Limit cycles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones

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1 This is a preprint of: Limit ccles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones Jaume Llibre Douglas D. Novaes Marco Antonio Teixeira Internat. J. Bifur. Chaos Appl. Sci. Engrg. vol pages 015. DOI: [10.114/S ] International Journal of Bifurcation and Chaos c World Scientific Publishing Compan Limit ccles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones JAUME LLIBRE Departament de Matemàtiques Universitat Autònoma de Barcelona Bellaterra Barcelona Catalonia Spain jllibre@mat.uab.cat DOUGLAS D. NOVAES * 1 Departament de Matemàtiques Universitat Autònoma de Barcelona Bellaterra Barcelona Catalonia Spain ddnovaes@mat.uab.cat Departamento de Matemática Universidade Estadual de Campinas Rua Sérgio Baruque de Holanda 651 Cidade Universitária Zeferino Vaz CEP Campinas São Paulo Brazil ddnovaes@ime.unicamp.br MARCO A. TEIXEIRA Departamento de Matemática Universidade Estadual de Campinas Rua Sérgio Baruque de Holanda 651 Cidade Universitária Zeferino Vaz CEP Campinas São Paulo Brazil teixeira@ime.unicamp.br Received to be inserted b publisher We stud a class of discontinuous piecewise linear differential sstems with two zones separated b the straight line x = 0. In x > 0 we have a linear saddle with its equilibrium point living in x > 0 and in x < 0 we have a linear differential center. Let p be the equilibrium point of this linear center when p lives in x < 0 we sa that its is real and when p lives in x > 0 we sa that it is virtual. We assume that this discontinuous piecewise linear differential sstems formed b the center and the saddle has a center q surrounded b periodic orbits ending in a homoclinic orbit of the saddle independent if p is real virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of sstems according to the position of p inside the class of all discontinuous piecewise linear differential sstems with two zones separated b x = 0. Let N be the maximum number of limit ccles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = when p is on x = 0 and N when p is a real or virtual center. Kewords: discontinuous differential sstem limit ccle piecewise linear differential sstem 1 present address and permanent address. 1

2 J. Llibre D.D. Novaes and M.A. Teixeira 1. Introduction and statement of the main results For a given differential sstem a limit ccle is a periodic orbit isolated in the set of all its periodic orbits of the sstem. One of the main problems of the qualitative theor of the differential sstems in the plane is the stud of their limit ccles. A center is a point having a neighborhood except itself filled b periodic solutions. A classical wa to produce and stud limit ccles is perturbing the periodic solutions of a center. This problem for the continuous differential sstems in the plane has been studied intensivel see for instance the hundred of references in the book [Christopher et al 007]. The main objective of this paper is to stud the limit ccles that can bifurcate from a center of a discontinuous piecewise linear differential sstems with two zones separated b the straight line x = 0 when the center is perturbed inside the class of all discontinuous piecewise linear differential sstems with two zones separated b x = 0. The stud of the piecewise linear differential sstems goes back to Andronov and coworkers [Andronov et al 1966] and in the present das still continues receiving a big attention b researchers. Thus in these last ears there has been a big interest from the mathematical communit in understanding their dnamical richness. On the other hand such sstems are widel used to model man real phenomena and different modern devices see for instance the books [di Bernardo et al 008] and [Simpson 010] the paper [Makarenkov & Lamb 01] and the references quoted in there. Of course the case of continuous piecewise linear sstems when the have onl two pieces separated b a straight line is the simplest possible configuration of piecewise linear sstems. We note that even in this simple case onl after a huge analsis it was possible to establish the existence of at most one limit ccle for such sstems see [Freire et al 1988] and for a recent shorter proof see [Llibre et al 01]. There are two reasons for that misleading simplicit of piecewise linear sstems. First even being easil the computations of the solutions in an linear region the time that each orbit requires to pass from one linear region to the other is not eas to compute and consequentl the matching of the corresponding solutions is a difficult problem. Second the number of parameters to consider in order to be sure that one controls all possible configurations is generall not small so the obtention of efficient canonical forms with fewer parameters is important. See also [Liang et al 013] and the references quoted there. In this paper we deal with discontinuous piecewise linear sstems having the form Ẋ = F X signxgx where X = x R and F and G are linear vector fields. Sstems of this kind have been studied recentl in [Giannakopoulos & Pliete 001; Han & Zhang 010; Shui et al 010; Freire et al 01; Huan & Yang 01; Llibre & Ponce 01; Llibre et al 01; Artés et al 013; Braga & Mello 013; Buzzi et al 013; Huan & Yang 013ab; Freire et al 014; Llibre et al 014; Novaes 014; Xiong & Han 014] among other papers. In [Han & Zhang 010] some results about the existence of two limit ccles surrounding a unique equilibrium point appeared and the authors conjectured that the maximum number of limit ccles surrounding a unique equilibrium point of piecewise linear differential sstems with a unique straight line of discontinuit is at most two. This conjecture is analogous to Conjecture 1 of [Tonnelier 003]. Later on in [Huan & Yang 01] the authors provide numerical evidence on the existence of three limit ccles surrounding a unique equilibrium for the discontinuous piecewise linear differential sstems with two linear zones separated b a straight line. In [Llibre & Ponce 01] it is proved the existence of such 3 limit ccles. Up to now there are results that some classes of discontinuous piecewise linear differential sstems with a unique straight line of discontinuit can have at least k limit ccles surrounding a unique equilibrium point. As far as we know there are no results that some classes of discontinuous piecewise linear differential sstems with a unique straight line of discontinuit have at most k limit ccles surrounding a unique equilibrium point. Probabl this is the first approach where such kind of results are stated. We consider planar discontinuous piecewise linear differential sstems with two zones separated b the

3 Limit ccles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones 3 straight line Σ = {x = 0} i. e. ẋ = ẏ x M x M u if x > 0 u if x < 0 1 where M and M are real matrices and x T u u R. Here the dot denotes derivative with respect to the independent variable t here called the time. Select the following sets of hpotheses: Ha = {Hak : k = 1 3} where Ha1 s in x > 0 is a saddle for the sstem ẋ ẏ T = M x T u Ha p in x < 0 is a center for the sstem ẋ ẏ T = M x T u Ha3 the singular point p is a center for the sstem 1 such that its period annulus formed b all the periodic orbits surrounding p ends in a homoclinic loop of the saddle s Hb = {Hbk : k = 1 3} where Hb1 s in x > 0 is a saddle for the sstem ẋ ẏ T = M x T u Hb p on x = 0 is a center for the sstem ẋ ẏ T = M x T u Hb3 the singular point p is a center for the sstem 1 such that its period annulus ends in a homoclinic loop of the saddle s and Hc = {Hck : k = 1 3} where Hc1 s in x > 0 is a saddle for the sstem ẋ ẏ T = M x T u Hc p in x > 0 is a virtual center for the sstem ẋ ẏ T = M x T u Hc3 r is a fold fold point which is a center for the sstem 1 such that its period annulus ends in a homoclinic loop of the saddle s. Roughl speaking a fold fold singularit for sstem 1 is a point on x = 0 at which two curves of fold points meet one from the solutions in x 0 and the other from the solutions in x 0. An affine change of variables in the plane preserving the straight line x = 0 will be called in what follows a Σ preserving affine change of variables. Proposition 1. The discontinuous piecewise linear differential sstems 1 satisfing assumptions Ha after a Σ preserving affine change of variables and a time rescaling can be written as x c A if x > 0 ẋ = ẏ x d A if x < 0 where A = 0 a A b 0 = and the parameters a b c and d are positive see Figure Proposition. The discontinuous piecewise linear differential sstems 1 satisfing assumptions Hb after an affine Σ preserving change of variables and a time rescaling can be written as sstem with the parameters a b and c positive and d = 0 see Figure. Proposition 3. The discontinuous piecewise linear differential sstems 1 satisfing assumptions Hc after an affine Σ preserving change of variables and a time rescaling can be written as sstem with the parameters a b and c positive and d negative see Figure 3.

4 4 J. Llibre D.D. Novaes and M.A. Teixeira Propositions 1 and 3 are proved in section. Σ Γ s p 0 s x u Fig. 1. Normal form of sstem 1 assuming the Hpotheses set Ha. Here Γ is the homoclinic orbit; the points 0 u and 0 s are the intersections between Γ and Σ; the point p is both a center of the sstem ẋ ẏ T = M x T u and a center of the sstem 1 with its period annulus ending in Γ; the point s is a saddle of the sstem ẋ ẏ T = M x T u ; and the origin 0 is a fold fold point of Σ. We consider the more general affine perturbation of sstem inside the class of discontinuous piecewise linear differential sstems with two zones separated b the straight line x = 0 namel x c x A ε B ẋ v if x > 0 = 3 ẏ x d x A ε B v if x < 0 where ε is a small parameter and B = b 1 b b 3 b 4 B = b 1 b b 3 b 4 v = v 1 v and v = v 1 v. Let N be the maximum number of limit ccles of perturbed sstem 3 which can bifurcate from the periodic solutions of the unperturbed sstem when ε 0 is sufficientl small. Theorem A. For ever a > 0 b > 0 c > 0 and d > 0 we have that N. Moreover we can find parameters b ± i for i = and v j ± for j = 1 such that sstem 3 satisfing Ha when ε = 0 has at least 0 1 or limit ccles. Theorem A is proved in section 3. In the next corollar we show that sstem 3 has at least 3 limit ccles for some values of the parameters for which the hpotheses of Theorem A hold. Corollar 1. We assume that a = b = c = 1/ d = 1/4 b 1 = 1171/5 b 1 = 1331 v 1 = 4496 and b ± = b± 3 = b± 4 = v 1 = v± = 0. Then for ε 0 sufficientl small sstem 3 has at least 3 limit ccles. Numericall we can see that these 3 limit ccles pass ε close through the points 0 i where i

5 Limit ccles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones 5 Σ Γ s p s x u Fig.. Normal form of sstem 1 assuming the Hpotheses set Hb. Here Γ is the homoclinic orbit; the points 0 u and 0 s are the intersections between Γ and Σ; the point p = 0 is both a center of the sstem ẋ ẏ T = M x T u and a center of the sstem 1 with its period annulus ending in Γ moreover it is a fold fold point of Σ; and the point s is a saddle of the sstem ẋ ẏ T = M x T u. for i = 1 3 are the solutions of the following equation π arctan 8 for 0 1/. Moreover and Corollar 1 is proved in section log 1 = 0 1 Theorem B. For ever a > 0 b > 0 c > 0 and d = 0 we have that N =. Moreover we can find parameters b ± i for i = and v j ± for j = 1 such that sstem 3 satisfing Hb when ε = 0 has exactl 0 1 or limit ccles. Theorem B is proved in section 3. Theorem C. For ever a > 0 b > 0 c > 0 and d < 0 we have that N. Moreover we can find parameters b ± i for i = and v j ± for j = 1 such that 3 satisfing Hc when ε = 0 has at least 0 1 or limit ccles. Theorem C is proved in section 3.. Proofs of Propositions 1 and 3 First we shall prove the normal forms given in the statements of Propositions 1 and 3 for the discontinuous piecewise linear differential sstems with two zones separated b the straight line Σ satisfing the assumptions either Hak or Hbk or Hck for k = 1 3 respectivel. Proof. [Proof of Proposition 1] From Hpothesis Ha p = d e with d > 0 is a center for the sstem

6 6 J. Llibre D.D. Novaes and M.A. Teixeira Σ Γ s 0 p s x u Fig. 3. Normal form of sstem 1 assuming the Hpotheses set Hc. Here Γ is the homoclinic orbit; the points 0 u and 0 s are the intersections between Γ and Σ; the point p is a center of the sstem ẋ ẏ T = M x T u ; the point s is a saddle of the sstem ẋ ẏ T = M x T u ; and the origin 0 is a fold fold point of Σ wich is a center of the sstem 1 with its period annulus ending in Γ. ẋ ẏ T = M x T u. So M = m1 m m 3 m 1 with m 1 m m 3 < 0. B translating the sstem through the straight line {x = d} which is a Σ preserving change of variables we can assume that e = m 1 d/m. Doing the Σ preserving change of variables x = ỹ m 1 m m 3 0 m 1 m x and rescaling the time b τ = m 1 m m 3 t sstem 1 can be written as x c x A if x > 0 ỹ c ỹ = x d A if x < 0 ỹ where A = ā a b b and A = with a b c d ā b c R c > 0 and d > 0. Now the prime indicates derivative with respect to the new time τ. From Hpothesis Ha1 s = c c is a saddle for the sstem x ỹ T = A x c ỹ c T. So A has two distincts real non zero eigenvalues λ 1 λ such that λ 1 λ < 0. It is eas to see that this last condition is equivalent to 4 ab > ā b. 5

7 Limit ccles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones 7 As usual W u s and W s s represent respectivel the unstable and stable sets of the saddle s. We denote us = Σ W us s. From Hpothesis Ha3 { u s } W u s W s s. Let Γ = W u s W s s containing u and s. Hence Γ ia a homoclinic loop. The center at d 0 of the sstem x ỹ T = A x d T induces a smmetr on 4 namel: the solution of 4 starting in Σ for > 0 has to return to Σ in for t < 0 so the same occurs for t > 0 and for ever between u and s because from Ha3 d 0 is a center of 4 such that its period annulus ends in the homoclinic loop Γ. From the above smmetr u = s which is equivalent to b = cā a c/c. 6 Now the origin 0 0 is a singularit for the sstem 4 because it is a point of tangenc. From Hpothesis Ha3 the origin 0 0 must be a fold-fold point. Moreover ever orbit distinct to 0 0 inside the region delimited b Γ reaching the line Σ have to cross Σ. Thus d 0 is a center with its period annulus ending in Γ. These conditions are satisfied if and onl if c = āc/a and a > 0. From 6 we conclude that b = ā Computing the solution of 4 we conclude from the above smmetr that ā = 0. Furthermore from 5 we have that b > 0. Hence 4 has a center at d 0 such that its period annulus ends in the homoclinic loop Γ if and onl if ā = b = c = 0 and a b c > 0. So we have conclude the proof. Proof. [Proof of Proposition ] B translating the sstem through the straight line Σ which is a Σ preserving change of variables we can assume that p = 0 0. From Hpothesis Hb 0 0 is a center for the sstem ẋ ẏ T = M x T u. So M m1 m = m 3 m 1 with m 1 m m 3 < 0. Doing the Σ preserving change of variables x m = 1 m m 3 0 x ỹ m 1 m and rescaling the time b τ = m 1 m m 3 t sstem 1 can be written as x c x A if x > 0 ỹ c ỹ = x A if x < 0 ỹ 7 where A = ā a b b and A = with a b c ā b c R and c > 0. From here the proof follows analogousl to the proof of Proposition 1. Proof. [Proof of Proposition 3] From the Hpothesis Hc p = d e with d < 0 is a center for the sstem ẋ ẏ T = M x T u. So M m1 m = m 3 m 1 with m 1 m m 3 < 0. B translating the sstem through the straight line {x = d} which is Σ preserving change of variables we can assume that e = m 1 d/m.

8 8 J. Llibre D.D. Novaes and M.A. Teixeira Doing the Σ preserving change of variables x m = 1 m m 3 0 x ỹ m 1 m and rescaling the time b τ = m 1 m m 3 t sstem 1 can be written as x c A if x > 0 ỹ c x ỹ = A x d ỹ if x < 0 8 where A = ā a b b and A = with a b c d ā b c R c > 0 and d > 0. From here the proof follows in a similar wa to the proof of Proposition Proofs of Theorems A B C and Corollar 1 The idea of the proofs of Theorems A B and C is to compute the Talor expansion at ε = 0 up to order 1 in ε of the Poincaré map associated to 3 and then appl the Implicit Function Theorem. Let I be a proper real interval and let f 0 f 1... f n : I R. We sa that f 0 f 1... f n are linearl independent functions if and onl if t I n α i f i t = 0 α 0 = α 1 = = α n = 0. i=0 Proposition 4. If f 0 f 1... f n are linearl independent then there exist t 1 t... t n I and α 0 α 1... α n R such that for ever j {1... n} n α i f i t j = 0. For a proof of Proposition 4 see for instance [Llibre & Świrszcz 011]. i=0 We sa that an ordered set of complex valued functions F = f 0 f 1... f n defined on I is an Extended Chebshev sstem or ET sstem on I if and onl if an nontrivial linear combination of the functions of F has at most n zeros counting multiplicities. We sa that F is an Extended Complete Chebshev sstem or an ECT sstem on I if and onl if for an k 0 k n f 0 f 1... f k is an ET sstem. For more details see the book of Karlin and Studden [Karlin & Studden 1966]. In order to prove that F is a ECT sstem on I is sufficient and necessar to show that W f 0 f 1... f k t 0 on I for 0 K n. Here W f 0 f 1... f k t denotes the Wronskians of the functions f 0 f 1... f k with respect to t. We recall the definition of the Wronskian. f 0 t f k t f 0 t f k t W f 0... f k t = f k 0 t f k t k

9 Limit ccles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones 9 Now consider the functions g 1 = 1 a b c b c a g = log b c a g 1 3 = d g3 = and g 3 3 = d 3π arctan π arctan d d d d. 9 We define the sets of functions G i = {g 1 g g3 i } for i = 1 3. In the proofs of Theorems A B and C we shall use the following lemma. Lemma 1. Assume that a > 0 b > 0 c > 0. a If d > 0 then G 1 is a set of functions linearl independent on the interval 0 b c/ a. b If d = 0 then G is a ECT-sstem on the interval 0 b c/ a. c If d < 0 then G 3 is a set of functions linearl independent on the interval 0 b c/ a. Proof. To prove statement a we compute the power series expansion in up to order of the functions g i for i = 1 and g3 1. So g 1 = 1 g = abc 4a3/ 3 bc O 3 and g 1 3 = 4d π 1 4d 4π 8 3d O Let C be the square matrix formed b the coefficients of i for i = 0 1 of 10 namel C = abc 0 4a3/ 3 bc. 4d 4π 8 3d Since detc = 16 a π/3 bc 0 we have that G 1 is a set of functions linearl independent for > 0 near 0 which implies that G 1 is a set of functions linearl independent in 0 b c/ a for ever a b c d > 0. This concludes the proof of statement a. The proof of statement c follows analogousl b considering the power series expansion in up to order of the functions g i for i = 1 and g 3 3. To prove statement b we compute the Wronskians W 1 = W g 1 W = W g 1 g3 and W 3 = W g 1 g3 g. So W 1 = 1 W = 1 and 4a W 3 = b c a 4 5 ab c b c 5a b c a log. b c a

10 10 J. Llibre D.D. Novaes and M.A. Teixeira Clearl W 1 0 and W 0 for > 0. Now since b c a b c a =1 = 1 and d b c a = d b c a abc bc a > 0 for 0 bc/ a it follows that log b c a b c a > 0 11 for 0 bc/ a. Now let P = abc a bc log b c a. b c a So W 3 = bc bc 3 a 5 P. Moreover P 0 = 8 a 3 /3 bc > 0 and P = 4abc log b c a b c a > 0. Since bc 3 a 5 > 0 for ever 0 bc/ a we conclude that W 3 > 0 for 0 bc/ a. Hence the lemma is proved. Proof. [Proof of Theorem A] The solutions of the discontinuous piecewise linear differential sstems and 3 can be easil computed because the are piecewise linear differential sstems. Let ϕ t x ε = ϕ 1 t x ε ϕ t x ε be the solution of 3 for x > 0 such that ϕ 0 x ε = x. Similarl let ϕ t x ε = ϕ 1 t x ε ϕ t x ε be the solution of 3 for x < 0 such that ϕ 0 x ε = x. For > 0 let t ε be the smallest positive time such that ϕ 1 t ε 0 ε = 0 and let t ε be the biggest negative time such that ϕ 1 t ε 0 ε = 0. Clearl there exists a limit ccle passing through if and onl if ϕ t ε 0 ε = ϕ t ε 0 ε. So we must stud the zeros of the function f ε = ϕ t ε 0 ε ϕ t ε 0 ε. 1

11 Limit ccles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones 11 Using some algebraic manipulator as Mathematica or Maple it is eas to see that t ε = 1 b c a ε b c a log ab b c a a b bc a 3 log b c a abb abb c 3 b 1 b 4 c b ab abb 3 ab abcb 1 b 4 ab abb aba b 3 3 ab a cb 3 v bc c b 1 b 4 v 1 b Oε and d t ε = arctan 3π d ε d d arctan d d b 1 b 4 d b b 3 d 3 π b 1 b 4 b 1 πb b 3 b 4 d πb b 3 v b b 3 πb 1 b 4 v1 d Oε. π 13 Substituting 13 in 1 and expanding 1 in Talor series at ε = 0 up to order 1 in ε we get f ε = εf 1 Oε where f 1 = k 1 g 1 k g k 3 g and the functions g i for i = 1 and g 1 3 are the ones defined in 9. Here the coefficients k m for m = 1 3 depend linearl on the parameters b ± i for i = and v ± j for j = 1 namel k 1 = c b 1 b 4 ad b 1 b 4 v 1 av1 a k = b 1 b 4 a 3 b and k 3 = b 1 b 4 4. Now let be a simple zero of 14. We note that the function f ε satisfies the hpothesis of the Implicit Function Theorem in a neighborhood of the point ε = 0 that is f 0 = 0 and z fz ε 0 0. Therefore from the Implicit Function Theorem there exists a unique analtic function ξε defined in a neighborhood of ε = 0 such that fξε ε = 0 and ξ0 =. So from the definition of the function 1 each simple zero of 14 is associated with a hperbolic limit ccle of 3. Hence appling Lemma 1a and Proposition 4 we conclude the proof of the theorem. In the proof of Theorem B we shall need the following lemma. Lemma. Assume that a > 0 b > 0 c > 0 and d = 0. If b 1 = b 4 b 1 = b 4 and v 1 = av 1 then sstem does not admit limit ccles.

12 1 J. Llibre D.D. Novaes and M.A. Teixeira Proof. From the assumptions the smallest positive time t ε such that ϕ 1 t ε 0 ε = 0 and the biggest negative time t ε such that ϕ 1 t ε 0 ε = 0 can be explicitl computed using some algebraic manipulator as Mathematica or Maple namel t ε = 1 b c a ab log 3 ε bb b c a ab b c a log a 3 b 3 b c a bcb a bcv 1 cb 3 v ab bc a ε log b c a b c a 3bb 3ab 3 ab b b 3 b 1 8 a 5 b 5 ab 3 5cb 3 8v 3 4b 3bc3 b 4a bc3 b v a c 4a 8cv 3c 3 b 3 8av1 cb 3 v 1 bc a 4b 3cb b 3 cb 1 4b v 4ab 1 v 1 c b 1 b b 3 8a v1 5b 15 8av 1 cv cb 1 b Oε 3 and b t ε = π ε 3 b ε π 8 π v 4b 1 3b b b 3 3b 3 v b 3 b v 1 b 1 v Oε 3. Expanding 1 in Talor series at ε = 0 up to order in ε we get f ε = ε v 1 b ab Oε 3. a Now suppose that for ε 0 sufficientl small there exists a continuous branch ȳ ε of solutions of the equation f ε = 0 such that ȳ ε ȳ when ε 0 for some 0 < ȳ < b c/ a. It implies that v1 = 0 or b = ab. However in both cases we can verif that f ε 0 i.e. the equilibrium point of sstem 3 is a center. This implies the non existence of limit ccles for the sstem 3. Proof. [Proof of Theorem B] Following the proof of Theorem A we have to estimate 1. Again using some algebraic manipulator as Mathematica or Maple it is eas to see that t ε is given b 13 and Oε. 16 t ε = π ε b b 3 π v Substituting t ε and 16 in 1 and expanding 1 in Talor series at ε = 0 up to order 1 in ε we get f ε = εf Oε where f = k 1 g 1 k g k 3 g 3 17

13 REFERENCES 13 and the functions g i for i = 1 and g3 are the ones defined in 9. Here the coefficients k m for m = 1 3 depend linearl on the parameters b ± i for i = and v j ± for j = 1 namel k 1 = c b 1 b 4 v 1 av1 a k = b 1 b 4 a 3 b and k 3 = πb 1 b 4. Note that f 0 if and onl if b 1 = b 4 b 1 = b 4 and v 1 = av 1. In this case from Lemma sstem does not admit limit ccles. Thus we can assume that the function 17 is not identicall zero. This implies that for ε 0 sufficientl small the zeros of the function 1 on a given bounded interval assuming the hpotheses of Theorem B are completel controlled b the function 17. From here appling Lemma 1b the definition of the ECT sstems and the Implicit Function Theorem the proof follows. Proof. [Proof of Theorem C] Here t ε is given b 13 and Here t ε is given b 13 and d t ε = arctan π d b d ε 1 b 4 b b 3 d b 1 b 4 v d v1 b b 3 d Oε. 18 This proof follows completel analogous to the proof of Theorem A b appling Lemma 1c Proposition 4 and the Implicit Function Theorem for the function f 3 = k 1 g 1 k g k 3 g where the functions g i for i = 1 and g 3 3 are the ones defined in 9. Here the coefficients k m for m = 1 3 depend linearl on the parameters b ± i for i = and v ± j for j = 1. Proof. [Proof of Corollar 1] For the given coefficients the function 14 from the proof of Theorem 13 becomes f 1 = π arctan log Evaluating this function in some points we can check that f 1 1/10 > 0 f 1 4/10 < 0 f 1 45/100 > 0 and f 1 499/100 < 0. So there exists 1 1/10 4/10 4/10 45/100 and 3 45/ /1000 such that f i = 0 for i = 1 3. Since from the proof of Theorem 13 f ε = εf 1 Oε so for ε 0 sufficientl small we also have that f1/10 ε > 0 f4/10 ε < 0 f45/100 ε > 0 and f499/100 ε < 0 which implies the existence of at least 3 limit ccles. Now plotting the graphic of f 1 and f 1 for 0 1/ see Figure 4 and estimating the solutions of the equation f 1 = 0 we obtain the numerical conclusion of the corollar. References Andronov A. Vitt A. & Khaikin S. [1966] Theor of Oscillations Pergamon Press Oxford. Artés J.C. Llibre J. Medrado J.C. & Teixeira M.A. [013] Piecewise linear differential sstems with two real saddles Math. Comp. Sim Berezin I.S. & Zhidkov N.P. [1964] Computing Methods Volume II Pergamon Press Oxford. Braga D.C. & Mello L.F. [013] Limit ccles in a famil of discontinuous piecewise linear differential sstems with two zones in the plane Nonlinear Dnamics

14 14 REFERENCES Fig. 4. The normal bold line is the graphic of the function f 1 for 0 1/ and the dashed bold line is the graphic of the function f 1 for 0 1/. We can see that these functions have no common zeros for 0 1/ that is the zeros i for i = 1 3 of f are simple. It implies that the persist under small perturbation. Buzzi C. Pessoa C. & Torregrosa J. [013] Piecewise linear perturbations of a linear center Discrete Continuous Dn. Sst Christopher C. & Li C. [007] Limit ccles of differential equations Birkhäuser Verlag Basel. di Bernardo M. Budd C.J. Champnes A. R. & Kowalczk P. [008] Piecewise-Smooth Dnamical Sstems: Theor and Applications Springer-Verlag London. Freire E. Ponce E. Rodrigo F. & Torres F. [1988] Bifurcation sets of continuous piecewise linear sstems with two zones Int. J. Bifurcation and Chaos Freire E. Ponce E. & Torres F. [01] Canonical Discontinuous Planar Piecewise Linear Sstems SIAM J. Applied Dnamical Sstems Freire E. Ponce E. & Torres F. [014] A general mechanism to generate three limit ccles in planar Filippov sstems with two zones Nonlinear Dnamics DOI /s Giannakopoulos F. & Pliete K. [001] Planar sstems of piecewise linear differential equations with a line of discontinuit Nonlinearit Han M. & Zhang W. [010] On Hopf bifurcation in non smooth planar sstems J. of Differential Equations Huan S.M. & Yang X.S. [01] The number of limit ccles in general planar piecewise linear sstems Discrete and Continuous Dnamical Sstems-A Huan S.M. & Yang X.S. [013] On the number of limit ccles in general planar piecewise linear sstems of node node tpes J. Math. Anal. Appl Huan S.M. & Yang X.S. [013] Existence of limit ccles in general planar piecewise linear sstems of saddle saddle dnamics Nonlinear Anal Karlin S.J. & Studden W.J. [1966] T-Sstems: With Applications in Analsis and Statistics New York London Sidne. Liang F. Han M. & Zhang X. [013] Bifurcation of limit ccles from generalized homoclinic loops in planar piecewise smooth sstems J. Differential Equations Llibre J. Novaes D.D. & Teixeira M.A. [014] On the birth of limit ccles for non smooth dnamical sstems Bulletin des Sciences Mathemàtiques DOI: /j.bulsci Llibre J. Ordóñez M. & Ponce E. [01] On the existence and uniqueness of limit ccles in planar continuous piecewise linear sstems without smmetr Nonlinear Analsis: Real World Applications Llibre J. & Ponce E. [01] Three nested limit ccles in discontinuous piecewise linear differential sstems with two zones Dnamics of Continuous Discrete and Impulsive Sstems Serie B Llibre J. & Świrszcz G. [011] On the limit ccles of polnomial vector fields Dnamics of Continuous

15 REFERENCES 15 Discrete and Impulsive Sstems Serie A Llibre J. Teixeira M.A. & Torregrosa J. [01] Lower bounds for the maximum number of limit ccles of discontinuous piecewise linear differential sstems with a straight line of separation Int. J. Bifurcation and Chaos Makarenkov O. & Lamb J.S.W. [01] Dnamics and bifurcations of nonsmooth sstems: A surve Phsica D Novaes D.D. [014] On nonsmooth perturbations of nondegenerate planar centers Publicacions Matemàtiques Vol. Extra Shui S. Zhang X. & J. Li [010] The qualitative analsis of a class of planar Filippov sstems Nonlinear Anal Simpson D.J.W. [010] Bifurcations in Piecewise Smooth Continuous Sstems World Scientific Singapore. Tonnelier A. [003] The McKean s caricature of the FitzHugh-Nagumo model I. The space-clamped sstem SIAM J. Appl. Math Xiong Y. & Han M. [014] Limit ccle bifurcations in a class of perturbed piecewise smooth sstems Appl. Math. Comput Acknowledgments We thank to the referees for their helpful comments and suggestions. The first author is partiall supported b a MINECO/FEDER grant MTM and MTM P an AGAUR grant number 013SGR-568 an ICREA Academia the grants FP7-PEOPLE-01- IRSES and and a FEDER-UNAB 10-4E-378. The second author is partiall supported b a FAPESP BRAZIL grant 013/ The third authors is partiall supported b a FAPESP BRAZIL grant 01/ The three authors are also supported b a CAPES CSF PVE grant / from the program CSF-PVE.