This is a preprint of: “Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields”, Joäo L. Cardoso, Jaume Llibre, Douglas D. Novaes, Durval J. Tonon, Dyn. Syst., 1–25, 2020. DOI: [10.1080/14689367.2020.1722064] SIMULTANEOUS OCCURRENCE OF SLIDING AND CROSSING LIMIT CYCLES IN PIECEWISE LINEAR PLANAR VECTOR FIELDS JOAO L. CARDOSO1 , JAUME LLIBRE2 , DOUGLAS D. NOVAES3 AND DURVAL J. TONON1 Abstract. In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line x = 0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has center, a real one for y < 0 and a virtual one for y > 0, and such that the real center is a global center. Then working with a first order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second order piecewise linear perturbation. Then imposing the existence of a sliding limit cycle we prove that only one more crossing limit cycle can appear. Finally, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov functions, the Extended Chebyshev systems and the Bendixson transformation. 1. Introduction and statement of the main results For a given differential system a limit cycle is a periodic orbit isolated in the set of all periodic orbits of the system. One of the main problems of the qualitative theory of planar differential systems is determining the existence of limit cycles. A center is a singular point p that possesses a neighborhood U such that U \ {p} is filled by periodic solutions. A classical way to produce and study limit cycles is by perturbing the periodic solutions of a center. This problem has been studied intensively for continuous planar differential systems, see for instance, [5] and the references therein. The main objective of this paper is to study the limit cycles that can bifurcate from a center of a discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0 when the center is perturbed inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. The study of the piecewise linear differential systems goes back to Andronov and co- authors [1]. In the present days these systems continue receiving a considerable attention, mainly due to their applications. Indeed, such systems are widely used to model many real processes and different modern devices, see for instance the book [3] and the references therein. The case of continuous piecewise linear systems with two regions separated by a straight line is the simplest possible configuration of piecewise linear systems. We note that even in this simple case, establishing the existence of at most one limit cycles was a difficult task, see [6], and [12] for a shorter proof. There are two reason for that misleading simplicity of 2010 Mathematics Subject Classification. 34C05, 34C07, 37G15. Key words and phrases. non–smooth differential system, limit cycle, piecewise linear differential system. 1 2 J. L. CARDOSO, J. LLIBRE, , D.D. NOVAES AND D.J. TONON piecewise linear systems: First, whereas it is easy to compute the solutions in any linear region, the time that each orbit requires to pass from one linear region to the other is usually not computable. Second, the increasing number of parameters. Discontinuous piecewise linear systems with two regions separated by a straight line have received a lot of attention during these last years, see for instance [9, 10, 13, 14, 16, 17] among other papers. In [9], the authors conjectured that piecewise linear systems with two regions separated by a straight line could have at most two limit cycles. Later on in [10], the authors provided numerical evidences on the existence of three limit cycles, which was analytically proved in [16]. In the present study, we consider planar piecewise linear vector fields with two zones separated by the straight line x = 0. Firstly, we provide a canonical form for these systems assuming that each linear system has center, a real one for y < 0 and a virtual one for y > 0, such that the real center is a global center. Then, proceeding with a first order piecewise linear perturbation, we study the bifurcation of crossing limit cycles. In short, it is shown that any configuration of 1, 2 or 3 limit cycles can bifurcate from the periodic solutions of the unperturbed system. We also study the stability of the higher amplitude crossing limit cycle and the infinity. Secondly, we see that a sliding cycle can be detected after a second order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one more crossing limit cycle appears. 1.1. Setting the problem. Consider the following planar piecewise linear differential sys- tems with two zones separated by the straight line Σ = {(x, y); x = 0}        x u+   M+   +  1,ε  if x ≥ 0,      ε  y u+ ẋ 2,ε (1)   = Wε (x, y) =     ẏ   −   x u   M−   +  1,ε  if x ≤ 0,   ε y u− 2,ε where Mε± is one-parameter family of 2 × 2 real matrices and u± ± 1,ε , u2,ε ∈ R. Here, the dot + denotes derivative with respect to the time variable t. We define Σ = {(x, y); x > 0} and Σ− = {(x, y); x < 0} and denote the piecewise linear vector field associated to system (1) by Wε = (Wε+ , Wε− ), where Wε+ and Wε− are defined in Σ+ and Σ− , respectively. Let p± be the singular points of W0± . We say that a singular point p± of W0± is real (respectively virtual) if p+ ∈ {(x, y); x ≥ 0} or p− ∈ {(x, y); x ≤ 0} (respectively if p+ ∈ {(x, y); x ≤ 0} or p− ∈ {(x, y); x ≥ 0}). Assume that system (1) satisfies (H1 ) p− is a center for the system W0− and p− ∈ Σ− , i.e, p− is a real singular point. (H2 ) p+ is a center for the system W0+ and p+ ∈ Σ− , i.e, p+ is a virtual singular point. (H3 ) p− is global center for the system W0 , see Figure 1. 1.2. Canonical Form. The next result provides a canonical form for the piecewise linear vector fields (1) having a center and it is proved in Section 3. Fixing ε = 0, we denote   m±11 m±12 M0 =  ± . m±21 −m11 ± 3 Σ • p− Figure 1. The global center for the system W0 . Proposition 1. Consider the piecewise linear vector field Wε (x, y) given in (1) and as- sume that Wε (x, y) satisfies hypotheses (H1 ), (H2 ) and (H3 ). Then, there exists a change of coordinates (t, x, y) 7→ (t̃, x, ỹ) where the piecewise linear vector field (1) can be written as (x0 , ỹ 0 )T = Zε (x, ỹ), where        x 0  +    Z (x, ỹ) = A+  +  if x ≥ 0,  0  ỹ d Z0 (x, ỹ) =         x 0   Z − (x, ỹ) = A−  +  if x ≤ 0,   0 ỹ e     q a b 0 −1 1 + A+ =   , A− =   with ρ = |(m− 2 − − 11 ) + m12 m21 |, a = m11 c −a 1 0 ρ   m− m+  m+ 1 (m− 2 + 11 ) m12 − + + + m+ u+ − 11 − 12 , b = − 12 ,c = 2 − 2m11 m11 − m12 m21 , d = − 12 2 and m12 m− 12 ρ m12− ρ m− − 12 u2 e=− . Furthermore, the parameters of this canonical form satisfy ρ b < 0, c > 0, d > 0, e > 0, a2 + bc < 0. In general we can write (2) Zε (x, y) = Z0 (x, y) + εZ1 (x, y) + ε2 Z2 (x, y) + O(ε3 ) where      x     B+   + v+ if x ≥ 0,   y Z1 (x, y) =       x   B−   + v− if x ≤ 0,   y 4 J. L. CARDOSO, J. LLIBRE, , D.D. NOVAES AND D.J. TONON and      x     C+   + w+ if x ≥ 0,   y Z2 (x, y) =       x   C−   + w− if x ≤ 0,   y         ± ± ± ± ± ± b 11 b 12  v 1  c 11 c 12  w 1  with B ± =  , v± =  , C± =  , w± =  . We b±21 b ± 22 v ± 2 c ± 21 c ± 22 w ± 2 denote the piecewise smooth vector field Zε by Zε = (Zε+ , Zε− ), where Zε± is defined in Σ± . 1.3. Crossing limit cycles. For the piecewise linear differential system (2) we obtain an upper bound for the number of crossing limit cycles bifurcating from the periodic solutions of the unperturbed system. We define  2d ξ 2 (2v1− b − be(b− − + + + 11 + b22 ) + 2v1 ) + bd(b11 + b22 ) K0 = , eξ K1 = beξ 2 (b− − 11 + b22 ), bd2 (b+ + 11 + b22 ) K2 = − , eξ where a2 + bc = −ξ 2 , with ξ > 0. Theorem A. Suppose Zε (x, y) is a piecewise linear vector field with two zones separated by the straight line x = 0 such that hypotheses (H1 ), (H2 ) and (H3 ) are satisfied by Z0 . Without loss of generality, assume that Zε writes as (2). Then, the following statements hold. (a) There exist elections of the parameters K0 , K1 , and K2 for which system Zε has any configuration of 1, 2, or 3 crossing limit cycles taking into account their multiplicity. (b) The highest amplitude limit cycle (when it exists) and the infinity have opposite stability. More precisely, the highest amplitude limit cycle is stable (resp. unstable) and the infinity is an unstable (resp. stable) periodic orbit provided that ξ(b− 11 + b− 22 ) + b + 11 + b + 22 < 0 (resp. ξ(b − 11 + b − 22 ) + b + 11 + b+ 22 > 0). (c) The lowest amplitude limit cycle (when it exists) is stable (resp. unstable) provided that b− − − − − + − − 11 + b22 < 0, or b11 + b22 = 0 and bv1 + v1 > 0 (resp. b11 + b22 > 0, or − − − + b11 + b22 = 0 and bv1 + v1 < 0). Theorem A is proved in Subsection 3.3. We note that it is well known that piecewise linear differential system with two regions separated by a straight line can exhibit 3 crossing limit cycles. The first numerical evidence of that was given in [10] by Huan and Yang. Then, in [16], Llibre and Ponce established an analytical proof of the existence of 3 limit cycles. After that, many authors have provided different examples of those systems having 3 limit cycles. For instance, in [4] Buzzi et al., using a seventh order piecewise linear perturbation, proved that 3 limit cycles bifurcate from a linear center. In [13], Llibre et al. proved that 3 limit cycles bifurcates from a 5 piecewise linear center, but using only a first order piecewise linear perturbation. In the present paper, Theorem A also guarantees the bifurcation of 3 crossing limit cycles after a first order perturbation. In the next subsection we shall see that a second order perturbation allows the study of sliding limit cycles. 1.4. Sliding and escaping limit cycles. We recall that a sliding/escaping limit cycle is a closed orbit composed by segments of orbits of the sliding vector field Z s (see Subsection 2.1) and Z + and/or Z − . We must mention that this kind of limit cycle has been considered before, see for instance [7]. Consider a piecewise linear vector field Z and assume that the sliding/escaping region (see Subsection 2.1) is bounded. A sliding/escaping limit cycle of Z is a closed trajectory composed by trajectories of the sliding vector field Z s , Z + , and/or Z − . More precisely, we say that a sliding/escaping limit cycle is: of Type I if it is composed by trajectories of Z s , and Zε+ or Zε− ; and of Type II if it is composed by trajectories of Z s , Zε+ and Zε− , see Figure 2. Σ Σ yf 1 yf 1 yf 2 yf 2 Type I Type II Figure 2. Types of sliding limit cycles. Since we are dealing with piecewise linear vector fields, then the number of fold points for each vector field Zε+ and Zε− is at most one. Therefore, the system can have at most one sliding/escaping limit cycle. Theorem B. Suppose Zε (x, y) is a piecewise linear vector field with two zones separated by the straight line x = 0 such that hypotheses (H1 ), (H2 ) and (H3 ) are satisfied by Z0 . Without loss of generality, assume that Zε writes as (2). (1) Additionally, suppose that b− − − + 11 = −b22 , bv1 + v1 < 0 and 0 < a < d + be. Then, the following statements hold. (v1− b + v1+ )2 (a) If 0 < c− − 11 + c22 < then system Zε admits a sliding cycle of Type 2b2 e2 π I. (v − b + v + )2 2(v1− b + v1+ )2 (b) If 1 2 2 1 < c− 11 + c − 22 < then system Zε admits a sliding 2b e π b2 e 2 π cycle of Type II. (2) Conversely, suppose that b− − − + 11 = −b22 , bv1 + v1 > 0, a < 0 and 0 < d + be. Then, the following statements hold. 6 J. L. CARDOSO, J. LLIBRE, , D.D. NOVAES AND D.J. TONON (v − b + v + )2 (a) If − 1 2 2 1 < c− − 11 + c22 < 0 then system Zε admits a escaping cycle of 2b e π Type I. (v1− b + v1+ )2 (b) If c− 11 + c − 22 < − then system Zε admits a escaping cycle of Type 2b2 e2 π II. Theorem B is proved in Subsection 3.4 1.5. Simultaneity. The next result provides conditions on the parameters of the vector field Zε for the simultaneous occurrence of crossing and sliding/escaping limit cycles. Theorem C. Under the assumptions of Theorem B the following statements hold. (v1− b + v1+ )2 (1.a) If 0 < c− − 11 + c22 < then Zε possesses a sliding limit cycle of type I and 2πb2 e2 at most one more crossing limit cycle. (v − b + v + )2 2(v1− b + v1+ )2 (1.b) If 1 2 2 1 < c− − 11 + c22 < then Zε possesses a sliding limit cycle 2b e π b2 e 2 π of type II and at most one more crossing limit cycle. (v − b + v + )2 (2.a) If − 1 2 2 1 < c− − 11 + c22 < 0 then Zε possesses a escaping limit cycle of type I 2b e π and at most one more crossing limit cycle. (v1− b + v1+ )2 (2.b) If c− 11 + c − 22 < − then Zε possesses a escaping limit cycle of type II and 2b2 e2 π at most one more crossing limit cycle. Moreover, there exist parameters of Zε for which the above crossing limit cycle exists, see Figure 3. Theorem C is proved in Subsection 3.5. Σ Σ yf 1 yf 1 yf 2 yf 2 Figure 3. Simultaneous occurrence of crossing and sliding limit cycles for a 2-order linear perturbation of a piecewise linear center. This paper is organized as follows. In Section 2 we provide some preliminary results. Section 3 is devoted to the proofs of our main results. Finally, in Section 4 we present some examples of piecewise linear vector fields for which the upper bounds of crossing and sliding/escaping limit cycles are reached. 7 2. Preliminaries This section is devoted to present some basic notions on nonsmooth vector fields and all the tools needed to prove our main results. Firstly, we introduce the concept of Fil- ippov system. Then, some results on extended Chebyshev systems are presented. This results are important to bound the number of limit cycles. Finally, we discuss the Bendixon transformation which allows the study of the stability of the infinity. 2.1. Filippov’s convention. Let h : R2 → R a differentiable function for which 0 is a regular value. Consider the piecewise smooth vector field   Z + (x, y), h(x, y) > 0 Z(x, y) =  Z − (x, y), h(x, y) < 0. The switching manifold is given by Σ = h−1 (0). Notice that h(x, y) = x for system (1). Consider the Lie’s derivative Z ± h(p) = hZ ± , ∇hi, where h·, ·i is the canonical inner product in R2 . Then, according to Filippov’s conventions (see [20]), the following regions on Σ are distinguished: Crossing Region: Σc = {p ∈ Σ; (Z + h)(p).(Z − h)(p) > 0}.; Sliding Region: Σs = {p ∈ Σ; (Z + h)(p) < 0, (Z − h)(p) > 0}; and Escaping Region: Σe = {p ∈ Σ; (Z + h)(p) > 0, (Z − h)(p) < 0}. The local trajectories of the points in Σs ∪ Σe follow the so called sliding vector field Z −h · Z + − Z +h · Z − (3) Zs = . Z −h − Z +h Then, the flow of Z is obtained by the concatenation of flows of Z + , Z − , and Z s . We say that p ∈ Σ is a fold point of Z ± when Z ± h(p) = 0 and (Z ± )2 h(p) = Z ± .(Z ± h)(p) 6= 0. Moreover, p is called a visible (resp. invisible) fold point of Z ± if Z ± h(p) = 0 and (Z ± )2 h(p) ≷ 0 (resp. (Z ± )2 h(p) ≶ 0). Remark 1. Assuming that p0 ∈ Σc is an invisible fold point for both vector fields Z + and Z − , a first return map is well defined in a neighborhood of p0 . Indeed, from the Implicit Function Theorem, there exists a neighborhood U of p0 such that for each p ∈ U ∩ Σ, there exists a smallest positive time t(p) > 0 such that the trajectory t 7→ φZ + (t, p) of Z + through p intercepts Σ at a point pe = φZ + (t(p), p). Then, define the positive half-return map associated to Z + by γZ + : (R, 0) → (R, 0) where γZ + (p) = pe. Analogously, we define the positive half-return map associated to Z − by γZ − : (R, 0) → (R, 0). Thus, the first return map ϕZ : (Σ, 0) → (Σ, 0) is defined by the composition ϕZ = γZ − ◦ γZ + . 2.2. Extended Chebyshev systems. An ordered set of complex–valued functions F = (g0 , g1 , . . . , gk ) defined on a proper real interval I is an Extended Chebyshev system or ET– system on I if and only if any nontrivial linear combination of functions in F has at most k zeros counting multiplicities. The set F is an Extended Complete Chebyshev system or an ECT–system on I if and only if for any s ∈ {0, 1, . . . , k} we have that (g0 , g1 , . . . , gs ) is an ET–system. For more details see the book of Karlin and Studden [11]. In order to prove that F is an ECT–system on I it is necessary and sufficient to show that W (g0 , g1 , . . . , gs )(t) 6= 0 on I for 0 ≤ s ≤ k, where W (g0 , g1 , . . . , gs )(t) denotes the 8 J. L. CARDOSO, J. LLIBRE, , D.D. NOVAES AND D.J. TONON Wronskian of the functions (g0 , g1 , . . . , gs ) with respect to the variable t. That is,   g0 (t) · · · gs (t)    0   g0 (t) · · · gs0 (t)  W (g0 , . . . , gs )(t) = det  . .. .. ..   . . .    (s) (s) g0 (t) · · · gs (t) In [21], the authors proved the following results: Theorem 2 ([21]). Let F = [g0 , g1 , . . . , gn ] be an ordered set of C ∞ functions gj : [a, b] → R for j = 0, 1, . . . , n such that there exists ξ ∈ (a, b) with W (g0 , g1 , . . . , gn−1 )(ξ) = Wn−1 (ξ) 6= 0. Then, the following statements hold. (a) If Wn (ξ) 6= 0, then for each configuration of m ≤ n zeros, taking into account their multiplicity, there exists f ∈ Span(F) with this configuration of zeros. (b) If Wn (ξ) = 0 and Wn0 (ξ) 6= 0, then for each configuration of m ≤ n + 1 zeros, taking into account their multiplicity, there exists f ∈ Span(F) with this configuration of zeros. Corollary 3 ([21]). Let F = [g0 , g1 , . . . , gn ] be an ordered set of C ∞ functions gj : [a, b] → R for j = 0, 1, . . . , n. Assume that all the Wronskians are nonvanishing except Wn (x), which has exactly one zero on (a, b) and this zero is simple. Then, Z(F) = n + 1 and for any configuration of m ≤ n + 1 zeros there exists an element in Span(F) realizing it. 2.3. The Bendixson transformation. The Bendixson transformation is a useful tool to analyze the stability of the infinity of planar vector fields. In what follows, following [8, 15], we shall discuss this transformation. Consider the differential systems (4) ẋ = f (x, y, ε), ẏ = g(x, y, ε), where f, g are Lipschitz functions in the variables (x, y) and ε > 0 is a small parameter. Applying to system (4) the Bendixson transformation defined as     u 1 x (5)   =  , v x2 + y 2 y we obtain an equivalent system whose local phase portrait at the origin is equivalent to the local phase portrait of system (4) in a neighborhood of the infinity. Composing the Bendixson change of variables (5) with the polar coordinates u = r cos θ, v = r sin θ, we get the polar Bendixson transformation x = (cos θ)/r, y = (sin θ)/r. Applying this last transformation, system (4) becomes       cos θ sin θ cos θ sin θ ṙ = R(r, θ, ε) = −r2 f , , ε cos θ + g , , ε sin θ , r r r r (6)       cos θ sin θ cos θ sin θ θ̇ = ξ(r, θ, ε) = −r f , , ε cos θ − g , , ε cos θ . r r r r We shall study the flow of system (6) contained in the half-cylinder R+ ×S1 = {(r, θ) : r ≥ 0, θ ∈ (−π, π)}. Notice that after multiplying (6) by a power of r, the system can be extended for r = 0. Therefore, the existence of a periodic orbit at infinity for system (4) is equivalent 9 to the existence of the periodic orbit r = 0 for system (6) on the cylinder. Now, consider the assumptions: (a) the functions R and ξ are Lipschitz functions in the variable r and they have period 2π in the variable θ. (b) R(0, θ, ε) = 0 and ξ(0, θ, ε) 6= 0 for all θ ∈ S1 and for every ε ≥ 0 sufficiently small. Notice that (a) and (b) are sufficient and necessary conditions in order to guarantee that system (4) has a periodic solution at infinity. Finally, taking θ as the new independent variable the differential system (6) can be written as the first order differential equation dr R(r, θ, ε) (7) r0 = = S(r, θ, ε) = . dθ ξ(r, θ, ε) Consequently, the Poincaré map defined on a neighborhood of r = 0 is given by Π(ρ) = r(2π, ρ, ε), where r(θ, ρ, ε) is the solution of (7) such that r(0, ρ, ε) = ρ. 3. Proof of the main results In this section we provide the proofs of Proposition 1 and Theorems A, B and C. 3.1. Proof of Proposition 1. We assume that the piecewise linear vector field W0 (x, y) satisfies the hypotheses (H1 ), (H2 ) and (H3 ). Then, the left and right linear differential systems are written as (ẋ, ẏ) = M0± (x, y) + (0, u± T 2 ) , where   ± ± m 11 m 12  M0± =  , m±21 −m ± 11 with (m± 2 ± ± ± 11 ) + m12 m21 < 0 and m12 6= 0. Then, applying the change of variables (x̃, ỹ) = ψ(x, y) = (x, −m− − 11 x − m12 y) we obtain the following piecewise linear differential system        x̃ 0       Ae+  +  if x̃ ≥ 0, 0   ỹ −m + + u x̃ 12 2 (8)  =     ỹ 0     e−  x̃ 0   A +  if x̃ ≤ 0,   ỹ −m+ u− 12 2 where   m− + 11 m12 m+  m+ 11 − − − 12  e+ =  m12 m− 12  A   (m− 2 + 1 ) m12 m− + 11 m12 − − 2m− + − + 11 m11 − m12 m21 − − m+ 11 m12 m12 and   0 −1 e− =  A . −(m− 2 − − 11 ) − m12 m21 0 Notice that the above change of variables fixes the switching manifold. 10 J. L. CARDOSO, J. LLIBRE, , D.D. NOVAES AND D.J. TONON q Now, let ρ = |(m− 2 − − − 2 − − 11 ) + m12 m21 |, where (m11 ) + m12 m21 < 0. Proceeding with the following change of variables and rescaling of time   x t (x̃, ỹ, t̃) 7→ , y, ρ ρ system (8) becomes      0 −1 x 0 Z0− (x, ỹ) =   + , for x ≤ 0 1 0 ỹ e and      a b x 0 Z0+ (x, ỹ) =   + , for x ≥ 0, c −a ỹ d where 1 + m− m+  a= m11 − 11 − 12 , ρ m12 m+12 b=− , m−12 1  (m− )2 m+  c= 11 12 − 2m− m + 11 11 − m+ m+ 12 21 , ρ2 m− 12 m+ + 12 u2 d=− and ρ m− − 12 u2 e=− > 0. ρ d The singular points of Z0− , Z0+ are given by p− = (−e, 0) and p+ = (−b, a), resp.. a2 + bc From (H1 ), (H2 ) and (H3 ) we conclude that b < 0, c > 0, d > 0, e > 0 and a + bc < 0.  2 3.2. Study of the infinity. Applying the Bendixson change of coordinates given in (2) to Zε , we obtain that the differential system in Σ− can be written du−    = −v(2eu + 1) u2 + v 2 + ε − u3 (v1− u + b− − 4 2 − − 11 ) + v1 v + uv (b11 − 2b22 ) dt  −u2 v(a2 + 2(v2− u + b− 3 − 21 )) + v (a2 − 2v2 u) − − , u=u ,v=v dv−  = eu4 − ev 4 + u3 + uv 2 + ε u2 v(−2v1− u − 2b− − 3 − − 11 + b22 ) − v (2v1 u + b22 ) dt  −uv 2 (2a2 + b− 3 − − − 4 21 ) + u (v2 u + b21 ) − v2 v , u=u− ,v=v − 11 and for the differential system in Σ+ can be written as du+  = −au3 + 3auv 2 − u2 v(b + 2(c + du)) + v 3 (b − 2du) + ε − u3 (v1+ u + b+ 11 ) dt  +v1+ v 4 + uv 2 (b+ + 2 3 11 − 2b22 ) − u v(c2 + 2(d0 u + d1 )) + v (c2 − 2d0 u) , + u=u ,v=v + dv +  = −3au2 v + av 3 − uv 2 (2b + c) + u3 (c + du) + ε u2 v(−2v1+ u − 2b+ + 11 + b22 ) dt  −v 3 (2v1+ u + b+ 2 3 22 ) − uv (2c2 + d1 ) + u (d0 u + d1 ) − d0 v 4 − dv 4 . + u=u ,v=v + Applying the polar change of coordinates the left differential system can be written as dr−  = −er2 sin θ − rε cos θ(v1− r + (a2 + b− − 2 21 ) sin θ) + b11 cos θ dt  + sin θ(v2− r + b− 22 sin θ) − − , r=r ,θ=θ dθ −  = 1 + er cos θ + ε − sin θ(v1− r + a2 sin θ) + cos θ((b− − 22 − b11 ) sin θ dt  +v2− r) + b− 2 21 cos θ , r=r − ,θ=θ − and, similarly, the right differential system can be written as dr+  = −r(a cos(2θ) + sin θ((b + c) cos θ + dr)) − rε cos θ(v1+ r dt  +(c2 + d1 ) sin θ) + b+ 2 + 11 cos θ + sin θ(d0 r + b22 sin θ) , r=r + ,θ=θ + dθ+  = cos θ(dr − 2a sin θ) − b sin2 θ + c cos2 θ + ε − sin θ(v1+ r + c2 sin θ) dt  + cos θ((b+ 22 − b + 11 ) sin θ + d 0 r) + d1 cos 2 θ . r=r + ,θ=θ + Considering the rescaling of the radius given by r± t = ε3 ρ± t and taking θ− as the new independent variable we obtain dρ−   − 2 − − 2 = −ερ b 11 cos θ + (a2 + b 21 ) sin θ cos θ + b 22 sin (θ) dθ−   +ε2 ρ (b− − 2 22 − b11 ) sin θ cos θ − a2 sin θ + b21 cos θ − 2   b− 11 cos 2 θ + (a 2 + b − 21 ) sin θ cos θ + b − 22 sin 2 θ + O(ε3 ) , ρ=ρ− ,θ=θ − and dρ+ ρ(2a cos(2θ) + (b + c) sin(2θ)) ρ =   −ε  2 dθ+ 2 2 a sin(2θ) + b sin θ − c cos2 θ 2 a sin(2θ) + b sin2 θ − c cos2 θ  − sin(2θ)(a(b+ + + + 11 + b22 ) + bd1 − cc2 ) + cos(2θ)(ac2 − ad1 + bb22 + cb11 )  −ac2 − ad1 − bb+ + 22 + cb11 + O(ε ) 2 . ρ=ρ+ ,θ=θ + 12 J. L. CARDOSO, J. LLIBRE, , D.D. NOVAES AND D.J. TONON dρ∓ Let ρ∓ (θ) = ρ∓ ∓ 2 0 (θ) + ερ1 (θ) + O(ε ) be the solution of satisfying the initial condition  π dθ∓ ρ± − = ρ0 . So, we have 2 ρ− 0 (θ) = ρ0 , r ρ0  2a sin(2θ) − (b + c) cos(2θ) + b − c  ρ+ 0 (θ) = √ , 2 b ρ0 ρ− 1 (θ) = (−2b− − − − − − 11 θ − b11 sin(2θ) + πb11 + a2 cos(2θ) + a2 + b21 cos(2θ) + b21 − 2b22 θ 4 +b− − 22 sin(2θ) + πb22 ), ρ0  a ρ+ 1 (θ) = − √ p 2b(b+ + 11 + b22 ) arctan 4bξ −2b −2a sin(2θ) + (b + c) cos(2θ) − b + c ξ b tan θ  + (−2a sin(2θ) + (b + c) cos(2θ) − b + c) + 2 sin(2θ)(πab(b+ + 11 + b22 ) ξ +2ac2 ξ + bξ(b+ + + + 22 − b11 )) − cos(2θ)(πb(b + c)(b11 + b22 ) + 2ξ(cc2 − bd1 ))  +πb(b − c)(b+ + 11 + b22 ) + 2bd1 ξ − 2cc2 ξ , where a2 + bc = −ξ 2 , with ξ > 0. Therefore, the displacement map writes − − ρ(ρ0 ) = ρ+ + 2 0 (π/2) − ρ0 (−3π/2) + ε(ρ1 (π/2) − ρ1 (−3π/2)) + O(ε ) 1  b+ +  11 + b22 = − επρ0 b− − 11 + b22 + + O(ε2 ). 2 ξ Consequently, if ξ(b− − + + − − + + 11 + b22 ) + b11 + b22 > 0 (resp. ξ(b11 + b22 ) + b11 + b22 < 0), then the infinity is a stable (resp. unstable) periodic solution. 3.3. Proof of Theorem A. The proof will be split in three steps. In the first one we prove that the number of crossing limit cycles of Z1,ε (X) is given by the zeros of the first order Melnikov function 1  −     M1 (y0 ) = 4v1 y0 − 2(b− − 2 2 11 + b22 ) π e + y0 + ey0 2y0    2  +(b− 11 + b − 22 ) e 2 + y 2 0 arccos 2e 2 e2 +y0 − 1 (9)    1 − 3 − 2bdy0 ξ(b+ + + 3 + + 2 2 2 11 + b22 ) − 4v1 y0 ξ + b(b11 + b22 ) d + y0 ξ bξ   2d2 arccos d2 +y 2 ξ2 − 1 , 0 were the Melnikov Function is given by M (y0 , ε) = M0 (y0 ) + M1 (y0 )ε + M2 (y0 )ε2 + O(ε)3 and Mi (y0 ) = Mi− (y0 ) − Mi+ (y0 ) for i = 0, 1, 2. 13 In the second one we prove that the upper bound of the number of zeros is three, and that this number is reached. Finally, in the third step we study the stability of the crossing limit cycles. Lemma 4. The zeros of M1 (y0 ) correspond to crossing limit cycles for Z1,ε (X). Proof. Let Zε be given by (2). From hypotheses we know that b < 0, d > 0 and a2 + bc < 0. So, denote a2 + bc = −ξ 2 , ξ > 0. Let (x± ± ε (t), yε (t)) be the trajectories of the linear vector ± − − fields Z0 satisfying xε (0) = 0, yε (0) = y0 > 0, and x+ + ε (0) = 0, yε (0) = y1 . So, for ε = 0, we compute x− 0 (t) = e(−1 + cos t) − y0 sin t, (10) y0− (t) = y0 cos t + e sin t, and b(d − d cos(sξ) + y1 ξ sin(sξ)) x+ 0 (s) = , ξ2 (11) 1    y0+ (s) = 2 − d + d + y1 ξ 2 cos(sξ) + ξ(d − y1 ) sin(sξ) . ξ Let tl,ε > 0 and tr,ε < 0 be the first return times to Σ of the above solutions, that is x− + ε (tl ) = xε (tr ) = 0. For ε = 0 we have  2e2  tl0 = 2π − arccos 2 − 1 e + y02 and   1 2d2 tr0 = − arccos −1 . ξ d2 + ξ 2 y12 Writing tl,ε = tl0 + tl1 ε + O(ε2 ) and tr,ε = tr0 + tr1 ε + O(ε2 ), the coefficients tl1 and tr1 can be computed by expanding the equations x− + ε (tl ) = 0 and xε (tr ) = 0 around ε = 0. So 1    tl1 =   2y0 (2v1− e − e(e(b−11 + b − 22 ) + y 0 (b − 21 − a 2 )) + 2v − 2 y0 ) − 2π e 2 + y 2 − 0 (e(b11 2 2y0 e + y0 2    2   +b− − 2 2 2e − − 22 ) + y0 (b21 − a2 )) + e + y0 arccos e2 +y 2 − 1 (e(b11 + b22 ) + y0 (b21 − a2 )) , − 0 and 1      tr1 =   − 2dy1 ξ y1 a2 c2 + ab(b+ 22 − b+ 11 ) − b2 d1 + bd(b + 11 + b + 22 ) 2by1 ξ 3 d2 + y12 ξ 2    +2y1 ξ 3 (2av1+ y1 + 2bd0 y1 − 2v1+ d − c2 dy1 ) + c2 y1 a2 + ξ 2 + b(−ab+ + 11 y1 + ab22 y1    2d2  +b+ + 2 11 d + db22 ) − b d1 y1 d2 + y12 ξ 2 arccos 2 −1 . d2 + y1 ξ 2 Replacing the expression of tl0 and tl1 in the expansion of the solution of (10) we get the positive half return map in Σ− , i.e.      2  4v1− y0 − 2(b− 11 + b− 22 ) πe2 + ey0 + πy02 + (b− 11 + b− 22 ) e2 + y02 arccos e22e +y 2 − 1 M1− = 0 . 2y0 14 J. L. CARDOSO, J. LLIBRE, , D.D. NOVAES AND D.J. TONON Analogously, replacing the expressions of tr0 and tr1 in the expansion of the solution of (11) we obtain the negative half return map in Σ+ , namely    2d2 b(b+ 11 + b+ 22 ) d2 + ξ 2 y12 arccos d2 +ξ 2 y2 − 1 − 2bdξy1 (b+ + + 3 11 + b22 ) − 4v1 ξ y1 M1+ = 1 . 2bξ 3 y1 The difference between M1− and M1+ provides the first order Melnikov function M1 (y0 ) = M1− (y0 ) − M1+ (y0 ) given in (9), and the simple zeros of M1 (y0 ) provide the crossing limit cycles of Z1,ε (X).  Lemma 5. The function M1 presented in (9) has at most three simple zeros. Furthermore, this upper bound is reached. Proof. Considering the change of coordinates and parameters given by y0 = αs0 /ξ, e = α/ξ and d = αβ in the function M1 , given in (9), we obtain 1   2 − + 2 + + − − M1 (s0 ) = − − 4βs 0 ξ (v 1 b + v 1 ) − 2αbβ s0 (b 11 + b 22 ) + αbξ(b 11 + b22 ) πβ 2 s20 2bβs0 ξ 2     2    +2βs0 + π + αbξ(b− − 2 2 11 + b22 ) β s0 + 1 arccos 1 − 2 2 + αbβ 2 s20 + 1 β s0 + 1  2  (b+ + 11 + b22 ) arccos −1 . s20 + 1 The positive zeros of M1 (s0 ) coincide with the zeros of 2bβs0 ξ 2 M1 (s0 ) = M f1 (s0 ). We have that   Mf1 (s0 ) = 2αbβ 2 s0 (b+ + b+ ) + 4βs0 ξ 2 (v − b + v + ) − αbξ(b− + b− ) πβ 2 s2 + 2βs0 + π 11 22 1 1 11 22 0    2    −αbξ(b− 11 + b− 22 ) β 2 2 s0 + 1 arccos 1 − − αbβ 2 s2 0 + 1 (b+ + 11 + b22 ) β 2 s20 + 1  2  arccos 2 −1 . s0 + 1 If   K0 = 2β 2ξ 2 (v1− b + v1+ ) − αbξ(b− 11 + b− 22 ) + αbβ(b+ 11 + b + 22 , ) K1 = αbξ(b− − 11 + b22 ), K2 = −bαβ 2 (b+ + 11 + b22 ), f1 can be rewritten as then M    2      2  f1 (s0 ) = K0 s0 +K1 β 2 s20 +1 arccos M −1 −2π +K2 s20 +1 arccos −1 . β 2 s20 +1 s20 +1 f1 (s0 ) is the linear combination Note that M f1 (s0 ) = K0 f0 (s0 ) − 2πK1 f1 (s0 ) + K2 f2 (s0 ) + K1 f3 (s0 ) M 15 of the functions f0 (s0 ) = g0 f1 (s0 ) = 1 + β 2 s20    2 f2 (s0 ) = s20 + 1 arccos 2 −1 s0 + 1  2  f3 (s0 ) = (β 2 s20 + 1) arccos − 1 . β 2 s20 + 1 Denoting Wk (s0 ) = Wk (f0 , f1 , ..., fk )(s0 ), we have W0 (s0 ) = s0 W1 (s0 ) = β 2 s20 − 1    2 2 4 β 2 + 1 s0 W2 (s0 ) = 2 β − 1 arccos −1 − s20 + 1 s20 + 1      2 2 16β 3 β 2 − 1 2s0 s20 − 1 + s20 + 1 arccos − 1 s20 + 1 W3 (s0 ) = 2 2 . (s20 + 1) (β 2 s20 + 1) Observe that the functions Wk (s0 ) for k = 0, 1, 2, 3 have not roots if s0 > 1/β. In fact, we have that 2 W3 (s0 ) β 2 s20 + 1 = W f3 (s0 ), where      2 2 16β 3 β 2 − 1 2s0 s20 − 1 + s20 + 1 arccos 2 −1 f3 (s0 ) = s0 + 1 W 2 . (s20 + 1) f3 (s0 ) we have Computing the derivative of W  dWf3 256β 3 β 2 − 1 s20 = 3 , ds0 (s20 + 1) which is strictly positive for all s0 6= 0 and β > 1, and strictly negative for all s0 6= 0 and 0 < β < 1. Therefore, since lims0 →0 W f3 (s0 ) = 0 and lims →∞ W f3 (s0 ) = 0, the function W f3 0 has no roots for s0 > 0. Hence, the function W3 has no roots if s0 > 0 and β 6= 1. In summary, the ordered set F = [f0 , f1 , f2 , f3 ] is an ET-Chebyschev System. By Theorem 2 we conclude that there exists a linear combination of the functions of F with at most three roots. So, the upper bound for the number of zeros of any function in the linear space of functions generated by the functions of F is three. In Example 4.1 of Section 4 we show that this upper bound for the zeros is reached.  Lemma 6. The highest amplitude limit cycle (when it exists) is stable (resp. unstable) provided that ξ(b− − + + − − + + 11 + b22 ) + b11 + b22 < 0 (resp. ξ(b11 + b22 ) + b11 + b22 > 0). The lowest amplitude limit cycle (when it exists) is stable (resp. unstable) provided that b− − 11 + b22 < 0 − − − + − − − − − + or b11 + b22 = 0 and bv1 + v1 > 0 (resp. b11 + b22 > 0 or b11 + b22 = 0 and bv1 + v1 < 0). 16 J. L. CARDOSO, J. LLIBRE, , D.D. NOVAES AND D.J. TONON Proof. In what follows we study the stability of the crossing limit cycles. In fact, this stability depends of the sign of M1 . Indeed, M1 (y0 ) = M1− (y0 ) − M1+ (y0 ), let y0∗ ∈ R+ such that M1 (y0 ) = 0, if M1 (y0 ) > 0 for y0 < y0∗ and M1 (y0 ) < 0 for y0 > y0∗ , then the crossing limit cycle defined by y0 is unstable. If these signs are reversed then it is stable. We get that f1 (s0 ) = −2παbξ(b− + b− ), and lim M lim M1 (s0 ) = ξ(b− − + + 11 22 s0 →∞ 11 + b22 ) + b11 + b22 . s0 →0   Therefore, since b < 0, α > 0 and ξ > 0 the sign lim M f1 (s0 ) = sign(b− + b− ), and 11 22   s0 →0 2 − f1 (s0 ) so sign lim M1 (s0 ) = −sign(b + b ), see Figure 4. So, the 2bβs0 ξ m11 (s0 ) = M − − 11 22 s0 →0 lemma follows.  Σ 0.20 y0∗ 0.15 0.10 0.05 1 2 3 4 5 - 0.05 Figure 4. The repealer crossing limit cycle and the graphic of M1 (y0 ) 1-order linear perturbation of a piecewise linear center. Now, we complete the proof of Theorem A analyzing the stability of the crossing limit cycles and of the periodic orbit at infinity. In fact, the stability of the periodic orbit at infinity is given by sign(ξ(b− − + + 11 + b22 ) + b11 + b22 ), see Section 3.2, and the stability of the crossing limit cycle (c.l.c.) of the biggest amplitude limit cycle is given in Table 1. sign(ξ(b− − + + 11 + b22 ) + b11 + b22 ) stability of the bigger c.l.c stability of ∞ −1 stable unstable 1 unstable stable Table 1. Stability of the highest amplitude crossing limit cycle and the infinity. 3.4. Proof of Theorem B. In order to study the sliding/escaping limit cycle we consider the second order linear perturbation of the vector field Z given in (2). The singular points and the spectrum of the systems in Σ− and Σ+ are given by p− = (−e, 0) and p+ = (−bd/(a2 + bc), −d/(a2 + bc)), respectively; and the eigenvalues of the unperturbed piecewise linear vector field Z2,0 (X) in Σ− and Σ+ are given by Spec− = {i, −i} and Spec+ = 17  √ √ − a2 + bc, a2 + bc , respectively. Consider Zε given in (2) and assume that b− − 11 = −b22 , − + the fold point in Σ and Σ are given by yf 1 = v1− ε + (A0 + v1− a2 )ε2 + O(ε3 ) and v1+ ε (−bC0 + v1+ c2 )ε2 yf 2 = − + + O(ε3 ), b b2 respectively. Under the assumptions e > 0 and db < 0 we get that yf 1 is a visible fold point v+ and yf 2 is an invisible fold point. Assuming that v1− > − 1 we have that yf 1 is over of yf 2 b and by (3) the expression of the sliding vector fields is Zεs (0, y) = y(ay − be − d) + ε(v1− (d − ay) − y(y(aa2 + bb− + − 22 + b22 ) − a2 d + bv2 + c2 e + d0 )  −v1+ e) + ε2 − y(aA0 − v1− b+ − − − + 22 − a2 d0 − A2 d + bw2 + v2 c2 + b22 v1 + C2 e  − − − − + +D0 ) + y 2 (−(aA2 − a2 b+22 + bc22 + b22 c2 + D2 )) + v1 d0 + A0 d − v2 v1 − C0 e . Any point in the sliding region is given as a convex combination of yf 1 and yf 2 as follows  v+ λ  ys (λ) = (1 − λ)yf 1 + λyf 2 = ε v1− − v1− λ − 1 , b where 0 < λ < 1. A necessary condition for the existence of a sliding/escaping limit cycle is that the sliding vector field is regular and points toward the visible fold point yf 1 . The pseudo-equilibrium is (0, y ∗ ) with y ∗ = (d + be)/a, which is reached when λ = λ∗ = −b(d + be)/(a(v1− b + v1+ )ε). Under the hypotheses a < 0 and d + be > 0, we obtain that (0, y ∗ ) ∈ / Σs , i.e., Zεs is regular. The direction of the sliding vector field is given by the sign of the derivative of Zεs evaluated at ys , that is by (v1− b + v1+ )(be + d)/b. From assumptions (d + be) > 0 and (v1− b + v1+ ) < 0 ((v1− b + v1+ ) > 0 resp.), we conclude that the sliding (escaping resp.) vector field points towards yf 1 (yf 2 resp.). In what follows we study the return maps passing through the fold point of Zε+ and Zε− . Our goal is to provide an order relation between the images by the flow of the fold points in a transverse section through yf 1 . This analysis not only provides a necessary condition for the existence of a sliding/escaping limit cycle, but also provides its distinct topological type. The negative half return map in a neighborhood of the invisible fold point yf 2 defines the involution γZε+ : I − → I + , where I + , I − is an open interval above, below, resp., of yf 2 . For more details about the construction of this involution, see [19]. In this way, we have that γZ−1+ (yf 1 ) = yf 3 ∈ I − . The line {(x, y); x = 0} is tangent to the fold ε points. Therefore we cannot use the Implicit Function Theorem in this case. However, we can obtain a condition for the existence of a sliding/escaping limit cycle studying the intersection of the trajectories of Zε in Σ− with initial conditions at yf 1 , yf 2 and yf 3 with the line Λ = {(x, y); x ≤ 0, y = yf 1 } ⊂ Σ− , which is a transversal section at the fold point yf 1 . Considering the smooth vector field Zε− and the initial conditions (0, yf 1 ), (0, yf 2 ) and (0, yf 3 ) the intersection of the flow of Zε− with Λ define the return maps S0 (ε), S1 (ε), S2 (ε) 18 J. L. CARDOSO, J. LLIBRE, , D.D. NOVAES AND D.J. TONON and S3 (ε), respectively, given by   2 1 S0 (ε) = −2e + ε(2b− 21 e − 2v − 2 ) + ε πe(c − 11 + c − 22 ) − 2 v1− b− − − 22 − v2 b21 + w2 −  2 +(b− 2 − 21 ) e − c21 e + O(ε3 ),    S1 (ε) = −2e + ε(2b− − 21 e − 2v2 ) + ε 2 − 2 v1− b− − − − − 2 22 − v2 b21 + w2 + (b21 ) e − c21 e − 1  − πe(c− − 11 + c22 ) + O(ε ), 3 2 ε2  − 2 2   S2 (ε) = −2e + ε(2b− − 21 e − 2v2 ) − 2 (v1 ) b + 2v1− b(2bb− + 2 − − 22 e + v1 ) + b e 4 − v2 b21 2b e    +w2− + (b− 2 − − − 21 ) e − c21 e − πe(c11 + c22 ) + (v1 ) + 2 + O(ε3 ), 2  ε S3 (ε) = −2e + ε(2b− − 21 e − 2v2 ) + − 4b2 e(v1− b− − − − − + 2 22 − v2 b21 + w2 ) − 4(v1 b + v1 ) 2b2 e   +πb2 e2 (c− 11 + c− 22 ) + 4b 2 2 e c − 21 − (b − 2 21 ) + O(ε3 ). By hypothesis (1 − a) If 0 < c− − − + 2 2 2 11 + c22 < (v1 b + v1 ) /(2b e π), then S3 < S2 < S1 < S0 and system Zε admits a sliding cycle of Type I. (1 − b) If (v1− b + v1+ )2 )/(2b2 e2 π) < c− − − + 2 2 2 11 + c22 < 2(v1 b + v1 ) /(b e π), then S3 < S1 < S2 < S0 , and system Zε admits a sliding cycle of Type II. S3 S1 S2 S0 yf 1 Λ yf 2 yf 3 Σ Figure 5. The point yf 3 and the return maps S0 (ε), S1 (ε), S2 (ε) and S3 (ε). Working similarly and assuming that a < 0, d + be > 0 and v1− b + v1+ > 0 we can conclude that (2 − a) If −(v1− b + v1+ )2 )/(2b2 e2 π) < c− − 11 + c22 < 0, then system Zε admits a escaping cycle of Type I. (2 − b) If c− − − + 2 2 2 11 + c22 < −(v1 b + v1 ) )/(2b e π), then system Zε admits a escaping cycle of Type II. This completes the proof of Theorem B. 19 3.5. Proof of Theorem C. We divide the proof in two steps. In the first we provide an upper bound for the number of crossing limit cycles, and in the second we study the stability of these limit cycles. Lemma 7. Assuming that b− − 11 = −b22 the function M1 , given in (9), has at most one simple zero. Moreover, there exists choice of parameters for which this upper bound is reached. Proof. Consider M1 (y0 ) given in (9) and assume that b− − 11 = −b22 , then we obtain    + + 2 2 2 2d2 2v + + + d(b11 + b22 ) (b11 + b 22 ) d + ξ y 0 arccos 2 2 d +ξ y02 − 1 (12) M1 (y0 ) = 2v1− + 1 + − . b ξ2 2y0 ξ 3 Proceeding with the change of coordinates y0 = ds0 /ξ and writing k0 = 2v1− + 2v1+ /b + d(b+ + 2 + + 2 11 + b22 )/ξ and k1 = −d(b11 + b22 )/(2ξ ) we get   d  2 s0 M1 (s0 ) = k0 s0 + k1 s20 + 1 arccos 1 − 2 . ξ s0 + 1 So, denoting f0 (s0 ) = s0    2 f1 (s0 ) = s20 + 1 arccos 1 − , s20 + 1 and computing their Wronskians we obtain W (f0 )(s0 ) = 1   2 W (f0 , f1 )(s0 ) = −2s0 + (s20 − 1) arccos 1 − 2 . s0 + 1 f   Let Wf1 (s0 ) = W1 /(s2 − 1). Therefore dW1 (s0 ) = 8s2 /( s2 − 1 2 s2 + 1 ) which is strictly 0 0 0 0 ds0 positive for all s0 > 0. So, W f1 is strictly increasing and W1 has at most one zero. The existence of a sliding limit cycle follows from Theorem B. In Example 4.2 we present a piecewise linear vector field that exhibits a crossing/sliding limit cycle.  In the previous case the stability of the crossing limit cycle is given by the following result. v1+ Lemma 8. The crossing limit cycle of (2) is unstable (resp. stable) if v1− + b > 0 (resp. v1+ v1− + b < 0). Proof. The stability of the crossing limit cycle is given by the sign of M1 . Indeed, M1 (y0 ) = M1− (y0 ) − M1+ (y0 ), let y0∗ ∈ R+ such that M1 (y0 ) = 0, M1 (y0 ) > 0 for y0 < y0∗ , and M1 (y0 ) < 0 for y0 > y0∗ . Therefore, the crossing limit cycle is unstable. If M1 (y0 ) < 0 for y0 < y0∗ and M1 (y0 ) > 0 for y0 > y0∗ , then the crossing limit cycle is stble. By (12) we have that   v+ lim+ M1 (y0 ) = 2 v1− + 1 . y0 →0 b So, the lemma follows.  20 J. L. CARDOSO, J. LLIBRE, , D.D. NOVAES AND D.J. TONON 4. Final remarks and some examples In this section we provide two examples of piecewise linear vector fields, the first one admitting three crossing limit cycles and the second one with a sliding and a crossing limit cycle. 4.1. Example 1. Consider the following piecewise linear vector field Z = (Z + , Z − ) where      0 −1 x 0 Z − (x, y) =   +  1 0 y 0.55      −1 0 x −2.65 +ε   +  0 −1 y 0 and         1 −1 x 0 0.21 0 x Z + (x, y) =   +  + ε    . 1.01 −1 y 0.1 0 0 y The first order Melnikov function associated to this system is given by −1     2  M1 (y0 ) = − 242π − 800πy02 + 420 y02 + 1 arccos 2 −1 400y  0    y0 + 1 2 242 + 400y0 + 121 arccos − 1 + 840y0 , 400y02 + 121 which has three zeros y0 = 1, y0 = 2 and by the Newton-Kantorovich method, see [2], we have the third zero in the neighborhood of y0 = 3.82781, see Figure 6. Each zero corresponds to a crossing limit cycle with alternating stability. In this case, the highest amplitude crossing limit cycle is unstable and the infinity is stable. 0.20 0.15 0.10 0.05 1 2 3 4 5 - 0.05 Figure 6. Graphic of M1 . Each zero correspond to a crossing limit cycle. 21 4.2. Example 2. Consider the following piecewise linear vector field Z = (Z + , Z − ) where           0 −1 x 0 1 0 x 0.2 Z − (x, y) =    +   + ε   +  1 0 y 1 0 −1 y 0    0.03 0 x +ε2    , 0 0.02 y           −1 −1 x 0 1.5 0 x −0.5 Z + (x, y) =   +  + ε   +  . 2 1 y 2 0 −0.4 y 0 The associated first order Melnikov function is given by     1.6 8 M1 (y0 ) = 2.2 − 0.1y0 + arccos − 1 . y0 0.25y02 + 4 The graph of M1 is given by Figure 7, the crossing limit cycle is located in a neighborhood of y0 = 7.94622, is repealer and the infinite is attractor. 1 2 3 4 5 -2 -4 -6 Figure 7. Graphic of M1 . The unique zero of M1 correspond to a repealer crossing limit cycle. Acknowledgements JL is partially supported by the Ministerio de Economı́a, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017 SGR 1617, and the European project Dynamics- H2020-MSCA-RISE-2017-777911. JLC is partially supported by Goiás Research Foundation (FAPEG) and CAPES. DDN is partially supported by a FAPESP grant 2016/11471-2. DJT is partially supported by PROCAD/CAPES grant 88881.0 68462/2014-01 and by CNPq- Brazil. 22 J. L. CARDOSO, J. LLIBRE, , D.D. NOVAES AND D.J. TONON References [1] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations Pergamon Press, Oxford, 1966. [2] I.K. Argyros, Convergence and applications of Newton-type iterations., Springer Science and Business Media, 2008. [3] M. di Bernardo, C.J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Appl. Math. Sci. 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A., On extended Chebyshev systems with positive accuracy, J. Math. Anal. Appl. 448 (2016), 171–186. 1 Institute of Mathematics and Statistics of Federal University of Goiás, Avenida Esperança s/n, Campus Samambaia, 74690-900, Goiânia, Goiás, Brazil Email address: joao.lopes@ifg.edu.br Email address: djtonon@ufg.br 2 Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 23 3 Departamento de Matematica, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária, 13083-859, Campinas, São Paulo, Brazil Email address: ddnovaes@ime.unicamp.br