This is a preprint of: “On the limit cycles of a class of discontinuous piecewise linear differential systems”, Jaume Llibre, Lucyjane de A. S. Menezes, Discrete Contin. Dyn. Syst. Ser. B, vol. 25(5), 1835–1858, 2020. DOI: [10.3934/dcdsb.2020005] ON THE LIMIT CYCLES OF A CLASS OF DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS JAUME LLIBRE1 AND LUCYJANE DE A. S. MENEZES2 Abstract. In this paper we consider discontinuous piecewise linear differ- ential systems whose discontinuity set is a straight line L which does not pass through the origin and it is formed by the two linear differential sys- tems ẋ = Ax ± b. We study the limit cycles of this class of discontinuous piecewise linear differential systems. We do this study analyzing the fixed points of the return map of the system defined on the straight line L. 1. Introduction and statement of the main results The theory of discontinuous piecewise differential systems is in constant development due to its applicability in different areas of the knowledge such as ecology, mechanic and electrical engineering, see for instance [6]. However even in the planar case there are important questions unsolved for this class of differential system as to know the number of their limit cycles. In 2010 Han and Zhang [4] conjecture that piecewise linear systems with only two regions have at most two limit cycles. In 2012 Huan and Yang [5] investigate the number of limit cycles of planar piecewise linear systems with two regions sharing the same equilibrium. Moreover they provide a numerical example to illustrate the existence of three limit cycles, thus have a negative answer to the conjecture by Han and Zhang. In 2012 Llibre and Ponce [10] provide a rigorous proof of the existence of such three limit cycles. This was the first example that a discontinuous differential piecewise linear systems with two regions can have three limit cycles. Many others researchers have analyzed the existence of limit cycles for a piecewise linear systems with two regions separated by a straight line. In [3] is proved that a such piecewise linear system has at most two limit cycles when the singularities are both virtual focus or center. In [1] the authors consider a piecewise linear system separated by a straight line with singularities of type real saddle and prove that this system has at most two limit cycles. 2010 Mathematics Subject Classification. 34C05, 34C07. Key words and phrases. discontinuous systems, Poincaré map, return map, limit cycles. 1 2 J. LLIBRE AND L.A.S. MENEZES Consider the 2 × 2 real matrix A+ and A− , and b+ , b− ∈ R2 . We define the planar piecewise discontinuous linear systems   X(x) = A+ x + b+ if x > 0, 1 (1) ẋ =  Y (x) = A− x + b− if x1 < 0, where x = (x1 , x2 ) ∈ R2 . Note that L = h−1 (0) = {x ∈ R2 ; x1 = 0} split the plane in two open regions S + = {x ∈ R2 ; x1 > 0} and S − = {x ∈ R2 ; x1 < 0}. We say that a limit cycle of system (1) is a crossing limit cycle if it share no points with the sliding set of the system. In [2] Cespedes studies systems (1) satisfying div(X)div(Y ) ≥ 0, i.e. the product of the divergences of the subsystems X and Y is non–negative, and show that such systems have at most two limit cycles. Moreover this author exhibited an example with exactly two crossing limit cycles. In this paper we consider S+ = {(x1 , x2 ) ∈ R2 x: 1 > 1} and S− = {(x1 , x2 ) ∈ R2 : x1 < 1} and piecewise linear systems with two zones given by   X(x) = Ax + b if x ∈ S , + (2) ẋ =  Y (x) = Ax − b if x ∈ S− , where b ∈ R2 \ {0}. In this case the discontinuity of system (2) is the straight line L = {(x1 , x2 ) ∈ R2 : x1 = 1}. Assuming that there are no singularities of system (2) in L we shall study the existence of crossing limit cycles. Set Φ± (t, x) the flow of the system X and Y , respectively. The flow Φ+ is said transversal to L at point p if X(p) is not contained in L. If X(p) ∈ L then the point p is called a contact point of the flow with L. Analogous definitions hold for system Y . We say that p ∈ R2 is a real singularity of system (2) if p = (x1 , x2 ) is such that either x1 > 1 and X(p) = 0, or x1 < 1 and Y (p) = 0. On the other hand p is a virtual singularity if p = (x1 , x2 ) is such that either x1 > 1 and Y (p) = 0, or x1 < 1 and X(p) = 0. It follows that the discontinuous system (2) can be of type virtual–virtual, virtual–real and real–real depending if the singularities of the systems X and Y are virtual or real. In this work d and t denote the determinant and trace of the matrix A, respectively. Our main results are the following. Theorem A. Consider the discontinuous piecewise linear differential system (2). If the contact points of system (2) with the straight line L coincides then system (2) has no limit cycles. Theorem B. Assume that the discontinuous piecewise linear differential sys- tem (2) is of type virtual–virtual. If the contact points of the systems X and Y ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 3 are distinct, then system (2) has at most one limit cycle. There exists necessary and sufficiently conditions for the existence of exactly one limit cycle. Observe that system (2) satisfy div(X)div(Y ) ≥ 0, but this work provides new results with respect the ones obtained in chapter 2 of [2]. One of the most important tools in the study of the periodic orbits are the Poincaré maps. These maps characterize the behavior of the flows in the neigh- borhood of periodic orbits. Moreover, there exists a correspondence between the limit cycles of a system and the fixed points of same Poincaré map. In [8] Llibre and Teruel studied Poincaré maps for piecewise linear differential sys- tems in Rn . In [9] these authors determine the Poincaré maps and analyze the existence of crossing limit cycles of planar piecewise linear differential systems defined in three regions separated by two straight lines L+ and L− which are symmetric with respect to the origin. In order to prove Theorems A and B we determine the Poincaré maps of the subsystems X and Y of system (2) with respect to the straight line L, and we use these maps to determine the return map of system (2) in L. After that we analyze the existence of fixed points of this return map. The paper is organized as follows. In section 2 we provide the basic results that we shall need to prove the mains results. Section 3 is split in three subsections, in subsection 3.1 and 3.2 we study existence of the Poincaré maps for the non–homogeneous systems X and Y , respectively. In subsection 3.3 we define the return map for the discontinuous piecewise linear differential system (2). In section 4 we study the existence of limit cycle for the discontinuous system (2) and prove our main results. 2. Preliminary results Consider a linear differential system (3) ẋ = Ax, where x = (x1 , x2 ) ∈ R . Given p, v ∈ R2 let L = {p + λv : λ ∈ R} be a 2 straight line that does not pass through the origin, and let n be a unit vector orthogonal to L such that nT p > 0, we say that n is oriented in the opposite sense to the origin. Furthermore if nq̇ 6= 0 the flow of system (3) is transversal to the line L at a point q ∈ L, where q̇ denotes the vector field of the differential system (3) evaluated at q. Otherwise q is a contact point of the flow with the straight line L. The following result is proved in Proposition 4.2.7 of [9]. Proposition 1. Consider the differential system (3) with A ∈ GL(R2 ). Let L be a straight line in the phase plane not passing through the origin, p a contact 4 J. LLIBRE AND L.A.S. MENEZES point of the flow in L, and x(t) the solution of the system such that x(0) = p. Then (a) if det(A) > 0 there exists ε > 0 such that {x(t); t ∈ (−ε, ε)} ⊂ S0 ∪ L, (b) if det(A) < 0 there exists ε > 0 such that {x(t); t ∈ (−ε, ε)} ⊂ S ∪ L, where S0 and S are the half–planes bounded by the straight line L, being S0 the half–plane containing the origin. A transversal flow to L at a point q ∈ L is said to have outside orientation if nT q̇ > 0, and it is said to have inside orientation if nT q̇ < 0. Thus we define the following subsets in L LI = {q ∈ L : nT q̇ ≤ 0} and LO = {q ∈ L : nT q̇ ≥ 0}. If det A 6= 0 and there exists a contact point p of the flow of system (3) in L, then LI and LO are two half–lines such that the flow over LI and LO has opposite sense and LI ∩ LO = p. This is part of the next result that is proved in Proposition 4.2.5 of [9]. Proposition 2. Consider the differential system (3) with A ∈ GL(R2 ). Let L be a straight line in the phase plane not passing through the origin. (a) There exists at most one contact point of the flow with L. (b) Let p be a contact point of the flow with L. If det A > 0, then LI = {p + λṗ : λ ≥ 0} and LO = {p + λṗ : λ ≤ 0}. If det A < 0, then LI = {p + λṗ : λ ≤ 0} and LO = {p + λṗ : λ ≥ 0}. (c) If LI 6= ∅ and LO 6= ∅, then there exists exactly one contact point of the flow with L. (d) If the flow has no contact points with L, then either LI = L and LO = ∅, or LI = ∅ and LO = L. Poincaré maps of a homogeneous linear system Consider the system (3) and two parallel straight lines in the plane L+ and L− which are symmetric with respect to origin. Notice that the lines L+ and L− split the plane in three regions S0 , S+ , and S− , where S0 is the open strip containing the origin and S+ and S− the half–planes bounded by L+ and L− , ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 5 respectively. Moreover, we can define the following subsets,  Dom++ = q ∈ L+ : ∃tq > 0 such that eAtq q ∈ L+ and either eAt q ⊂ S+ or eAt q ⊂ S0 ∀t ∈ (0, tq ) } ∪ CP+ ,  Dom+− = q ∈ L+ : ∃tq > 0 such that eAtq q ∈ L− and eAt q ⊂ S0 ∀t ∈ (0, tq ) } , (4)  Dom−− = q ∈ L− : ∃tq > 0 such that eAtq q ∈ L− and either eAt q ⊂ S− or eAt q ⊂ S0 ∀t ∈ (0, tq ) ∪ CP− ,  Dom−+ = q ∈ L− : ∃tq > 0 such that eAtq q ∈ L+ , and eAt q ⊂ S0 ∀t ∈ (0, tq ) , where CP+ and CP− are either empty, sets or consist of contact points of the flow with the lines L+ and L− , respectively. We have the following result that is proved in Lemma 4.3.2 of [9]. Proposition 3. Let ẋ = Ax be a planar linear differential system with A non identically zero. Consider L+ and L− two parallel straight lines symmetric with respect to the origin and let Domjk be the sets defined in (4). Assume that for some j, k ∈ {+, −} the set Domjk 6= ∅. Then there exists a unique contact point p+ of the flow in L+ , and p− = −p+ is the unique contact point of the flow in L− . Provided that Domjk 6= ∅, for j, k ∈ {+, −}, we define the Poincaré map Πjk : Domjk ⊂ Lj −→ Lk of the linear differential system (3) associated to the lines Lj and Lk as Πjk (q) = eAtq q. In what follows we present some results on the domains of definition of these Poincaré maps and necessary and sufficient conditions for the existence of these maps. Notice that when Dom++ 6= ∅ and Dom−− 6= ∅, the contact points p+ and p− split L+ and L− into respective half–lines LI+ , LO I O + , L− , and L− . We have the following results. Proposition 4. Consider a planar linear differential system ẋ = Ax with A non identically zero. Set L+ and L− two parallel straight lines symmetric with respect to the origin and let Domjk be the sets defined in (4). Suppose that Domjk 6= ∅ for every j, k ∈ {+, −}. 6 J. LLIBRE AND L.A.S. MENEZES (i) If det A > 0, then Π++ : Dom++ ⊂ LO I + −→ L+ , Π+− : Dom+− ⊂ LI+ −→ LO −, (5) Π−− : Dom−− ⊂ LO I − −→ L− , Π−+ : Dom−+ ⊂ LI− −→ LO +. (ii) If det A < 0, then Π++ : Dom++ ⊂ LI+ −→ LO +, Π+− : Dom+− ⊂ LI+ −→ LO −, (6) Π−− : Dom−− ⊂ LI− −→ LO −, Π−+ : Dom−+ ⊂ LI− −→ LO +. (iii) If det A = 0 then Dom++ = {p+ }, Dom−− = {p− }, Π−− (p− ) = {p− }, and Π++ (p+ ) = {p+ } Π+− : Dom+− ⊂ LI+ −→ LO −, (7) Π−+ : Dom−+ ⊂ LI− −→ LO +. Proposition 5. Let ẋ = Ax be a planar linear differential system with A non identically zero. Consider L+ and L− two parallel straight lines symmetric with respect to the origin. The Poincaré maps which appear in the statement of Proposition 4 are defined if and only if the flow of the system has a unique contact point with L+ . The proofs of the two previous propositions can be found in Propositions 4.3.3 and 4.3.4 of [9], respectively. 2.1. Qualitative behavior of the Poincaré maps. In this subsection we analyze the qualitative behavior of the Poincaré maps defined by the flow of system (3) and associated to straight lines L+ and L− . Assume that the Poincaré maps Πjk are defined, for j, k ∈ {+, −}. By Proposition 5 there exists a unique contact point p of the flow of system (3) with the straight line L+ and, therefore −p is the unique contact point of the flow of system (3) with the straight line L− . Thus L+ = {p + λṗ : λ ∈ R} and L− = {−p + λṗ : λ ∈ R}. ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 7 Observe that, by Proposition 2(b), if det A > 0 then LI+ = {p + aṗ : a ≥ 0}, LO + = {p − aṗ : a ≥ 0}, (8) LI− = {−p − aṗ : a ≥ 0}, LO − = {−p + aṗ : a ≥ 0}, and if det A < 0 then LI+ = {p − aṗ : a ≥ 0}, LO + = {p + aṗ : a ≥ 0}, (9) LI− = {−p + aṗ : a ≥ 0}, LO − = {−p − aṗ : a ≥ 0}. It follows that given any point q on L+ or on L− we can associated a unique a ≥ 0, called the coordinate of q. Consider q1 ∈ Lj and q2 ∈ Lk such that Πjk (q1 ) = q2 , where j, k ∈ {+, −}. Let a1 and a2 be the coordinates of q1 and q2 , respectively. Then we define the Poincaré maps πjk by πjk (a1 ) = a2 . Thus to know the qualitative behavior of the map πjk is equivalent to know the qualitative behavior of the Poincaré map Πjk . The next results, proved in Lemma 4.3.5 and Proposition 4.3.7 of [9], re- spectively, provide some properties of the Poincaré maps πjk . Proposition 6. Consider the linear differential system (3) and let L+ and L− two parallel straight lines which are symmetric with the respect to the origin. Suppose that the Poincaré maps πjk with j, k ∈ {+, −} are defined. Then (a) the maps π++ and π−− coincides. (b) the maps π+− and π−+ coincides. ∗ (c) the Poincaré maps πjk associated to the flow of the system ẋ = −Ax ∗ −1 and to the lines L+ and L− are defined, and they satisfy πjk = πjk . −1 (d) πjk and πjk are analytic functions. Proposition 7. Consider the linear differential system (3) and let L+ and L− two parallel straight lines which are symmetric with the respect to the origin. Assume that the Poincaré maps πjk with j, k ∈ {+, −} are defined. If M ∈ GL(R2 ), then the maps πjk are invariant under the change of coordinates y = M x. Assume that the maps π++ and π+− are defined. Since these maps are invariant under linear changes of coordinates, see Proposition 7, in what follows we consider A given in its real Jordan normal form. Moreover we denote the eigenvalues of A by λ1 and λ2 . In what follows we characterize the behavior of the Poincaré map π++ depending on t and d. Moreover we characterize the behavior of composition π e++ = π−+ ◦ π−− ◦ π+− when d > 0 and t2 − 4d < 0. Saddle: d < 0. 8 J. LLIBRE AND L.A.S. MENEZES The next two results are proved in Proposition 4.4.15 and Corollary 4.4.16 of [9], respectively. Proposition 8. Assume that d < 0, t ≥ 0. Then the eigenvalues of the matrix A satisfy λ1 > 0 > λ2 . Let π++ be the Poincaré map defined by the flow of the linear system ẋ = Ax and associated to the parallel straight lines L+ and L− symmetric with  respect to the origin. If t = 0, then π++ is the identity map on the interval 0, λ−1 1 ), and if t > 0, then  −1 (a) π++ : 0, λ1 ) → [0, |λ2 |−1 ), π++ (0) = 0, lim−1 π++ (a) = |λ2 |−1 , and a%λ1 π++ (a) > a in (0, λ−1 1 ). (b) if a ∈ (0, λ−1 1 ), then 0 π++ (a) 0 > 1. Furthermore, lim π++ (a) = 1 and a&0 0 lim−1 π++ (a) = +∞. a%λ1 (c) if a ∈ (0, λ−1 00 1 ), then π++ (a) > 0. (d) the graph of π++ has a vertical asymptote at a = λ−11 . (e) π++ is implicitly defined by the equation (10) √ √ t + t2 − 4d √   √ 2 + π++ (a)(t − t2 − 4d) t − t2 − 4d 2 + π++ (a)(t + t2 − 4d) √ = √ . 2 − a(t − t2 − 4d) 2 − a(t + t2 − 4d) (f) The qualitative behavior of the graph of π++ is represented in Figure 1-(a). b b π++ (a) b=a b=a |λ2 |−1 |λ2 |−1 π++ (a) λ−1 1 a λ−1 a 1 (a) (b) Figure 1. Qualitative behavior of the Poincaré map π++ ; (a) t > 0 and (b) t < 0. Proposition 9. Assume that d < 0, t < 0. Then the eigenvalues of the matrix A satisfy λ1 > 0 > λ2 . Let π++ be the Poincaré map defined by the flow of the linear system ẋ = Ax and associated to the parallel straight lines L+ and L− ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 9 symmetric with  respect to the origin. If t = 0, then π++ is the identity map on the interval 0, λ−1 1 ), and if t > 0, then  (a) π++ : 0, λ−1 1 ) → [0, |λ2 | −1 ), π++ (0) = 0, lim−1 π++ (a) = |λ2 |−1 , and a%λ1 π++ (a) < a in (0, λ−1 1 ). (b) if a ∈ (0, λ−1 0 0 1 ), then 0 < π++ (a) < 1 and lim π++ (a) = 1. a&0 (c) if a ∈ (0, λ−1 1 ),then 00 π++ (a) < 0. (d) the graph of π++ has a horizontal asymptote at a = λ−1 1 when a tends to +∞. (e) π++ is implicitly defined by equation (10). (f) The qualitative behavior of the graph of π++ is represented in Figure 1-(b). Diagonal node: d > 0 and t2 − 4d > 0. The following results are proved in Proposition 4.4.1 and Corollary 4.4.2 of [9], respectively. Proposition 10. Assume that d > 0, t > 0, and t2 − 4d > 0. Then the eigenvalues of the matrix A satisfy λ1 > λ2 > 0. Let π++ be the Poincaré map defined by the flow of the linear system ẋ = Ax and associated to the parallel straight lines L+ and L− symmetric with respect to the origin. Then  (a) π++ : 0, λ−1 1 ) → [0, +∞), π++ (0) = 0, lim −1 π++ (a) = +∞, and a%λ1 π++ (a) > a in (0, λ−1 1 ). −1 0 0 (b) if a ∈ (0, λ1 ), then π++ (a) > 1 and lim π++ (a) = 1. a&0 (c) if a ∈ (0, λ−1 1 ),then 00 π++ (a) > 0. (d) the graph of π++ has a vertical asymptote at a = λ−1 1 . (e) π++ is implicitly defined by the equation (11) √ √ t + t2 − 4d √   √ 2 + π++ (a)(t − t2 − 4d) t − t2 − 4d 2 + π++ (a)(t + t2 − 4d) √ = √ . 2 − a(t − t2 − 4d) 2 − a(t + t2 − 4d) (f) The qualitative behavior of the graph of π++ is represented in Figure 2-(a). Proposition 11. Assume that d > 0, t < 0, and t2 − 4d > 0. Then the eigenvalues of the matrix A satisfy λ2 < λ1 < 0. Let π++ be the Poincaré map defined by the flow of the linear system ẋ = Ax and associated to the parallel straight lines L+ and L− symmetric with respect to the origin. Then 10 J. LLIBRE AND L.A.S. MENEZES b b π++ (a) b=a b=a |λ2 |−1 π++ (a) λ−1 1 a a (a) (b) Figure 2. Qualitative behavior of the Poincaré map π++ ; (a) t > 0 and (b) t < 0. (a) π++ : [0, +∞ ) → [0, |λ2 |−1 ), π++ (0) = 0, lim π++ (a) = |λ2 |−1 , and a%+∞ −1 π++ (a) < a in (0, |λ2 | ). 0 0 (b) if a ∈ (0, +∞), then 0 < π++ (a) < 1 and lim π++ (a) = 1. a&0 00 (c) if a ∈ (0, +∞), then π++ (a) < 0. (d) the graph of π++ has a horizontal asymptote at b = |λ2 |−1 . (e) π++ is implicitly defined by equation (11). (f) The qualitative behavior of the graph of π++ is represented in Figure 2-(b). Non–diagonal node: d > 0 and t2 − 4d = 0. The next two results are proved in Proposition 4.4.6 and Corollary 4.4.7 of [9], respectively. Proposition 12. Assume that d > 0, t > 0, and t2 − 4d = 0. Then the eigenvalues of the matrix A satisfy λ1 = λ2 = λ. Suppose that A does not diagonalizable and let π++ be the Poincaré map defined by the flow of the linear system ẋ = Ax and associated to the parallel straight lines L+ and L− symmetric with respect to the origin. Then (a) π++ : [0, λ−1 ) → [0, +∞ ), π++ (0) = 0, lim π++ (a) = +∞, and a%λ−1 π++ (a) > a in the interval (0, λ−1 ). (b) if a ∈ (0, λ−1 ), then π++ 0 0 (a) > 1 and lim π++ (a) = 1. a&0 (c) if a ∈ (0, λ−1 1 ), then 00 π++ (a) > 0. (d) the graph of π++ has a vertical asymptote at a = λ−1 . ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 11 (e) π++ is implicitly defined by the equation 2t(π++ (a) + a) tπ++ (a) + 2 (12) = e (tπ++ (a) + 2)(2 − at) . 2 − at (f) The qualitative behavior of the graph of π++ is represented in Figure 2-(a). Proposition 13. Suppose that d > 0, t < 0, and t2 − 4d = 0. Then the eigenvalues of the matrix A satisfy λ2 = λ1 = λ < 0. Assume that A does not diagonalize and let π++ be the Poincaré map defined by the flow of the linear system ẋ = Ax and associated to the parallel straight lines L+ and L− symmetric with respect to the origin. Then (a) π++ : [0, +∞) → [0, |λ|−1 ), π++ (0) = 0, lim π++ (a) = |λ|−1 , and a%+∞ π++ (a) < a on the interval (0, +∞). 0 0 (b) if a ∈ (0, +∞), then 0 < π++ (a) < 1 and lim π++ (a) = 1. a&0 00 (c) if a ∈ (0, +∞), then π++ (a) < 0 in (0, +∞). (d) when a tends to +∞, the graph of π++ has a horizontal asymptote at b = |λ|−1 . (e) π++ is implicitly defined by equation (12). (f) The qualitative behavior of the graph of π++ is represented in Figure 2-(b). Degenerate node: d = 0. In this case A has one null eigenvalue and other equal to t. Hence the matrix A has two different real Jordan normal forms. One being for t 6= 0 and the other one for t = 0. In any case the behavior of the Poincaré map is defined only in a contact point, see Proposition 4-(iii). Thus the map π++ is defined only at zero and π++ (0) = 0. Center and focus: d > 0 and t2 − 4d < 0. The results for the centers and foci are proved on sections 4.4 and 4.5 of [9]. The following results are Proposition 4.4.11 and Corollary 4.4.12 of [9], re- spectively. Proposition 14. Suppose that d > 0, t ≥ 0, and t2 − 4d < 0. Then the matrix A has a pair of complex eigenvalues. Let π++ be the Poincaré map defined by the flow of the linear system ẋ = Ax and associated to the parallel straight lines L+ and L− symmetric with respect to the origin. If t = 0 then π++ is the identity in [0, +∞), and if t > 0 then 12 J. LLIBRE AND L.A.S. MENEZES (a) π++ : [0, +∞) → [0, +∞), π++ (0) = 0, lim π++ (a) = +∞, and a%+∞ π++ (a) > a on the interval (0, +∞). 0 0 (b) if a ∈ (0, +∞), then π++ (a) > 1 and lim π++ (a) = 1. a&0 00 (c) if a ∈ (0, +∞), then π++ (a) > 0. γπ (d) when a tends to +∞, the graph γπ √ of π++ has an asymptote at b = ae − t(1 + e )/d, where γ = t/ 4d − t . 2 (e) π++ is implicitly defined by the equation   (a + π++ (a))β 2γ arctan  1 + tπ++ (a) + dπ++ (a)2 (π++ (a) − a)α + 1 − adπ++ (a) . (13) = e 1 − ta + da2 (f) The qualitative behavior of the graph of π++ is represented in Figure 3-(a). b b b = eγπ − dt (1 + eγπ ) b = eγπ − dt (1 + eγπ ) b=a b=a π++ (a) π++ (a) a a (a) (b) Figure 3. Qualitative behavior of the Poincaré map π++ ; (a) t > 0 and (b) t < 0. Proposition 15. Suppose that d > 0, t < 0, and t2 − 4d < 0. Then the matrix A has a pair of complex eigenvalues. Let π++ be the Poincaré map defined by the flow of the linear system ẋ = Ax and associated to the parallel straight lines L+ and L− symmetric with respect to the origin. Then (a) π++ : [0, +∞) → [0, +∞), π++ (0) = 0, lim π++ (a) = +∞, and a%+∞ π++ (a) < a on the interval (0, +∞). 0 0 (b) if a ∈ (0, +∞), then 0 < π++ (a) < 1 and lim π++ (a) = 1. a%0 00 (c) if a ∈ (0, +∞), then π++ (a) < 0. (d) when a tends to +∞, the straight line b = √ aeγπ − t(1 + eγπ )/d is an asymptote of the graph of π++ , where γ = t/ 4d − t2 . ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 13 (e) π++ is implicitly defined by equation (13). (f) The qualitative behavior of the graph of π++ is represented in Figure 3-(b). e++ = π+− ◦ π−− ◦ π−+ . The proof Next results characterizes the behavior of π can be found in Proposition 4.5.7 and Corollary 4.5.8 of [9], respectively. Proposition 16. Consider a matrix A ∈ GL(R2 ) such that d > 0, t ≥ 0, and t2 − 4d < 0, and a vector b ∈ R2 \ {0}. Let π e++ be the Poincaré map defined by the flow of the linear system ẋ = Ax + b and associated to the parallel straight lines L+ and L− symmetric with respect to the origin. If t = 0 then π e++ is the identity in [0, +∞). On the other hand if t > 0 then (a) there exist a value b∗ > 0 such that π e++ : [0, +∞) → [b∗ , +∞ ) and e++ (0) = b∗ . Furthermore, lim π π e++ (a) = +∞ and πe++ (a) > a in a%+∞ (0, +∞). 0 0 (b) if a ∈ (0, +∞), then π e++ e++ (a) > 0 and lim π (a) = 0. a&0 00 (c) if a ∈ (0, +∞), then πe++ (a) > 0. e++ has an asymptote (d) the graph of π √ at b = aeγπ + t(1 + eγπ )/d when a tends to +∞, where γ = t/ 4d − t2 . e++ is implicitly defined by the equation (e) π   e++ (a))β (a + π 2γ arctan  1 − te π++ (a)2 π++ (a) + de π++ (a) − a)α − 1 + ade (e π++ (a) . (14) =e 1 + ta + da2 e++ is represented in Fig- (f) The qualitative behavior of the graph of map π ure 4–(a). Proposition 17. Consider a matrix A ∈ GL(R2 ) such that d > 0, t < 0, and t2 − 4d < 0, and a vector b ∈ R2 \ {0}. Let π e++ be the Poincaré map defined by the flow of the linear system ẋ = Ax + b and associated to the parallel straight lines L+ and L− symmetric with respect to the origin. (a) there exist a value a∗ > 0 such that π e++ : [a∗ , +∞ ) → [0, +∞ ) and ∗ e++ (a ) = 0. Furthermore, lim π π e++ (a) = +∞ and π e++ (a) < a in a%+∞ (a∗ , +∞). (b) if a ∈ (a∗ , +∞), then π 0 e++ 0 e++ (a) > 0 and lim∗ π (a) = +∞. a&a (c) if a ∈ (a∗ , +∞), then π 00 e++ (a) < 0. (d) the graph of πe++ has an asymptote √ at b = aeγπ + t(1 + eγπ )/d when a tends to +∞, where γ = t/ 4d − t2 . e++ is implicitly defined by equation (14). (e) π 14 J. LLIBRE AND L.A.S. MENEZES e++ is represented in Fig- (f) The qualitative behavior of the graph of map π ure 4–(b). b b = eγπ a + dt (1 + eγπ ) b e++ (a) π b = eγπ a + dt (1 + eγπ ) b=a b=a b∗ e++ (a) π a a∗ a (a) (b) e++ ; Figure 4. Qualitative behavior of the Poincaré map π (a) t > 0 and (b) t < 0. 3. Return map of discontinuous differential system Consider the following discontinuous piecewise linear differential system   Ax + b, if x > 1, 1 (15) ẋ =  Ax − b, if x1 < 1, where x = (x1 , x2 ) ∈ R2 and b ∈ R2 \ {0}. The discontinuity of this system is the straight line L = {(x1 , x2 ) ∈ R2 , x1 = 1}. Assume that the singularities of system (15) not belong to L. If d = 0. In this section we analyze the Poincaré maps defined by the flow of system (15) and associated to the straight line L and we define the return map for this system. In what follows we denote by n the unit orthogonal vector to the line L which is oriented in the direction opposite to the origin, and S+ = {(x1 , x2 ) ∈ R2 , x1 > 1} and S− = {(x1 , x2 ) ∈ R2 , x1 < 1} denote the half–planes bounded by L. In the study of the Poincaré maps associated to the flow of the linear differ- ential systems ẋ = Ax ± b with respect to the straight line L it by we denote L± to specify what system we are considering. 3.1. Poincaré maps of linear differential system ẋ = Ax + b. In this sub- section we study the Poincaré maps defined by the flow of the linear differential system (16) ẋ = Ax + b ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 15 and associated to the straight line L+ . Since L+ does not pass through the origin it can be split into the subsets LI+ = {q ∈ L : nT q̇ ≤ 0} and LO + = {q ∈ T I O L : n q̇ ≥ 0}, where q̇ = Aq + b. Then the set CP+ = L+ ∩ L+ consists of contact points of the flow of system (16) with L+ . Set Φ+ (t, q) the flow of system (16) such that Φ+ (0, q) = q. Define in L+ the subset  Dom++ = q ∈ LO I + : ∃tq ≥ 0, Φ+ (tq , q) ∈ L+ and Φ+ (t, q) ⊂ S+ ∀t ∈ (0, tq ) ∪ CP+ . Assuming that Dom++ 6= ∅ we define the Poincaré map of the linear differ- ential system (16) associated to the straight line L+ by Π+ : Dom++ ⊂ LO I + −→ L+ (17) q 7−→ Φ+ (tq , q). Suppose that the flow of system (16) has a unique contact point p with L+ . We have that Π+ (p) = p because p ∈ LI+ and p ∈ LO +. Let e+ = −A−1 b be the singularity of system (16). Applying the translation y = x − e+ we rewrite system (16) as (18) ẏ = Ay, and the straight line L+ is transformed into the straight line L∗+ . If the Poincaré map Π+ , given in (17), is defined then it induces a Poincaré map Π∗++ asso- ciated to the flow of system (18) and to the straight line L∗+ . Clearly the converse statement is also true and, therefore the behavior of the map Π+ can be obtained from the behavior of the map Π∗++ . We have the following result that are proved in Propositions 4.5.1 of [9]. Proposition 18. Consider L+ a straight line in the plane not passing through the origin and such that e+ = −A−1 b ∈ / L+ . The Poincaré map Π+ associated to the flow of system ẋ = Ax + b and to the straight line L+ is defined if and only if there exists a unique contact point p+ of the flow with L+ , and LI+ and LO I + are non–empty half–lines such that L+ = L+ ∪ L+ . O Since e+ ∈/ L+ we have that L∗+ does not pass through the origin, and we can define the symmetric line −L∗+ = L∗− = {−q; q ∈ L∗+ }. Therefore the results of section 2 can be applied to study the Poincaré maps of system (18) associated to the straight lines L∗+ and L∗− symmetric with respect to the origin. Set Π∗++ the Poincaré map induced by translation y = x − e+ , and Πjk with j, k ∈ {+, −} the Poincaré maps defined for homogeneous case, see Proposition 4. 16 J. LLIBRE AND L.A.S. MENEZES Proposition 19. Consider L+ a straight line in the plane not passing through the origin and such that e+ = −A−1 b ∈ / L+ . Assume that the Poincaré map Π+ associated to the flow of system ẋ = Ax + b and to the straight line L+ is defined. Let Π∗++ be the Poincaré map of system (18). (i) Suppose that e+ ∈ S− . – If det A > 0, then Π∗++ is the Poincaré map Π++ . – If det A < 0, then Π∗++ is trivial, i.e. the map Π∗++ is only defined in a contact point of the L∗+ . (ii) Suppose that e+ ∈ S+ . – If det A > 0 and t2 − 4d ≥ 0, then Π∗++ is trivial. – If det A > 0 and t2 − 4d < 0, then Π∗++ coincides with the compo- sition Π−+ ◦ Π−− ◦ Π+− . – Assume that det A < 0. Then Π∗++ is the Poincaré map Π++ . Proof. (i) We have that the translation y = x − e+ transforms LI+ into L∗I + ∗O and LO+ into L+ because e+ ∈ S− , see Figure 5-(a). Furthermore the domain Dom++ of Π∗++ is contained in L∗O + . Assuming that det A > 0, we have that the Poincaré map defined by a linear flow having the domain contained in L∗O+ is the map Π++ defined in subsection 2, see Proposition 4(i). On the other hand if det A < 0 then, by Proposition 4(ii), the domain of the map Π∗++ is contained in L∗I + and, therefore the domain of definition of the map Π++ is contained in the intersection L∗I ∗ ∗O + ∩ L+ . That is ∗ the map Π++ is defined only in the contact point. Consequently the behavior of Π∗++ is trivial. LI+ = L∗I + LI+ = L∗O + S− S+ S− S+ p+ e+ p+ e+ (a) ∗O (b) ∗I LO + = L+ LO + = L+ ∗O Figure 5. Relation between the half–lines LO I + , L+ , L+ , and ∗I L+ depending on (a) e+ ∈ S− , (b) e+ ∈ S+ . ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 17 (ii) By hypothesis we have that e+ ∈ S+ , thus the translation y = x − e+ transforms LI+ into L∗O O ∗I + and L+ into L+ , see figure 5-(b). Assuming that det A > 0 Proposition 4(i) implies that there is no Poincaré map Π++ , associated to the flow of a homogeneous linear system and to two parallel straight lines L+ and L− symmetric with respect to the origin, defined on LI+ with the image contained on LO + . This implies that either the behavior ∗ ∗ of Π++ is trivial or Π++ = Π−+ ◦ Π−− ◦ Π+− . Notice that in the last case the orbits have to surround the origin. It follows that Π∗++ is trivial for t2 − 4d ≥ 0 and Π∗++ = Π−+ ◦ Π−− ◦ Π+− for t2 − 4d < 0. Assuming that det A < 0 we have that Π∗++ coincides either with the Poincaré map Π++ or with the composition Π−+ ◦ Π−− ◦ Π+− , see Proposition 4(ii). But, in the last case we need that the orbits surround the origin and this is a contradiction with det A < 0. Therefore, Π∗++ is the map Π++ .  By Proposition 19 the behavior of the map Π∗++ depends on whether e+ ∈ S− , or e+ ∈ S+ and t2 −4d < 0. In order to distinguish between these situations we will denote Πe ++ the map Π−+ ◦ Π−− ◦ Π+− . Consequently we reduce the study of the Poincaré map Π+ associated to the flow of system (16) and to the straight line L+ to study the Poincaré maps Π++ and Π e ++ defined by the flow of the homogeneous linear system (18) and associated to the lines L∗+ and L∗− . Therefore, to know the qualitative behavior of the Poincaré map Π+ defined by system (16) is equivalent to know the behavior of Poincaré maps π++ and e++ = π−+ ◦ π−− ◦ π+− . π 3.2. The Poincaré maps of the linear differential system ẋ = Ax − b. In this subsection we consider the linear differential equation (19) ẋ = Ax − b, which is defined in the region S− and we study the Poincaré map defined by the flow of the system and associated to the straight line L = L− . Notice that L− does not pass through the origin, thus we split it into the the subsets LI− = {q ∈ L : nT q̇ ≤ 0} and LO T − = {q ∈ L : n q̇ ≥ 0}, where q̇ = Aq − b. The set CP− = LI− ∩ LO− consists of contact points of the flow of system (19) with L− . Set Φ− (t, q) the flow of system (19) such that Φ− (0, q) = q. Then we define in L− the following subset  Dom−− = q ∈ LI− ; ∃tq ≥ 0, Φ− (tq , q) ∈ LO − and Φ− (t, q) ⊂ S− ∀t ∈ (0, tq ) ∪ CP− . 18 J. LLIBRE AND L.A.S. MENEZES Suppose that Dom−− 6= ∅ then we can define the Poincaré map of the linear differential system (19) associated to the straight line L− by Π− : Dom−− ⊂ LI− −→ LO − (20) q 7−→ Φ− (tq , q). Suppose that the flow of system (19) has a unique contact point p with L− . We have that Π− (p) = p because p ∈ LI− and p ∈ LO −. Let e− = A−1 b be the singularity of system (19). Notice that the translation y = x − e− transform system (19) in (21) ẏ = Ay, and the line L− is transformed into the straight line L∗+ . Therefore if the Poincaré map Π− , given in (20), is defined then it induces a Poincaré map Π∗++ associated to the flow of system (21) and to the straight line L∗+ . Of course the converse statement is also true. Moreover, the behavior of the map Π− can be obtained from the behavior of the map Π∗++ . We have the following result that can be proved in a similar way to proof of Proposition 18. Proposition 20. Consider L− a straight line in the plane not passing through the origin and such that e− = A−1 b ∈/ L− . The Poincaré map Π− associated to the flow of the system ẋ = Ax − b and to the straight line L− is defined if and only if there exists a unique contact point p− of the flow in L− , and LI− and LO I − are non–empty half–lines such that L− = L− ∪ L− . O Since e− ∈/ L− then L∗+ does not pass through the origin and we can define the symmetric line L∗− = {−q; q ∈ L∗+ }. Therefore the results of section 2 can be applied to study the Poincaré maps Πjk , with j, k ∈ {+, −}, associated to the system (21) and to the straight lines L∗+ and L∗− symmetric with respect to the origin. Set Π∗++ the Poincaré map induced by translation y = x − e− , we have the following proposition. Proposition 21. Consider L− a straight line in the plane not passing through the origin and such that e− = A−1 b ∈/ L− . Assume that the Poincaré map Π− associated to the flow of the system ẋ = Ax − b and to the straight line L− is defined. Let Π∗++ be the Poincaré map induced by the translation y = x − e− . (i) Suppose that e− ∈ S− . – If det A > 0 and t2 − 4d < 0, then Π∗++ coincides with the compo- sition Π−+ ◦ Π−− ◦ Π+− . – If det A > 0 and t2 − 4d ≥ 0, then Π∗++ is trivial. – Suppose that det A < 0. Then Π∗−− is the Poincaré map Π++ . ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 19 (ii) Suppose that e− ∈ S+ . – If det A > 0 then Π∗++ is the Poincaré map Π++ . – If det A < 0, then Π∗++ is trivial. Proof. In case (i) the translation y = x − e− transforms LI− into L∗I O + and L− ∗O I ∗O into L+ , and in case (ii) the translation y = x − e− transforms L− into L+ and LO ∗I − into L+ . The rest of proof is similar to proof of Proposition 19.  By Proposition 21 the behavior of the map Π∗++ depends on whether e− ∈ S− , or e− ∈ S+ . In order to distinguish between these situation we will denote the map Π−+ ◦ Π−− ◦ Π+− by Π e ++ . We reduce the study of the Poincaré map Π− associated to the flow of system (19) and to the straight line L− to study the Poincaré maps Π++ and Π e ++ defined by the flow of the homogeneous linear system (21) and associated to two parallel lines L∗− and L∗+ . Then to know the qualitative behavior of the Poincaré map Π− , defined by system (19), is equivalent to know the behavior of of Poincaré maps π++ and π e++ = π−+ ◦ π−− ◦ π+− . Remark 22. By Proposition 6, we have that π++ = π−− and π e++ = πe−− . In order to not confuse the Poincaré maps πjk , where j, k ∈ {+, −}, associated to system (19), with the maps studied on the previous section, we will use the notation π−− and πe−− . 3.3. Return Map of the discontinuous system (15). In this subsection we describe the return map Π defined by the flow of the discontinuous piecewise linear differential system (15) and associated to the straight line L as the composition of the Poincaré maps Π+ and Π− defined in subsections 3.1 and 3.2. Notice that the half–lines LI+ , LO I O + , L− , and L− are contained in the straight line L. Furthermore, when the Poincaré map Π+ is defined we have that its domain is Dom++ ⊂ LO I + and its image is contained in L+ . Moreover, when the Poincaré map Π− is defined we have that its domain is Dom−− ⊂ LI− and its image is contained in LO I I O − . Then, consider the subset L+ ∩ L− and L+ ∩ L− O of the straight line L and define  Dom = q ∈ LO O O O + ∩ L− ; ∃tq > 0, Φ(tq , q) ∈ L+ ∩ L− and ∀t ∈ (0, tq ), / LO Φ(t, q) ∈ O + ∩ L− , where Φ(t, q) is the solution of the discontinuous system (15). Observe that if d = 0 then Dom = ∅ and therefore the return map is not defined. In what follows we assume that A ∈ GL(R2 ). 20 J. LLIBRE AND L.A.S. MENEZES When Dom 6= ∅ we define the return map associated to the flow of the discontinuous system(15) and to the straight line L by Π : Dom ⊂ LO O O O + ∩ L− −→ L+ ∩ L− (22) q 7−→ Φ(tq , q). In what follows we refer Π+ and Π− the restriction of the Poincaré maps define in subsections 3.1 and 3.2 to subsets LO O I I + ∩ L− and L+ ∩ L− , respectively. Theorem 23. Consider the discontinuous piecewise linear differential system (15). Assume that the return map Π associated to the flow of system (15) and to the straight line L is defined. (i) Suppose that e+ ∈ S− and e− ∈ S+ . – If det A > 0, then Π = Π− ◦ Π+ . – If detA < 0, then the return map Π is not defined. (ii) Suppose that e+ ∈ S− and e− ∈ S− . – If det A > 0 and t2 − 4d < 0, then the map Π = Π− ◦ Π+ . – If det A < 0 or det A > 0 and t2 − 4d ≥ 0 then return map Π is not defined. (iii) Suppose that e+ ∈ S+ and e− ∈ S− . – If det A < 0 or det A > 0 and t2 − 4d < 0, then Π = Π− ◦ Π+ . – If det A > 0 and t2 − 4d ≥ 0, then the return map Π is not defined. Proof. Notice that Dom 6= ∅ implies that Dom++ 6= ∅ and Dom−− 6= ∅. Thus there exists a contact point p+ of the flow of system (16) with the line L and a contact point p− of the flow of system (19) with the line L. Therefore the Poincaré maps Π+ and Π− are defined, see Propositions 18 and 20. The result follows from Propositions 19 and 21.  In what follows we assume that the return map of the discontinuous system (15) is defined. That is Dom 6= ∅. Let p+ and p− be the contact point of the system ẋ = Ax ± b, respectively. By definition of L∗O ∗I + and L− and from Figure 5 we have the following: • If det A > 0, e+ ∈ S− and e− ∈ S+ , then LI+ = {p+ + ap˙+ : a ≥ 0}, LO + ˙+ + = {p − ap : a ≥ 0}, (23) LI− = {p− − ap˙− : a ≥ 0}, LO − ˙− − = {p + ap : a ≥ 0}. ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 21 • If det A > 0, e+ ∈ S− and e− ∈ S− , then LI+ = {p+ + ap˙+ : a ≥ 0}, LO + ˙+ + = {p − ap : a ≥ 0}, (24) LI− = {p− + ap˙− : a ≥ 0}, LO − ˙− − = {p − ap : a ≥ 0}. • If det A > 0, e+ ∈ S+ and e− ∈ S− , then LI+ = {p+ − ap˙+ : a ≥ 0}, LO + ˙+ + = {p + ap : a ≥ 0}, (25) LI− = {p− + ap˙− : a ≥ 0}, LO − ˙− − = {p − ap : a ≥ 0}. • If det A < 0, e+ ∈ S+ and e− ∈ S− , then LI+ = {p+ + ap˙+ : a ≥ 0}, LO + ˙+ + = {p − ap : a ≥ 0}, (26) LI− = {p− − ap˙− : a ≥ 0}, LO − ˙− − = {p + ap : a ≥ 0}. Since the behavior of maps Π+ and Π− are determined by the Poincaré e++ and π maps π++ , π−− , π e−− , we will study the behavior of the map Π via so–defined Poincaré maps πjk , where j, k ∈ {+, −}. Note that the behavior of those maps were studied in subsection 2.1. We will use π± to identify which of the linear system ẋ = Ax ± b define the maps πjk and we denote p± the respective contact points with respect to the straight line L. Thus either π+ = π++ or π+ = π e++ , and either π− = π−− or π− = πe−− . Given q ∈ Dom, let r and r0 be the coordinates of q and Π(q) on the half– line LO + , respectively. We define the return map π as π(r) = r0 , that is the return map transforms the coordinate of q into the coordinate of Π(q). Consider p ∈ LI+ ∩ LI− and q ∈ LO O + ∩ L− , set r and s the coordinates of p on I I the half–lines L+ and L− , respectively, and m and n the coordinates of q on the half–lines LO O I + and L− , respectively. By previous definition of L± we have that • if det A > 0, e+ ∈ S− and e− ∈ S+ then p = p+ + rp˙+ = p− − sp˙− and q = p+ − mp˙+ = p− + np˙− . Thus we can define the maps p− − (p+ + rp˙+ ) p+ − (p− + np˙− ) (27) f1 (r) = , h1 (n) = . p˙− p˙+ • if det A > 0, e+ ∈ S− and e− ∈ S− , then p = p+ + rp˙+ = p− + sp˙− and q = p+ − mp˙+ = p− − np˙− . 22 J. LLIBRE AND L.A.S. MENEZES Thus we can define the maps (p+ + rp˙+ ) − p− p+ − (p− − np˙− ) (28) f2 (r) = , h2 (n) = . p˙− p˙+ • if det A > 0, e+ ∈ S+ and e− ∈ S− , then p = p+ − rp˙+ = p− + sp˙− and q = p+ + mp˙+ = p− − np˙− . Thus we can define the maps (p+ − rp˙+ ) − p− (p− − np˙− ) − p+ (29) f3 (r) = , h3 (n) = . p˙− p˙+ • if det A < 0, e+ ∈ S+ and e− ∈ S− , then p = p+ + rp˙+ = p− − sp˙− and q = p+ − mp˙+ = p− + np˙− . Thus we can define the maps p− − (p+ + rp˙+ ) p+ − (p− + np˙− ) (30) f4 (r) = , h4 (n) = . p˙− p˙+ Remark 24. The map fi transform the coordinate of p in LI+ into the coordi- nate of p in LI− and hi transform the coordinate of q in LO + into the coordinate O −1 −1 of q in L− . Clearly the inverse maps fi (s) and hi (m), i = 1, 2, 3, 4, are well defined. Suppose that the contact points satisfy p− 6= p+ . If p− belongs to LI+ , then the maps fi and hi satisfy (31) fi : [r∗ , ∞ ) −→ [0, ∞ ) , hi : [s∗ , ∞) −→ [0, ∞ ) , where kp− − p+ k kp+ − p− k (32) r∗ = , s∗ = , p˙+ p˙− are the coordinates of the contact points p− and p+ with respect to the straight lines L+ and L− , respectively. On the other hand, if p− belongs to LO + then the maps fi and hi satisfy (33) fi : [0, ∞ ) −→ [s∗ , ∞ ) , hi : [0, ∞ ) −→ [r∗ , ∞ ) , with r∗ and s∗ given in (32). ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 23 Finally, assuming that p− = p+ we get (34) fi : [ 0, ∞) −→ [0, ∞ ) , hi : [0, ∞ ) −→ [0, ∞ ) , Observe that fi and hi , i = 1, 2, 3, 4, are increasing linear maps that tends to infinity when r and s tends to infinity, respectively. Moreover, we have the following results. Lemma 25. Consider the inverse map of hi for i = 1, 2, 3, 4. If p− = p+ then h−1 − + i (r) = fi (r). If p 6= p and −1 • p− ∈ L O ∗ + then fi (r) > hi (r) for every r ∈ [ r , ∞). − I −1 ∗ • p ∈ L+ then fi (r) < hi (r) for every r ∈ [ r , ∞). Proof. Consider e+ ∈ S− and e− ∈ S− . In this case we have p− − (p+ − rp˙+ ) h−1 2 (r) = . p˙− Therefore p− = p+ implies that h−1 2 (r) = f2 (r), see (28). On the other hand, if p− 6= p+ , taking r ∈ [ r∗ , ∞) we have that p1 = p+ + rp˙+ ∈ LI+ and p2 = p+ − rp˙+ ∈ LO + are symmetric points with respect to contact point p . + Furthermore kp1 − p− k kp− − p2 k f2 (r) = , h−1 2 (r) = . p˙− p˙− It follows that, if p− ∈ LO − − + we get that kp − p2 k < kp1 − p k and there- fore h−1 2 (r) < f2 (r). Otherwise, if p − ∈ LI+ we get kp1 − p− k < kp− − p2 k. −1 Therefore f2 (r) < h2 (r). The others cases can be proved in a similar way.  The following result provides the return map π as compositions of the Poincaré maps π+ , π− and the maps fi and hi , i = 1, 2. Theorem 26. Consider the discontinuous piecewise linear differential system (15). (1) Assume that e+ ∈ S− and e− ∈ S+ . If d > 0, then π(r) = h1 (π−− (f1 (π++ (r)))). (2) Assume that e+ ∈ S− and e− ∈ S− . If d > 0 and t2 − 4d < 0, then π(r) = h2 (e π−− (f2 (π++ (r)))). (3) Assume that e+ ∈ S+ and e− ∈ S− . • If d > 0 and t2 − 4d < 0, then π(r) = h3 (e π−− (f3 (e π++ (r)))). • If d < 0 then π(r) = h4 (π−− (f4 (π++ (r)))). 24 J. LLIBRE AND L.A.S. MENEZES In the others cases we have that the return map π does not defined. Proof. Given q ∈ Dom, we have that q ∈ LO O I I + ∩ L− , Π+ (q) ∈ L+ ∩ L− , and O O Π(q) = Π− (Π+ (q)) ∈ L+ ∩ L− . Assume that r and r0 are the coordinate of q and Π(q) in LO + , respectively. By definitions of π+ and remark 24, it follows that π+ (r) and fi (π+ (r)) are the coordinates of Π+ (q) in LI+ and LI− , respectively. Furthermore, by definitions of π− and remark 24, we have that π− (fi (π+ (r))) and hi (π− (fi (π+ (r)))) = r0 are the coordinates of Π(q)) in LO − and LO + , respectively. Consequently, the return map π is given by π(r) = r0 = hi (π− (fi (π+ (r)))). Assume that e+ ∈ S− and e− ∈ S+ . By Theorem 23-(i) the return map of the discontinuous system (15) is given by Π = Π− ◦ Π+ for d > 0 and it is not defined when d < 0. Furthermore Proposition 19 implies that the behavior of the Poincaré map Π+ is equivalent to behavior of Π++ for d > 0 and Π+ is trivial for d < 0. The Proposition 21 and remark 22 implies that the behavior of the Poincaré map Π− is equivalent to behavior of Π−− for d > 0 and Π− is trivial for d < 0. Passing to Poincaré maps πjk , j, k ∈ {+, −}, we have that π+ = π++ and π− = π−− for d > 0 and π− , π+ are defined only at zero for d < 0. Since e+ ∈ S− , e− ∈ S+ and d > 0, we have that the maps f1 , h1 are defined, see equation (27). Therefore the return map is given by π(r) = h1 (π−− (f1 (π++ (r)))) for d > 0 and it is not defined for d < 0. We have proved the statement (1). The others cases can be proved in a similar way and the result follows.  4. Existence of limit cycle for discontinuous system (15) From the definition of the return map π we have that the existence of periodic orbits for system (15) is equivalent to the existence of a fixed point of π. Moreover any limit cycle is associated to an isolated fixed point of π. Theorem 27. Consider the discontinuous piecewise linear differential system (15). If one of the statements below is satisfied, then system (15) has no limit cycles. (1) e+ ∈ S− , e− ∈ S+ , and d < 0. (2) e+ ∈ S− , e− ∈ S− , and either d < 0 or d > 0 and t2 − 4d ≥ 0. (3) e+ ∈ S+ , e− ∈ S− , d > 0 and t2 − 4d ≥ 0. Proof. On these assumptions the return map π is not defined, see Theorem 26. Then system (15) has no limit cycles.  ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 25 In the rest of this paper we assume that the return map π for the discontin- uous piecewise linear differential system (2) is defined. Consider the following maps (35) g(r) = π− (fi (π+ (r))) − h−1 −1 −1 i (r), g1 (r) = fi (π+ (r)) − π− (hi (r)), i = 1, 2, 3, 4. We have the following result. Lemma 28. The maps g and g1 have a zero at r0 if and only if r0 is a fixed point of the return map π. Proof. Suppose that r0 is a zero of the map g(r). Then π− (fi (π+ (r0 ))) = h−1 −1 i (r0 ) and this implies that π(r0 ) = hi (π− (fi (π+ (r0 )))) = hi (hi (r0 )) = r0 . Reciprocally, if r0 is a fixed point of π then hi (π− (fi (π+ (r0 )))) = r0 from which follows that π− (fi (π+ (r0 ))) = h−1 i (r0 ). The proof to g1 is similar.  Suppose that p− = p+ , that is the contact points of system (16) and (19) coincide. We have the following results. Theorem 29. Consider the discontinuous piecewise linear differential system (15), assume that p− = p+ and the return map π is defined. • If t 6= 0 then π has not fixed points. • If t = 0 then π is the identity map. Proof. According to Theorem 26 we should consider the following cases: (i) e+ ∈ S− and e− ∈ S+ , (ii) e+ ∈ S− and e− ∈ S− , and (iii) e+ ∈ S+ and e− ∈ S− . We shall prove the case (i), the others cases can be proved in a similar way, observing that the behavior of the maps π e++ is characterized in Propositions 16 and 17. (i) In this case the return map is given by π(r) = h1 (π−− (f1 (π++ (r)))), see Proposition 26-(1) if d > 0, and it is not defined if d < 0. Furthermore by Lemma 25 we have that f1 (r) = h−1 1 (r). Then we can define g(r) = −1 π−− (f1 (π++ (r))) − h1 (r). Observe that if d > 0 the singularities of the piecewise linear differential system (15) can be diagonal nodes, non–diagonal nodes, centers or foci. Assume that the singularities of system (15) are diagonal nodes, i.e. d > 0, t2 − 4d > 0, and t 6= 0. If t > 0 then Proposition 10 implies that π++ (r) > r 26 J. LLIBRE AND L.A.S. MENEZES and π−− (r) > r. Thus (36) π−− (f1 (π++ (r))) > f1 (π++ (r)) > f1 (r), in the second inequality we have used that f1 is increasing. It follows that g(r) > 0. On the other hand if t < 0 Proposition 11 implies that π++ (r) < r and π−− (r) < r. Thus (37) π−− (f1 (π++ (r))) < f1 (π++ (r)) < f1 (r), and it follows that g(r) < 0. Assume that the singularities of system (15) are non–diagonal nodes, i.e. d > 0, t2 − 4d = 0, and t 6= 0. If t > 0 then Proposition 12 implies that π++ (r) > r and π−− (r) > r. Thus (38) π−− (f1 (π++ (r))) > f1 (π++ (r)) > f1 (r), in the second inequality we have used that f1 is increasing. It follows that g(r) > 0. On the other hand if t < 0 Proposition 13 implies that π++ (r) < r and π−− (r) < r. Thus (39) π−− (f1 (π++ (r))) < f1 (π++ (r)) < f1 (r), and it follows that g(r) < 0. Finally assume that the singularities of system (15) are centers or foci, i.e. d > 0 and t2 − 4d < 0. If t > 0 then Proposition 14 implies that π++ (r) > r and π−− (r) > r. Then (40) π−− (f1 (π++ (r))) > f1 (π++ (r)) > f1 (r), in the second inequality we have used that f1 is increasing. It follows that g(r) > 0. On the other hand if t < 0 Proposition 15 implies that π++ (r) < r and π−− (r) < r. Thus (41) π−− (f1 (π++ (r))) < f1 (π++ (r)) < f1 (r), and it follows that g(r) < 0. If t = 0, by Proposition 14 we have that the Poincaré maps π−− and π++ are the identity map. Moreover by Lemma 25 we have that f1 (r) = h−1 1 (r). Therefore g(r) = f1 (r) − h−1 1 (r) = 0 and π(r) = r, i.e. the return map is the identity, see Lemma 28. Consequently if t 6= 0 then either g(r) > 0, or g(r) < 0. Therefore the return map has not fixed points, see Lemma 28.  Proof of Theorem A. By Theorem 29 we conclude that either system (2) has no closed orbits, or it contains a continuum of closed orbits. Therefore system (2) has no limit cycle.  ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 27 In what follows we assume that p− 6= p+ . Observe that if fi and h−1 i are defined at r then f1 (r) = f2 (r) = f4 (r) = h−1 3 (r)and f3 (r) = h−1 2 (r) = h−1 −1 1 (r) = h4 (r). Suppose that A = (aij ) and b = (b1 , b2 )T , we can rewrite the maps f2 and h−1 2 as |(a11 a22 − a12 a21 − a12 b2 + a22 b1 )r + 2b1 | (42) f2 (r) = , |a11 a22 − a12 a21 + a12 b2 − a22 b1 | |(a11 a22 − a12 a21 − a12 b2 + a22 b1 )r − 2b1 | (43) h−1 2 (r) = . |a11 a22 − a12 a21 + a12 b2 − a22 b1 | Then if the singularities of system (15) are such that e+ ∈ S− and e− ∈ S+ , we get that a12 b2 − a22 b1 < −1. a11 a22 − a12 a21 Therefore a11 a22 − a12 a21 + a12 b2 − a22 b1 < 0, a11 a22 − a12 a21 − a12 b2 + a22 b1 > 0. −1 Since f1 and h1 are increasing maps, we conclude that a11 a22 − a12 a21 − a12 b2 + a22 b1 (44) f10 (r) = (h−1 0 1 ) (r) = . −a11 a22 + a12 a21 − a12 b2 + a22 b1 If e+ ∈ S− and e− ∈ S− we have that a12 b2 − a22 b1 −1 < < 1, a11 a22 − a12 a21 from which it follows that a11 a22 − a12 a21 + a12 b2 − a22 b1 > 0, a11 a22 − a12 a21 − a12 b2 + a22 b1 > 0. Therefore a11 a22 − a12 a21 − a12 b2 + a22 b1 (45) f20 (r) = (h−1 0 2 ) (r) = . a11 a22 − a12 a21 + a12 b2 − a22 b1 Finally assume that e+ ∈ S+ and e− ∈ S− , thus a12 b2 − a22 b1 > 1, a11 a22 − a12 a21 and this implies that a11 a22 − a12 a21 − a12 b2 + a22 b1 < 0. Furthermore if d > 0 we obtain a11 a22 − a12 a21 + a12 b2 − a22 b1 > 0 28 J. LLIBRE AND L.A.S. MENEZES and if d < 0 then a11 a22 − a12 a21 + a12 b2 − a22 b1 < 0. Since fi and h−1 i , i = 3, 4, are increasing maps, we conclude that −a11 a22 + a12 a21 + a12 b2 − a22 b1 (46) f30 (r) = (h−1 0 3 ) (r) = , a11 a22 − a12 a21 + a12 b2 − a22 b1 −a11 a22 + a12 a21 + a12 b2 − a22 b1 (47) f40 (r) = (h−1 0 4 ) (r) = . −a11 a22 + a12 a21 − a12 b2 + a22 b1 4.1. Virtual–virtual case. In this subsection we assume that the singular- ities of discontinuous piecewise linear differential system (15) are such that e+ ∈ S− and e− ∈ S+ and we analyze the existence of limit cycles. On the next results we study the existence of limit cycle for the discontinuous system (15). Theorem 30. Suppose that d > 0 and p− ∈ LO + . If t ≥ 0, then π has not fixed points. If t < 0, then π has one fixed point. Proof. By Theorem 26 we have that the return map π is given by π(r) = h1 (π−− (f1 (π++ (r)))). Then we get that (48) g(r) = π−− (f1 (π++ (r))) − h−1 1 (r). Observe that, by equation (33), the map f1 is define on the interval [0, ∞) and its image is [s∗ , ∞). Moreover, the map h−1 ∗ 1 is defined in [r , ∞) whose image is the interval [0, ∞). In order to prove the result we consider two cases: (i) t2 − 4d ≥ 0; (ii) t2 − 4d < 0. (i) Assume that t > 0, by Propositions 10 and 12 there exist α1 > 0 and α2 > 0 such that π++ : [0, α1 ) −→ [0, ∞) , π−− [0, α2 ) −→ [0, ∞) . Observe that the return map is only defined if r∗ < α1 and s∗ < α2 because r∗ and s∗ is the coordinate of p− and p+ with respect to L+ and L− , respectively. −1 Thus g is defined for r ∈ (r∗ , a), where a = π++ (f1−1 (α2 )). Furthermore the maps satisfy π++ (r) > r and π−− (r) > r and, therefore π−− (f1 (π++ (r))) > f1 (π++ (r)) > f1 (r) > h−1 1 (r), where we use that f1 is an increasing map and Lemma 25. This implies that g(r) > 0, and by Lemma 28 there are not fixed points of π. ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 29 Now, if t < 0 then Propositions 11 and 13 imply that there exist α1 > 0 and α2 > 0 such that π++ : [0, ∞) −→ [0, α1 ) , π−− [0, ∞) −→ [0, α2 ) . Furthermore, π++ and π−− are increasing maps and satisfy π++ (r) < r, π−− (r) < r, lim π++ (r) = α1 and lim π−− (r) = α2 . It follows that the r%+∞ r%+∞ ∗ map g(r) is defined for r ∈ (r , ∞) and satisfies lim g(r) = π−− (f1 (π++ (r∗ ))) > 0, r&r∗ lim g(r) = lim π−− (f1 (π++ (r))) − lim h−1 1 (r) = −∞. r%+∞ r%+∞ r%+∞ Therefore there exists a r0 ∈ (r∗ , +∞) such that g(r0 ) = 0. Since h−1 1 is linear, Propositions 11-(c) and 13-(c) imply that g 00 (r) < 0. (ii) If t = 0, by Propositions 14 and 15, we have that π++ and π−− are the identity in [ 0, +∞) . Then g(r) = f1 (r) − h−1 ∗ 1 (r) is defined for r ∈ (r , +∞) and, by Lemma 25, g(r) > 0. Therefore the return map π has no fixed points, see Lemma 28. In what follows we assume that t 6= 0. In this case we have that π++ : [0, ∞) −→ [0, ∞) , π−− [0, ∞) −→ [0, ∞) , are increasing maps such that lim π++ (r) = +∞ and lim π−− (r) = +∞ r%+∞ r%+∞ see Propositions 14 and 15. Then it follows that the map g(r) is defined for r ∈ (r∗ , ∞). Assume that t > 0, by Proposition 14 we get that π++ (r) > r and π−− (r) > r. Therefore π−− (f1 (π++ (r))) > f1 (π++ (r)) > f1 (r) > h−1 1 (r), where we use that f1 is an increasing map and Lemma 25. This implies that g(r) > 0 and, by Lemma 28 there are not fixed points of π. 0 Now, for t < 0 Proposition 15 implies that π++ (r) < r, π−− (r) < r, π++ (r) < 0 1, and π−− (r) < 1. It follows that (π−− ◦ f1 ◦ π++ )0 (r) = π−− 0 (f1 (π++ (r)))f10 (π++ (r))π++ 0 (r) < f10 (π++ (r)), where f10 (π++ (r)) = f10 (r) = (h−1 0 1 ) (r) is a constant, see equation (44). Conse- quently, lim g(r) = lim [π−− (f1 (π++ (r))) − h−1 1 (r)] = −∞. r%+∞ r%+∞ 30 J. LLIBRE AND L.A.S. MENEZES Furthermore, we have that lim∗ g(r) = π−− (f1 (π++ (r∗ ))) > 0. Therefore there r&r exist r0 ∈ (r∗ , ∞) such that g(r0 ) = 0. By Proposition 15-(c) we have that g 00 (r) < 0, and therefore r0 is unique. By Lemma 28, we conclude that the return map π has a unique fixed point.  Theorem 31. Suppose that d > 0 and p− ∈ LI+ . If t > 0 then the return map π has one fixed point, and if t ≤ 0 then π has not fixed points. Proof. By Theorem 26-(1) we have that the return map π is given by π(r) = h1 (π−− (f1 (π++ (r)))). Then we get that (49) g(r) = π−− (f1 (π++ (r))) − h−1 1 (r). By equation (31) the map f1 is defined on the interval [r∗ , ∞) and its image is [0, ∞) . Moreover the map h−1 1 is defined in [0, ∞) whose image is the interval [s∗ , ∞) . In order to prove the result we consider two cases: (i) t2 − 4d ≥ 0; (ii) 2 t − 4d < 0. (i) In this case we have that t 6= 0. Assume that t > 0, Propositions 10 and 12 imply that there exist α1 > 0 and α2 > 0 such that π++ : [0, α1 ) −→ [0, ∞) , π−− [0, α2 ) −→ [0, ∞) . Moreover π++ and π−− are increasing maps and satisfy π++ (r) > r, π−− (r) > r, lim π++ (r) = ∞ and lim π−− (r) = ∞. So it follows that the map g is r&α1 r&α2 −1 −1 defined for r ∈ (a, b), where a = π++ (r∗ ) and b = π++ (f1−1 (α2 )), and satisfies lim g(r) = −h−1 1 (a) < 0, r&a lim g(r) = lim π−− (f1 (π++ (r))) − h−1 1 (b) = +∞. r%b r%b Then there exists r0 ∈ (a, b) such that g(r0 ) = 0. Moreover, Propositions 10-(c) and 12-(c) imply that g 00 (r) > 0, thus we conclude that r0 is unique. Therefore, by Lemma 28, the return map π has one fixed point r0 . If t < 0 then there exist α1 > 0 and α2 > 0 such that π++ : [0, ∞) −→ [0, α1 ) , π−− [0, ∞) −→ [0, α2 ) , see Propositions 11 and 13. It follows that the map g(r) is defined for r ∈ −1 (a, ∞), where a = π++ (r∗ ). Furthermore π++ and π−− are increasing maps which satisfy π++ (r) < r, π−− (r) < r and, therefore π−− (f1 (π++ (r))) < f1 (π++ (r)) < f1 (r) < h−1 1 (r). ON THE LIMIT CYCLES VIA SO DEFINED POINCARÉ MAPS 31 On the last inequality we use the result of Lemma 25. Consequently g(r) < 0 for every r ∈ (a, ∞), and the return map has not fixed point, see Lemma 28. (ii) If t = 0 by Propositions 14 and 15 we have that π++ and π−− are the identity in [ 0, +∞) . Then g(r) = f1 (r) − h−1 ∗ 1 (r) is defined for r ∈ (r , +∞) and, by Lemma 25, g(r) < 0. Therefore the return map π has no fixed point, see Lemma 28. Assume that t 6= 0 then we have that the maps π++ : [0, ∞) −→ [0, ∞) , π−− [0, ∞) −→ [0, ∞) , are increasing, lim π++ (r) = +∞ and lim π−− (r) = +∞, see Propositions r%+∞ r%+∞ 14 and 15. Then it follows that the map g(r) is defined for r ∈ (a, ∞), where −1 a = π++ (r∗ ). 0 If t > 0 Proposition 14 implies that π++ (r) > r, π−− (r) > r, π++ (r) > 1, 0 and π−− (r) > 1. It follows that (π−− ◦ f1 ◦ π++ )0 (r) = π−− 0 0 (f1 (π++ (r)))f10 (π++ (r))π++ (r) > f10 (π++ (r)), where f10 (π++ (r)) = f10 (r) = (h−1 0 1 ) (r) is a constant, see equation (44). It follows that lim g(r) = lim [π−− (f1 (π++ (r))) − h−1 1 (r)] = +∞. r%+∞ r%+∞ Furthermore we have that lim g(r) = −h−1 1 (a) < 0 and, therefore there exists r&a r0 ∈ (a, ∞) such that g(r0 ) = 0. Since g 00 (r) > 0, see statement (c) of Propo- sition 14, we conclude that r0 is unique. By Lemma 28 we conclude that the return map π have one fixed point. Finally if t < 0 then π++ (r) < r and π−− (r) < r for every r ∈ [0, ∞), see Proposition 15. Therefore π−− (f1 (π++ (r))) < f1 (π++ (r)) < f1 (r) < h−1 1 (r), where we use that f1 is an increasing map and Lemma 25 on the second and third inequality, respectively. This implies that g(r) < 0 and, by Lemma 28 there is no fixed point of π.  Proof of Theorem B. Observe that if the contact point of X satisfy p− ∈ LO +, by Theorem 30 we have that the discontinuous piecewise linear differential system (2) has one limit cycle if t < 0, and it has no limit cycles if t ≥ 0. On the other hand, if the contact point of vector field X satisfies p− ∈ LI+ then Theorem 31 implies that the discontinuous piecewise linear differential system (15) has one limit cycle if t > 0, and it has no limit cycles if t ≤ 0, respectively. So Theorem B follows.  32 J. LLIBRE AND L.A.S. MENEZES Acknowledgements This work the first author is partially supported by the Ministerio de Economı́a, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016- 77278-P (FEDER) and grant MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Re- search Council grant MSCA-RISE-2017-777911. The second author is par- tially supported by the Coordenação de Aperfeiçoamento de Pessoal de Nı́vel Superior-Brazil(CAPES) grant PDSE-88881.133794/2016-01 and a CAPES fi- nance code 001 grant. References [1] J. C. Artés, J. Llibre, J. C. R. Medrado, M. Teixeira, Piecewise linear differ- ential systems with two real saddles. 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Teruel, Existence of Poincaré maps in piecewise linear differ- ential systems in Rn . Int. J. Bifurcation and Chaos 14 (2004), 2843–2851. [9] J. Llibre and A. E. Teruel, Introduction to the qualitative theory of differential systems. Springer, Basel, 2014. [10] J. Llibre , H. Ponce, Three limit cycles in discontinuous piecewise linear differential systems with two zones. Dyn. of Cont., Discr. Impul. Syst., Series B 19 (2012), 325–335. [11] J. C. R. Medrado, J. Torregrosa, Uniqueness of limit cycles for sewing piecewise linear systems. J. Math. Anal. Appl. 431 (2015), 529-544. 1 Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat 2 Instituto de Matemática e Estatı́stica, Universidade Federal de Goiás, 74690-900, Goiânia, GO, Brazil E-mail address: lucyjanedealmeida@gmail.com