introduced in Section 2, 2 V. Carmona, F. Fernández-Sánchez and D.D. Novaes Communications in Nonlinear Science and Numerical Simulation 123 (2023) 107257 where some properties of these half-maps, interesting for our study, are collected. As usual, the study of the crossing limit cycles for a two-zonal piecewise linear differential system is done by means of the analysis of zeros of an appropriate displacement function given as the difference between the Poincaré half-maps associated to the switching line. Section 3 is devoted to introduce and analyze the displacement function by using the results stated in Section 2. More specifically, we provide suitable expressions for its first and second derivatives. The local behavior of the displacement function around monodromic singularities on Σand, when it is appropriate, at the infinity is established in Section 4. Some results on the existence of limit cycles are given in Section 5. In particular, Corollary 2 provides an extension to sewing piecewise linear differential systems of the results about existence of limit cycles for continuous differential systems given in [15]. In Section 6, we show the uniqueness of hyperbolic limit cycles (see Proposition 12) for piecewise linear differential systems without sliding regions. Moreover, the stability of this unique limit cycles, if it exists, is determined by a simple condition in terms of the parameters of the system. Finally, Section 7is dedicated to showing that the considered differential systems do not admit degenerate limit cycles (see Proposition 13). 2. Liénard canonical form and Poincaré half-maps First of all, notice that if aL 12 =0, then the first equation of system (1) for x1<0 becomes ˙ x1=aL 11x1+bL 1, whose solutions are all monotonic and prevents the existence of periodic solutions for system (1). With a similar reasoning, aR 12 =0 also prevents the existence of limit cycles for system (1). Thus, the condition aL 12aR 12 = 0 is necessary for the existence of periodic solutions of system (1). Under this hypothesis, the non-sliding conditions (2) and (3) are equivalent to aL 12aR 12 >0 and aL 12 bR 1=aR 12 bL 1.(4) From now on, we assume that the differential system (1) satisfies condition (4). A natural first step in the analysis of any parameterized differential system consists in writing it in a suitable normal form. The following proposition, whose proof appeared in [16] for a more general case, is devoted to it. Proposition 1. Under assumption (4), the differential system (1) is reduced into the following Liénard canonical form {˙ x=TLx−y ˙ y=DLx−aL for x <0,{˙ x=TRx−y ˙ y=DRx−aR for x >0,(5) by a homeomorphism preserving the switching line Σ= {(x,y)∈R2:x=0}. Furthermore, aL=aL 12bL 2−aL 22bL 1, aR=aR 12bR 2−aR 22bR 1, and TL, TRand DL, DRare, respectively, the traces and determinants of the matrices ALand AR. As said before, the study of the crossing limit cycles for the differential system (5) is done by means of the analysis of the zeros of an appropriate displacement function given as the difference between the Poincaré half-maps associated to the switching line Σ. In what follows we introduce such maps, namely, the Forward Poincaré Half-Map yL:IL⊂ [0,+∞)−→ (−∞,0]and the Backward Poincaré Half-Map yR:IR⊂ [0,+∞)→(−∞,0], whose graphs are contained in the fourth quadrant Θıv := {(y0,y1)∈R2:y0⩾0,y1⩽0}. The forward Poincaré half-map takes a point (0,y0), with y0⩾0, and maps it to a point (0,yL(y0)) by following the forward flow of system (5). More specifically, let ϕ(t;y0)=(ϕ1(t;y0), ϕ2(t;y0)) be the orbit of system (5) satisfying ϕ(0;y0)=(0,y0). If there exists a value τL(y0)>0 such that ϕ1(τL(y0);y0)=0 and ϕ1(t;y0)<0 for every t∈(0, τL(y0)), we define yL(y0)=ϕ2(τL(y0);y0)⩽0 (see Fig. 1(a)). In addition, if for every ε > 0 there exist y0∈(0, ε) and y1∈(−ε, 0) such that yL(y0)=y1, the left Poincaré half-map can be extended to y0=0 with yL(0) =0, even if the above positive time τL(0) does not exist (see Fig. 1(b)). Analogously, the backward Poincaré half-map takes a point (0,y0), with y0⩾0, and maps it to a point (0,yR(y0)) by following the backward flow of (5). More specifically, if there exists a value τR(y0)<0 such that ϕ1(τR(y0);y0)=0 and ϕ1(t;y0)>0 for every t∈(τR(y0),0), we define yR(y0)=ϕ2(τR(y0);y0)⩽0. Again, if for every ε > 0 there exist y0∈(0, ε) and y1∈(−ε, 0) such that yR(y0)=y1, the right Poincaré half-map can be extended with yR(0) =0, even if the above negative time τR(0) does not exist. In Fig. 3, we illustrate the Poincaré half-maps defined above and their intervals of definition when the left linear system has a real focus and the right linear system has a real saddle. In the next Theorems 1 and 2, we will present an integral characterization of the Poincaré half-maps above which was introduced in [5]. For that, we will need the following concept of Cauchy Principal Value: PV {∫y0 y1 f(y)dy}:= lim ε↘0(∫−ε y1 f(y)dy +∫y0 ε f(y)dy), for y1<0<y0and fcontinuous in [y1,y0]\{0}(see, for instance, [17]). Note that if fis also continuous in 0, then the Cauchy Principal Value coincides with the definite integral. 3 V. Carmona, F. Fernández-Sánchez and D.D. Novaes Communications in Nonlinear Science and Numerical Simulation 123 (2023) 107257 Fig. 1. Definition of the forward Poincaré half-map. Since the flow of (5) is oriented in anticlockwise direction, the forward Poincaré half-map, yL, is determined by the following linear differential system {˙ x=TLx−y, ˙ y=DLx−aL,(6) which matches the left linear system of (5). Accordingly, its definition, its domain IL, and its analyticity are given by Theorem 19, Corollary 21, Corollary 24, and Remark 16 of [5]. In the following theorem, we summarize the mentioned results. Theorem 1. Let us consider system (6) and define the polynomial WL(y)=DLy2−aLTLy+a2 L.(7) The forward Poincaré half-map yLis well defined if, and only if, aL⩽0and 4DL−T2 L>0, or aL>0. In this case, its interval of definition IL:= [λL, µL)⊂ [0,+∞)is non-empty and the following statements hold: (a) The right endpoint µLof the interval ILis the smallest strictly positive root of the polynomial WL, if it exists. Otherwise, µL= +∞. (b) The left endpoint λLof the interval ILis strictly positive if, and only if, aL<0,4DL−T2 L>0, and TL<0. In this case, yL(λL)=0. (c) The left endpoint of the interval yL(IL)is the largest strictly negative root of the polynomial WL, if it exists. In the opposite case, this left endpoint is −∞. (d) The right endpoint of the interval yL(IL), that is yL(λL), is strictly negative if, and only if, aL<0,4DL−T2 L>0, and TL>0. In this case, λL=0. (e) The polynomial WLsatisfies WL(0) =a2 L>0for aL= 0and WL(0) =0for aL=0. Moreover, WL(y)>0for y∈ch(IL∪yL(IL)) \{0}, where ch(·)denotes the convex hull of a set. (f) The forward Poincaré half-map yLis the unique function yL:IL⊂ [0,+∞)−→ (−∞,0]that satisfies PV {∫y0 yL(y0) −y WL(y)dy}=qL(aL,TL,DL)for y0∈IL,(8) where qL(aL,TL,DL)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0 if aL>0, πTL DL√4DL−T2 L if aL=0, 2πTL DL√4DL−T2 L if aL<0. (9) (g) The graph of the forward Poincaré half-map, oriented according to increasing y0, is the portion included in the fourth quadrant Θıv of a particular orbit of the cubic vector field XL(y0,y1)= −(y1WL(y0),y0WL(y1)).(10) 4 V. Carmona, F. Fernández-Sánchez and D.D. Novaes Communications in Nonlinear Science and Numerical Simulation 123 (2023) 107257 Equivalently, the forward Poincaré half-map is a solution of the differential equation y1WL(y0)dy1−y0WL(y1)dy0=0.(11) (h) The forward Poincaré half-map yLis analytic in Int(IL). Moreover, yLis analytic in ILif, and only if, λL=0. Now, because of the anticlockwise direction of the flow of (5) again, the backward Poincaré half-map, yR, is determined by the following linear differential system {˙ x=TRx−y, ˙ y=DRx−aR,(12) which matches the right linear system of (5). Thus, its definition, its domain IR, and its analyticity are obtained from Theorem 1 by means of the change of variables (t,x)↦→ (−t,−x) and taking (aL,DL,TL)=(−aR,DR,−TR) in system (6). Theorem 2. Let us consider system (12) and define the polynomial WR(y)=DRy2−aRTRy+a2 R.(13) The backward Poincaré half-map yRis well defined if, and only if, aR⩾0and 4DR−T2 R>0, or aR<0. In this case, its interval of definition IR:= [λR, µR)⊂ [0,+∞)is non-empty and the following statements hold: (a) The right endpoint µRof its definition interval IRis the smallest strictly positive root of the polynomial WR, if it exists. Otherwise, µR= +∞. (b) The left endpoint λRof the interval IRis strictly positive if, and only if, aR>0,4DR−T2 R>0, and TR>0. In this case, yR(λR)=0. (c) The left endpoint of the interval yR(IR)is the largest strictly negative root of the polynomial WR, if it exists. In the opposite case, this left endpoint is −∞. (d) The right endpoint of the interval yR(IR), that is yR(λR), is strictly negative if, and only if, aR>0,4DR−T2 R>0, and TR<0. In this case, λR=0. (e) The polynomial WRsatisfies WR(0) =a2 R>0for aR= 0and WR(0) =0for aR=0. Moreover, WR(y)>0for y∈ch(IR∪yR(IR)) \{0}. (f) The backward Poincaré half-map yRis the unique function yR:IR⊂ [0,+∞)−→ (−∞,0]that satisfies PV {∫y0 yR(y0) −y WR(y)dy}=qR(aR,TR,DR)for y0∈IR,(14) where qR(aR,TR,DR)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0 if aR<0, −πTR DR√4DR−T2 R if aR=0, −2πTR DR√4DR−T2 R if aR>0. (15) (g) The graph of the backward Poincaré half-map, oriented according to increasing y0, is the portion included in the fourth quadrant Θıv of a particular orbit of the cubic vector field XR(y0,y1)= −(y1WR(y0),y0WR(y1)).(16) Equivalently, the backward Poincaré half-map is a solution of the differential equation y1WR(y0)dy1−y0WR(y1)dy0=0.(17) (h) The backward Poincaré half-map yRis analytic in Int(IR). Moreover, yRis analytic in IRif, and only if, λR=0. In the next remark, we provide a characterization to easily distinguish, under the generic condition aLTLaRTR= 0, when the Poincaré half-maps transform 0 to 0. This will be used in the proof of Propositions 8 and 12. Remark 1. Under the condition aLTL= 0, by assuming the existence of yL, it follows from the statements (b) and (d) of Theorem 1that 0∈ILand yL(0) =0if, and only if, aL>0. Analogously, under the condition aRTR= 0, by assuming the existence of yR, it follows from the statements (b) and (d) of Theorem 2that 0∈IRand yR(0) =0if, and only if, aR<0. Now, we establish some fundamental properties of the Poincaré half-maps which will be useful for the proof of the main result. The proofs of these properties can be found in [18]. From (11) and (17), it is straightforward to obtain explicit expressions for the first and second derivatives of yLand yR. 5 V. Carmona, F. Fernández-Sánchez and D.D. Novaes Communications in Nonlinear Science and Numerical Simulation 123 (2023) 107257 Proposition 2. The first and second derivatives of the Poincaré half-maps yLand yRare given by y′ L(y0)=y0WL(yL(y0)) yL(y0)WL(y0)for y0∈int(IL), y′ R(y0)=y0WR(yR(y0)) yR(y0)WR(y0)for y0∈int(IR), y′′ L(y0)= −a2 L(y2 0−(yL(y0))2)WL(yL(y0)) (yL(y0))3(WL(y0))2for y0∈int(IL),(18) and y′′ R(y0)= −a2 R(y2 0−(yR(y0))2)WR(yR(y0)) (yR(y0))3(WR(y0))2for y0∈int(IR).(19) Now, we show the first coefficients of the Taylor expansions of the Poincaré half-maps at the origin (see Proposition 3) and at infinity (see Proposition 4), under suitable conditions. These are the essentials for our study. More coefficients and other details can be seen in [18]. Proposition 3. Assume that aL= 0(resp. aR= 0) and 0∈IL(resp. 0∈IR). If yL(0) =0(resp. yR(0) =0), then the Taylor expansion of yL(resp. yR) around the origin writes as yL(y0)= −y0−2TL 3aL y2 0+O(y3 0)(resp.yR(y0)= −y0−2TR 3aR y2 0+O(y3 0)). Notice that if 4DL−T2 L>0 and 4DR−T2 R>0, then, from the statements (a) and (c) of Theorem 1 and the statements (a) and (c) of Theorem 2, the intervals ILand IRare unbouded and yL(y0) and yR(y0) tend to −∞ as y0→ +∞. Proposition 4. The following statements hold. (1) If 4DL−T2 L>0, then the forward Poincaré half-map yLsatisfies lim y0→+∞ yL(y0) y0= −exp ⎛ ⎝πTL √4DL−T2 L ⎞ ⎠. (2) If 4DR−T2 R>0, then the backward Poincaré half-map yRsatisfies lim y0→+∞ y0 yR(y0)= −exp ⎛ ⎝πTR √4DR−T2 R ⎞ ⎠. We conclude this section with two results that establish the relative position between the graph of the Poincaré halfmaps and the bisector of the fourth quadrant, that is, the half straight line y1= −y0,y0⩾0. In Fig. 2, we show the possible relative positions between the graph of the Poincaré map yLand the bisector of the fourth quadrant by varying the trace TLfor three illustrative cases. A similar figure could be given for the map yRby varying the trace TR. Proposition 5. The following statements hold. (a) The forward Poincaré half-map yLsatisfies the relationship sign (y0+yL(y0))= −sign(TL)for y0∈IL\{0}. In addition, when 0∈ILand yL(0) = 0or when TL=0, then the relationship above also holds for y0=0. (b) The backward Poincaré half-map yRsatisfies the relationship sign (y0+yR(y0))=sign(TR)for y0∈IR\{0}. In addition, when 0∈IRand yR(0) = 0or when TR=0, then the relationship above also holds for y0=0. Corollary 1. The following statements are true. (a) If TL=0(resp. TR=0), then the graph of the Poincaré half-map yL(resp. yR) is included in the bisector of the fourth quadrant. (b) If TL= 0(resp. TR= 0), then the graph of the Poincaré half-map yL(resp. yR) does not intersect the bisector of the fourth quadrant except perhaps at the origin. 6 V. Carmona, F. Fernández-Sánchez and D.D. Novaes Communications in Nonlinear Science and Numerical Simulation 123 (2023) 107257 Fig. 2. Three illustrative cases of relative positions between the graph of the Poincaré half-map yLand the bisector of the fourth quadrant in terms of the trace TL. The left figure corresponds to a real focus (that is, negative x-coordinate) with DL=1, aL= −1/2, and TL∈ {−1/10,0,1/5}; the middle one corresponds to a virtual focus (that is, positive x-coordinate) with DL=1, aL=1/2, and TL∈ {−2/5,0,1/2}; and the right one corresponds to a real saddle with DL= −1, aL=6/5, and TL∈ {−4/5,0,1}. The black circles represent the starting point of the graphs of yLwhile, in the right figure, the white circles represent the ending point of the graph of yL, whose coordinates are the roots of the polynomial WL(y)=DLy2−aLTLy+a2 L provided in (7) (which also correspond to the y-coordinates of the intersection points between the invariant straight lines of the real saddle with the switching line Σ). Moreover, in the right figure, the dashed line corresponds to the part of the bisector of the fourth quadrant which is not hidden by the graph of the Poincaré half-map. Fig. 3. An example to illustrate the displacement function. It corresponds to a piecewise linear system composed by a (left) system with a real focus and a (right) system with a real saddle. 3. Displacement function: fundamental properties In this section, as it is usual for the analysis of limit cycles, a displacement function whose zeros correspond to the periodic orbits of system (5) will be defined by means of the difference of the Poincaré half-maps. In addition, its lower order derivatives will be computed to later provide us with the hyperbolicity and stability of the periodic orbits. Notice that any limit cycle of the differential system (5) is anticlockwise oriented and transversally crosses the switching line Σtwice. Indeed, a limit cycle cannot be entirely contained in the closure of a zone of linearity and cannot pass through the origin which is the unique tangency point of the flow of the differential system (5) in the switching line. Moreover, from now on till the end of this paper, we assume IL∩IR= ∅ because this is a necessary condition for the existence of crossing periodic orbits of the differential system (5). Notice that this trivially implies the existence of the Poincaré half-maps and includes the conditions a2 L+D2 L= 0 and a2 R+D2 R= 0. The displacement function δis now defined in the interval I:= IL∩IR= [λ0, µ0)= ∅ as δ:I−→ R y0↦−→ δ(y0)=yR(y0)−yL(y0),(20) where λ0=max{λL, λR}and µ0=min{µL, µR}.InFig. 3, we illustrate the displacement function for a scenario with a real focus and a real saddle. Notice that a crossing periodic orbit exists and passes through (0,y∗ 0), with y∗ 0∈int(I), if, and only if, y∗ 0is a zero of displacement function δ. Such a periodic orbit is a hyperbolic limit cycle provided that δ′(y∗ 0)= 0. In this case, it is attracting (resp. repelling) provided that δ′(y∗ 0)<0 (resp. δ′(y∗ 0)>0). In the next result, a relevant expression for the sign of this derivative will be deduced from Proposition 2. 7 V. Carmona, F. Fernández-Sánchez and D.D. Novaes Communications in Nonlinear Science and Numerical Simulation 123 (2023) 107257 Proposition 6. Suppose that y∗ 0∈int(I)satisfies δ(y∗ 0)=0. Denote y∗ 1=yR(y∗ 0)=yL(y∗ 0)<0and define c0:= aRaL(aRTL−aLTR), c1:= aRTRDL−aLTLDR, c2:= a2 LDR−a2 RDL. (21) Then, the following statements hold: (a) The derivative of the displacement function δdefined in (20) verifies sign(δ′(y∗ 0)) =sign(F(y∗ 0,y∗ 1)),(22) being F(y0,y1)=c0+c1y0y1+c2(y0+y1).(23) (b) Moreover, if δ′(y∗ 0)=0, then the second derivative of δverifies sign(δ′′(y∗ 0)) =sign (TL(c2y∗ 0+c0))= −sign (TR(c2y∗ 1+c0)).(24) Proof. Let us fix y∗ 0∈int(I) with y∗ 1=yR(y∗ 0)=yL(y∗ 0). From Proposition 2, it is straightforward to get δ′(y∗ 0)=y′ R(y∗ 0)−y′ L(y∗ 0)=C(y∗ 0,y∗ 1)F(y∗ 0,y∗ 1), where Cand Fare given by C(y0,y1)=−y0(y0−y1) y1WR(y0)WL(y0) and F(y0,y1)=WL(y1)WR(y0)−WL(y0)WR(y1) y0−y1 .(25) By substituting expressions (7) and (13) of polynomials WLand WRin expression (25), we see that, for y0= y1, F(y0,y1)=c0+c1y0y1+c2(y0+y1), where the coefficients c0,c1, and c2are given in (21). Since y∗ 0>0,y∗ 1<0, WR(y∗ 0)>0, and WL(y∗ 0)>0 (see the statements (e) from Theorems 1 and 2), then C(y∗ 0,y∗ 1)>0 and item (a) is proven. Now, from Eqs. (18) and (19) of Proposition 2, it follows that δ′′(y∗ 0)=y′′ R(y∗ 0)−y′′ L(y∗ 0)=(y∗ 0)2−(y∗ 1)2 (y∗ 1)3[a2 LWL(y∗ 1) WL(y∗ 0)2−a2 RWR(y∗ 1) WR(y∗ 0)2].(26) Taking into account equalities (22) and (25), it is clear that δ′(y∗ 0)=0 if, and only if, WL(y∗ 1)WR(y∗ 0)=WL(y∗ 0)WR(y∗ 1). Therefore, the assumption δ′(y∗ 0)=0 allows to write the second derivative (26) as δ′′(y∗ 0)=(y∗ 0)2−(y∗ 1)2 (y∗ 1)3 WR(y∗ 1)WL(y∗ 0) (WL(y∗ 0)WR(y∗ 0))2[a2 LWR(y∗ 0)−a2 RWL(y∗ 0)] =(y∗ 0+y∗ 1)y∗ 0−y∗ 1 (y∗ 1)3 WR(y∗ 1) WL(y∗ 0)(WR(y∗ 0))2y∗ 0[c2y∗ 0+c0]. Since y∗ 0>0,y∗ 1<0, WR(y∗ 1)>0, WL(y∗ 0)>0, from Proposition 5 we obtain sign(δ′′(y∗ 0))= −sign((y∗ 0+y∗ 1)(c2y∗ 0+c0))=sign (TL(c2y∗ 0+c0)). Analogous computations allow to prove the second equality of Eq. (24). Hence, the proof of item (b) is finished. □ Remark 2. Notice that the coefficient c0given in (21) is the product of three factors. The factors aLand aRare the nonhomogeneous terms of the differential system (5) and ξ:= aRTL−aLTR(27) has already appeared in Theorem Aand, as will be proven, determines the stability of the unique limit cycle, if any. Remark 3. Let us give some comments about the zero set γ=F−1({0}), for the function F given in (23). When c1(c2 2−c1c0)= 0, the set γis either the empty set, the whole plane, a straight line, or a pair of secant straight lines. We shall see that these 8 V. Carmona, F. Fernández-Sánchez and D.D. Novaes Communications in Nonlinear Science and Numerical Simulation 123 (2023) 107257 non-generic cases will not be relevant in our analysis. Therefore, in this Remark, we assume that c1(c2 2−c1c0)= 0and, consequently, the set γis a non-degenerate hyperbola. The asymptotes of γare given by y0= −c2/c1and y1= −c2/c1. On the one hand, except for the case c2=0(when the asymptotes coincide with the coordinate axes), the two intersection points between the hyperbola and the coordinate axes are (−c0/c2,0) and (0,−c0/c2). On the other hand, the center of γ, (that is, the intersection point between its asymptotes) is the point (−c2/c1,−c2/c1). Note that a center like that must lay either on the origin or in the first or third quadrant. Thus, at most one branch of the hyperbola may intersect the fourth quadrant Θıv, which would then be divided into two connected components. Regarding the intersection between γand the bisector of the fourth quadrant, notice that it only occurs when c0c1⩾0. In this case, the point of intersection is (s y0,s y1)=(√c0 c1 ,−√c0 c1).(28) Furthermore, it can be seen that if c0= 0then the intersection is transversal, and if c0=0then the point of intersection is the origin and the intersection is non-transversal. Finally, it is easy to see that the function y1=φ(y0), which describes the hyperbola γas a graph, is increasing when c0c1−c2 2>0and decreasing when c0c1−c2 2<0. 4. Stability of monodromic singularities and the infinity This section will be devoted to presenting some results about stability for monodromic singularities. In broad terms, the concept of monodromy is related to the rotation of the flow of the differential system. In order to precisely establish this concept for a point of the phase plane of the differential system (5), it is convenient to distinguish if it is located in the switching line Σor not: •A point of the differential system (5) outside the switching line Σis said to be a monodromic singularity when it is a focus or a center. •Due to the crossing behavior of the flow of the differential system (5) on Σ, the origin is the unique point of Σ around which rotation is possible and, moreover, this rotation is a consequence of the half-rotations for the half-planes {(x,y)∈R2:x⩽0}and {(x,y)∈R2:x⩾0}. A glimpse of the vector field of the differential system (5) allows to assert that the half-rotation around the origin in the half-plane {(x,y)∈R2:x⩽0}only occurs when aL=0 and 4DL−T2 L>0 or when aL>0. Analogously, the half-rotation around the origin in the half-plane {(x,y)∈R2:x⩾0}only happens when aR=0 and 4DR−T2 R>0 or when aR<0. Therefore, the origin is said to be a monodromic singularity if, and only if, one of the four feasible combinations above holds. The next result provides the stability of the origin when it is a monodromic singularity of (5) in the case aL>0 and aR<0. Proposition 7. Let ξbe defined as in expression (27). Consider the differential system (5) with aL>0and aR<0. Then, the Poincaré half-maps satisfy yR(0) =yL(0) =0and the origin is an attracting (resp. repelling) monodromic singularity provided that ξ > 0(resp. ξ < 0). Proof. When aL>0, from expression (8), it follows that yL(0) =0. In the same manner, if aR<0, then yR(0) =0. Since aL>0 and aR<0, the origin is a monodromic singularity. Its stability is obtained by computing the power series of the displacement function (20) around y0=0. Accordingly, from Proposition 3, we get δ(y0)=(2TL 3aL−2TR 3aR)y2 0+O(y3 0)=2ξ 3aLaR y2 0+O(y3 0).(29) Therefore, the monodromic singularity is attracting (resp. repelling) provided that ξ > 0 (resp. ξ < 0). □ Remark 4. Taking expression (29) into account we see that, under the hypotheses of Proposition 7, the first Lyapunov coefficient of the monodromic singularity of (5) at the origin (see, for instance, [15,19]) is given by the expression 2ξ/(3aLaR). Thus, under suitable assumptions, the parameter ξcan be used, among other things, to generate a limit cycle through a Hopf–like bifurcation. The next result states that, in the case of a unique monodromic singularity and TLTR<0, the parameter ξalso provides its stability, even if it is outside Σ. Proposition 8. Assume that both (forward and backward) Poincaré half-maps exist. Let c0and ξbe as given in (21) and (27), respectively. Consider the differential system (5) with TLTR<0and c0= 0. If the differential system (5) admits a unique monodromic singularity, then it is attracting (resp. repelling) provided that ξ > 0(resp. ξ < 0). Moreover, if the displacement function δcan be defined, then sign(δ(y0)) = −sign(ξ)for y0close enough to the left endpoint λ0of the interval of definition I of the displacement function δ. 9 V. Carmona, F. Fernández-Sánchez and D.D. Novaes Communications in Nonlinear Science and Numerical Simulation 123 (2023) 107257 Fig. 7. In case (C2), if the one of the Poincaré half-maps enters the region Ωit cannot leave this region. Since Ωis bounded and, in this case, the Poincaré half-maps are unbounded, they cannot enter the region Ω. Therefore, such maps do not intersect γ, which implies the uniqueness of hyperbolic limit cycles. The dashed curve represents the curve γ; the arrows represent the vectors fields XLand XR, which are proportional over γ; and the continuous curve represents one of the graphs of the Poincaré half-maps which is trapped in the colored region. By applying the change of variables y0=1/Y0,y1=1/Y1, for Y0>0 and Y1<0, to function Fgiven in (23), we obtain F(y0,y1)=c1+c0Y0Y1+c2(Y0+Y1) Y0Y1 . From Proposition 6, since Y∗ 0Y∗ 1<0, then sign(∆′(Y∗ 0)) = −sign(δ′(y∗ 0)) = −sign(F(y∗ 0,y∗ 1)) =sign(˜ F(Y∗ 0,Y∗ 1)), where ˜ F(Y0,Y1)=c1+c0Y0Y1+c2(Y0+Y1). Now, since ∆(0) =0, sign (∆′(0+))=sign(c1), and ˜ F(0,0) =c1= 0, one has •∆(0) =0; •˜ F(0,0) =c1; •and sign(∆(Y0)) =sign(c1) for Y0>0 sufficiently small. Hence, an analogous reasoning to case (B) provides that a hyperbolic limit cycle corresponding to a point (Y∗ 0,Y∗ 1)= (1/y∗ 0,1/y∗ 1) satisfies sign(∆′(Y∗ 0)) = −sign(c1), therefore it is unique and asymptotically stable (resp. unstable) provided that ξ < 0 (resp. ξ > 0), because sign(δ′(y∗ 0)) = −sign(∆′(Y∗ 0)) =sign(c1)= −sign(c0)=sign(ξ). (C2) Suppose that c0c1>0. Without loss of generality, we can assume that TL>0 and TR<0. Indeed, the case TR>0 and TL<0 can be reduced to the previous one by the change of variables and time rescaling (t,y)↦→ (−t,−y). Since aL<0 and TL>0, from (8), it follows that yL(0) <0. Analogously, since aR>0 and TR<0, from (14), it follows that yR(0) <0. Thus, according to Corollary 1, the graphs of both Poincaré half-maps must be located below the bisector of the fourth quadrant. Now, we will prove that the graphs of the Poincaré half-maps are included in H−. From relationship (31), one obtains c2 c1 aLTL= −a2 L−c0 c1 DLand c2 c0 aRTR= −c1 c0 a2 R−DR. Hence, c1c2>0 and c0c2>0. As a consequence, from Remark 3, the center of the hyperbola γis located at the third quadrant, γintersects the axis y0=0 for the value y1= −c0/c2<0 and the bisector of the fourth quadrant at the point given in (28). In this way, the curve γ, the axis y0=0, and the bisector of the fourth quadrant define a bounded region Ω⊂H+. From expressions (35) and (36), the orbits of the vector fields XLand XR, as y0increases, cross γfrom H−to H+only for 0<y0<√c0/c1. Accordingly, Ωis a bounded trapping region as y0increases for the graphs of the Poincaré half-maps. However, since these graphs are unbounded, they cannot enter Ωand, therefore, they do not intersect γ(see Fig. 7). This implies that they are included in H−and the conclusion of case (C2) follows by a similar reasoning to case (A). 7. Nonexistence of degenerate limit cycles From Proposition 12, we have proven that if a hyperbolic limit cycle exists for piecewise linear differential system (5), then it is unique and its stability is determined by the sign of ξdefined in (27). Hence, in this section, we conclude the proof of Theorem A by providing the nonexistence of degenerate limit cycles of the differential system (5). 16 V. Carmona, F. Fernández-Sánchez and D.D. Novaes Communications in Nonlinear Science and Numerical Simulation 123 (2023) 107257 Proposition 13. The differential system (5) does not admit degenerate limit cycles. Proof. Assume, by contradiction, that there exists a choice of the parameters (aL,aR,TL,TR,DL,DR)=(a∗ L,a∗ R,T∗ L,T∗ R,D∗ L,D∗ R) for which the differential system (5) admits a degenerate limit cycle passing through (0,y∗ 0) and (0,y∗ 1), with y∗ 1<0<y∗ 0, that is δ(y∗ 0)=δ′(y∗ 0)=0. A contradiction will be obtained by showing the existence of a saddle–node bifurcation. That is, we will see that, at y∗ 0 and (a∗ L,a∗ R,T∗ L,T∗ R,D∗ L,D∗ R), the second derivative of the displacement function δwith respect to y0and the first derivative with respect to a parameter are distinct from zero. Consequently, two simple zeros of the displacement function δwill bifurcate from the degenerate zero y∗ 0. These zeros correspond to two hyperbolic limit cycles of (5), which contradicts Proposition 12. Notice that, from Proposition 10, the inequality (a∗ L)2+(a∗ R)2= 0 holds. We shall prove the proposition assuming a∗ L= 0. An analogous reasoning can be done for the case a∗ R= 0. Let c∗ 0and c∗ 2as defined in (21) for the above fixed parameters. Since δ(y∗ 0)=δ′(y∗ 0)=0, from Proposition 6, one obtains that sign(δ′′(y∗ 0)) =sign (T∗ L(c∗ 2y∗ 0+c∗ 0))=sign (T∗ R(c∗ 2y∗ 1+c∗ 0)), and so sign(δ′′(y∗ 0)) = 0. Indeed, y∗ 0y∗ 1<0 and, from Proposition 10, (c∗ 0)2+(c∗ 2)2= 0 and T∗ LT∗ R= 0. Now, depending on the sign of a∗ L, we choose either TLor aLas the bifurcation parameter in order to unfold two limit cycles. First, suppose that a∗ L>0. Assume that the parameter TLis taken in a small neighborhood of T∗ Land that the other parameters are fixed as aL=a∗ L,aR=a∗ R,TR=T∗ R,DL=D∗ L, and DR=D∗ R. Notice that the corresponding displacement function δ, the Poincaré half-map yL, and the polynomial function WLvary with the parameter TL. Since a∗ L>0, the forward Poincaré half-map yLis provided by expression (8) as ∫y0 yL(y0;TL) −y WL(y;TL)dy =0, being WL(y;TL)=D∗ Ly2−a∗ LTLy+(a∗ L)2. Therefore, ∂yL ∂TL (y∗ 0;T∗ L)=a∗ L WL(y∗ 1;T∗ L) y∗ 1∫y∗ 0 y∗ 1(y WL(y;T∗ L))2 dy = 0. Since the displacement function (20) satisfies ∂δ ∂TL(y∗ 0;T∗ L)= −∂yL ∂TL(y∗ 0,T∗ L)= 0, the proof follows for the case a∗ L>0. Finally, suppose that a∗ L<0. Assume that the parameter aLis taken in a small neighborhood of a∗ Land that the other parameters are fixed as aR=a∗ R,TL=T∗ L,TR=T∗ R,DL=D∗ L, and DR=D∗ R. Notice that the corresponding displacement function δ, the Poincaré half-map yL, and the polynomial function WLnow vary with the parameter aL. In this case, the inequality 4D∗ L−(T∗ L)2>0 holds and the expression (8) writes as ∫y0 yL(y0;aL) −y WL(y;aL)dy =2πT∗ L D∗ L√4D∗ L−(T∗ L)2,(38) being WL(y;aL)=D∗ Ly2−aLT∗ Ly+a2 L. The changes of variable Y=y/aLtransforms Eq. (38) into the equation ∫y0/aL yL(y0;aL)/aL −Y WL(Y;1)dY =2πT∗ L D∗ L√4D∗ L−(T∗ L)2. Thus, it is easy to see that ∂yL ∂aL (y∗ 0;a∗ L)=(y∗ 0−y∗ 1)(T∗ Ly∗ 0y∗ 1−a∗ L(y∗ 0+y∗ 1)) y∗ 1WL(y∗ 0;a∗ L). Observe that sign(T∗ Ly∗ 0y∗ 1)= −sign(T∗ L) and, from Proposition 5, sign (−a∗ L(y∗ 0+y∗ 1))= −sign(T∗ L). Hence, sign (T∗ Ly∗ 0y∗ 1−a∗ L(y∗ 0+y∗ 1))= −sign (T∗ L)= 0, 17 V. Carmona, F. Fernández-Sánchez and D.D. Novaes Communications in Nonlinear Science and Numerical Simulation 123 (2023) 107257 which implies that ∂δ ∂aL(y∗ 0;a∗ L)= −∂yL ∂aL(y∗ 0;a∗ L)= 0. It concludes this proof. □ CRediT authorship contribution statement Victoriano Carmona: Conceptualization, Methodology, Formal analysis, Investigation, Writing – original draft, Writing – review & editing. Fernando Fernández-Sánchez: Conceptualization, Methodology, Formal analysis, Investigation, Writing – original draft, Writing – review & editing. Douglas D. Novaes: Conceptualization, Methodology, Formal analysis, Investigation, Writing – original draft, Writing – review & editing. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability No data was used for the research described in the article Acknowledgments We thank the referees for the helpful comments and suggestions. The authors thank Espaço da Escrita – Pró-Reitoria de Pesquisa – UNICAMP for the language services provided. VC is partially supported by the Ministerio de Ciencia, Innovación yUniversidades, Spain, Plan Nacional I+D+I cofinanced with FEDER funds, in the frame of the project PGC2018-096265-B-I00. FFS is partially supported by the Ministerio de Economía yCompetitividad, Spain, Plan Nacional I+D+I cofinanced with FEDER funds, in the frame of the project MTM2017-87915-C2-1-P. VC and FFS are partially supported by the Ministerio de Ciencia e Innovación, Spain, Plan Nacional I+D+I cofinanced with FEDER funds, in the frame of the project PID2021-123200NB-I00. VC and FFS are partially supported by the Consejería de Educación yCiencia de la Junta de Andalucía, Spain (TIC-0130, P12-FQM-1658). VC and FFS are partially supported by the Consejería de Economía, Conocimiento, Empresas yUniversidad de la Junta de Andalucía, Spain (US-1380740, P20-01160). DDN is partially supported by São Paulo Research Foundation (FAPESP), Brazil grants 2022/09633-5, 2021/10606-0, 2019/10269-3, and 2018/13481-0, and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil grants 438975/2018-9 and 309110/2021-1. References [1] Filippov AF. Differential equations with discontinuous righthand sides. Mathematics and its applications (Soviet series), vol. 18, Dordrecht: Kluwer Academic Publishers Group; 1988, p. x+304. [2] Lum R, Chua LO. Global properties of continuous piecewise linear vector fields. Part I: Simplest case in R2. Int J Circuit Theory Appl 1991;19(3):251–307. [3] Freire E, Ponce E, Rodrigo F, Torres F. Bifurcation sets of continuous piecewise linear systems with two zones. Internat J Bifur Chaos Appl Sci Engrg 1998;8(11):2073–97. [4] Carmona V, Fernández-Sánchez F, Novaes DD. A new simple proof for Lum–Chua’s conjecture. Nonlinear Anal Hybrid Syst 2021;40:100992. [5] Carmona V, Fernández-Sánchez F. 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