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

We study a class of discontinuous piecewise linear differential systems with two zones separated by 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 say that its is real, and when p lives in x > 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential systems formed by the center and the saddle has a center q surrounded by 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 systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N ≥ 2 when p is a real or virtual center.


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 its periodic orbits of the system.One of the main problems of the qualitative theory of the differential systems in the plane is the study of their limit cycles.A center is a point having a neighborhood, except itself, filled by periodic solutions.A classical way to produce and study limit cycles is perturbing the periodic solutions of a center.This problem for the continuous differential systems in the plane has been studied intensively, see, for instance, the hundred of references in the book [Christopher et al, 2007].
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 coworkers [Andronov et al, 1966], and in the present days still continues receiving a big attention by researchers.Thus, in these last years there has been a big interest from the mathematical community in understanding their dynamical richness.On the other hand such systems are widely used to model many real phenomena and different modern devices, see for instance the books [di Bernardo et al, 2008] and [Simpson, 2010], the paper [Makarenkov & Lamb, 2012], and the references quoted in there.
Of course, the case of continuous piecewise linear systems, when they have only two pieces separated by a straight line is the simplest possible configuration of piecewise linear systems.We note that even in this simple case, only after a huge analysis it was possible to establish the existence of at most one limit cycle for such systems, see [Freire et al, 1988], and for a recent shorter proof see [Llibre et al, 2012].There are two reasons for that misleading simplicity of piecewise linear systems.First, even being easily the computations of the solutions in any linear region, the time that each orbit requires to pass from one linear region to the other is not easy to compute, and consequently 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 generally not small, so the obtention of efficient canonical forms with fewer parameters is important.See also [Liang et al, 2013] and the references quoted there.
In this paper we deal with discontinuous piecewise linear systems having the form Ẋ = F (X) + sign(x)G(X), where X = (x, y) ∈ R 2 and F and G are linear vector fields.Systems of this kind have been studied recently in [Giannakopoulos & Pliete, 2001;Han & Zhang, 2010;Shui et al, 2010;Freire et al, 2012;Huan & Yang, 2012;Llibre & Ponce, 2012;Llibre et al, 2012;Artés et al, 2013;Braga & Mello, 2013;Buzzi et al, 2013;Huan & Yang, 2013a,b;Freire et al, 2014;Llibre et al, 2014;Novaes, 2014;Xiong & Han, 2014], among other papers.In [Han & Zhang, 2010] some results about the existence of two limit cycles surrounding a unique equilibrium point appeared, and the authors conjectured that the maximum number of limit cycles surrounding a unique equilibrium point of piecewise linear differential systems with a unique straight line of discontinuity is at most two.This conjecture is analogous to Conjecture 1 of [Tonnelier, 2003].Later on in [Huan & Yang, 2012] the authors provide numerical evidence on the existence of three limit cycles surrounding a unique equilibrium for the discontinuous piecewise linear differential systems with two linear zones separated by a straight line.In [Llibre & Ponce, 2012] it is proved the existence of such 3 limit cycles.
Up to now there are results that some classes of discontinuous piecewise linear differential systems with a unique straight line of discontinuity can have at least k limit cycles surrounding a unique equilibrium point.As far as we know there are no results that some classes of discontinuous piecewise linear differential systems with a unique straight line of discontinuity have at most k limit cycles surrounding a unique equilibrium point.Probably this is the first approach where such kind of results are stated.
We consider planar discontinuous piecewise linear differential systems with two zones separated by the straight line Σ = {x = 0}, i. e.
where M + and M − are 2 × 2 real matrices, and (x, y) T , u + , u − ∈ R 2 .Here the dot denotes derivative with respect to the independent variable t, here called the time.
Select the following sets of hypotheses: (Ha) = {(Hak) : k = 1, 2, 3}, where (Ha1) s in x > 0 is a saddle for the system ( ẋ, ẏ) T = M + (x, y) T + u + , (Ha2) p in x < 0 is a center for the system ( ẋ, ẏ) T = M − (x, y) T + u − , (Ha3) the singular point p is a center for the system (1) such that its period annulus (formed by all the periodic orbits surrounding p) ends in a homoclinic loop of the saddle s, , where (Hb1) s in x > 0 is a saddle for the system ( ẋ, ẏ) T = M + (x, y) T + u + , (Hb2) p on x = 0 is a center for the system ( ẋ, ẏ) T = M − (x, y) T + u − , (Hb3) the singular point p is a center for the system (1) such that its period annulus ends in a homoclinic loop of the saddle s, and (Hc) = {(Hck) : k = 1, 2, 3}, where (Hc1) s in x > 0 is a saddle for the system ( ẋ, ẏ) T = M + (x, y) T + u + , (Hc2) p in x > 0 is a virtual center for the system ( ẋ, ẏ) T = M − (x, y) T + u − , (Hc3) r is a fold-fold point which is a center for the system (1) such that its period annulus ends in a homoclinic loop of the saddle s.
Roughly speaking, a fold-fold singularity for system (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 systems (1) satisfying assumptions (Ha) after a Σ-preserving affine change of variables and a time-rescaling can be written as where and the parameters a, b, c, and d are positive (see Figure 1).
Proposition 2. The discontinuous piecewise linear differential systems (1) satisfying assumptions (Hb) after an affine Σ-preserving change of variables and a time-rescaling can be written as system (2) with the parameters a, b and c positive and d = 0 (see Figure 2).
Proposition 3. The discontinuous piecewise linear differential systems (1) satisfying assumptions (Hc) after an affine Σ-preserving change of variables and a time-rescaling can be written as system (2) with the parameters a, b and c positive and d negative (see Figure 3).Here Γ is the homoclinic orbit; the points (0, y u ) and (0, y s ) are the intersections between Γ and Σ; the point p is both a center of the system ( ẋ, ẏ) T = M − (x, y) T + u − and a center of the system (1) with its period annulus ending in Γ; the point s is a saddle of the system ( ẋ, ẏ) T = M + (x, y) T + u + ; and the origin 0 is a fold-fold point of Σ.
We consider the more general affine perturbation of system (2) inside the class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0, namely where ε is a small parameter and Let N be the maximum number of limit cycles of perturbed system (3) which can bifurcate from the periodic solutions of the unperturbed system (2) when |ε| = 0 is sufficiently small.Theorem A. For every a > 0, b > 0, c > 0 and d > 0, we have that N ≥ 2. Moreover we can find parameters b ± i for i = 1, 2, 3, 4 and v ± j for j = 1, 2 such that system (3), satisfying (Ha) when ε = 0, has at least 0, 1 or 2 limit cycles.
Theorem A is proved in section 3.
In the next corollary we show that system (3) has at least 3 limit cycles for some values of the parameters for which the hypotheses of Theorem (A) hold.
Corollary 1.We assume that a Then for |ε| = 0 sufficiently small system (3) has at least 3 limit cycles.Numerically, we can see that these 3 limit cycles pass ε-close through the points (0, y i ), where y i Here Γ is the homoclinic orbit; the points (0, y u ) and (0, y s ) are the intersections between Γ and Σ; the point p = 0 is both a center of the system ( ẋ, ẏ) T = M − (x, y) T + u − and a center of the system (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 system ( ẋ, ẏ) T = M + (x, y) T + u + .
Corollary 1 is proved in section 3.
Theorem B is proved in section 3.
Theorem C is proved in section 3.

Proofs of Propositions 1, 2, and 3
First we shall prove the normal forms given in the statements of Propositions 1, 2, and 3 for the discontinuous piecewise linear differential systems with two zones separated by the straight line Σ satisfying the assumptions either (Hak), or (Hbk), or (Hck) for k = 1, 2, 3, respectively.Here Γ is the homoclinic orbit; the points (0, y u ) and (0, y s ) are the intersections between Γ and Σ; the point p is a center of the system ( ẋ, ẏ) T = M − (x, y) T + u − ; the point s is a saddle of the system ( ẋ, ẏ) T = M + (x, y) T + u + ; and the origin 0 is a fold-fold point of Σ wich is a center of the system (1) with its period annulus ending in Γ.
with m 2 1 + m 2 m 3 < 0. By translating the system through the straight line {x = d}, which is a Σ-preserving change of variables, we can assume that e = m 1 d/m 2 .
Doing the Σ-preserving change of variables and rescaling the time by τ = −m 2 1 − m 2 m 3 t, system (1) can be written as where 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 Hypothesis (Ha1), s = (c, −c) is a saddle for the system (x ′ , ỹ′ ) T = A + (x − c, ỹ + c) T .So A + has two distincts real non-zero eigenvalues λ 1 , λ 2 such that λ 1 λ 2 < 0. It is easy to see that this last condition is equivalent to ab > āb . (5) As usual, W u (s) and W s (s) represent respectively the unstable and stable sets of the saddle s.We denote y u,s = Σ ∩ W u,s (s).From Hypothesis (Ha3), {y u , y s } ⊂ W (s) ∩ W s (s).Let Γ = W u (s) W s (s) containing y u and y s .Hence Γ ia a homoclinic loop.
The center at (−d, 0) of the system (x ′ , ỹ′ ) T = A − (x + d, y) T induces a symmetry on (4), namely: the solution of (4) starting in Σ for y > 0 has to return to Σ in −y for t < 0, so the same occurs for t > 0 and for every y between y u and y 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 symmetry y u = −y s , which is equivalent to b = (cā − 2ac)/c.( 6) Now, the origin (0, 0) is a singularity for the system (4), because it is a point of tangency.From Hypothesis (Ha3) the origin (0, 0) must be a fold-fold point.Moreover, every orbit distinct to (0, 0), inside the region delimited by 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 only if c = āc/a and a > 0. From (6) we conclude that b = −ā Computing the solution of (4) we conclude, from the above symmetry, 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 only if ā = b = c = 0 and a, b, c > 0. So we have conclude the proof.
Proof.[Proof of Proposition 2] By translating the system through the straight line Σ, which is a Σpreserving change of variables, we can assume that p = (0, 0).From Hypothesis (Hb2), (0, 0) is a center for the system ( ẋ, ẏ) and rescaling the time by τ = −m 2 1 − m 2 m 3 t, system (1) can be written as where with a, b, c, ā, b, c ∈ R, and c > 0.
From here, the proof follows analogously to the proof of Proposition 1.
Proof.[Proof of Proposition 3] From the Hypothesis (Hc2), p = (−d, e), with d < 0 is a center for the system ( ẋ, ẏ) with m 2 1 + m 2 m 3 < 0. By translating the system through the straight line {x = −d}, which is Σ-preserving change of variables, we can assume that e = m 1 d/m 2 .Doing the Σ-preserving change of variables and rescaling the time by τ = −m 2 1 − m 2 m 3 t, system (1) can be written as where From here the proof follows in a similar way to the proof of Proposition 1.

Proofs of Theorems A, B, C and Corollary 1
The idea of the proofs of Theorems A, B, and C is to compute the Taylor expansion at ε = 0 up to order 1 in ε of the Poincaré map associated to (3) and then apply the Implicit Function Theorem.
We say that an ordered set of complex-valued functions F = (f 0 , f 1 , . . ., f n ) defined on I is an Extended Chebyshev system or ET-system on I if and only if any nontrivial linear combination of the functions of F has at most n zeros counting multiplicities.We say that F is an Extended Complete Chebyshev system or an ECT-system on I if and only if for any k, 0 ≤ k ≤ n, (f 0 , f 1 , . . ., f k ) is an ET-system.For more details, see the book of Karlin and Studden [Karlin & Studden, 1966].
In order to prove that F is a ECT-system on I is sufficient and necessary 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. .
October 8, 2014 8:48 LC*two*zones*IJBC˙revised Limit cycles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones 9 Now consider the functions g 2 3 (y) = y, and We define the sets of functions G i = {g 1 , g 2 , g i 3 } for i = 1, 2, 3.In the proofs of Theorems A, B, and C we shall use the following lemma.Proof.To prove statement (a) we compute the power series expansion in y up to order 2 of the functions g i for i = 1, 2 and g 1 3 .So ), and Let C be the square matrix formed by the coefficients of y i for i = 0, 1, 2 of (10), namely = 0, we have that G 1 is a set of functions linearly independent for y > 0 near 0, which implies that G 1 is a set of functions linearly independent in (0, √ b c/ √ a) for every a, b, c, d > 0. This concludes the proof of statement (a).
Hence the lemma is proved.
Proof.[Proof of Theorem A] The solutions of the discontinuous piecewise linear differential systems ( 2) and ( 3) can be easily computed, because they are piecewise linear differential systems.
From here, applying Lemma 1(b), the definition of the ECT-systems, and the Implicit Function Theorem the proof follows.
Proof.[Proof of Theorem C] Here t + (y, ε) is given by ( 13) and Here t + (y, ε) is given by (13) and This proof follows completely analogous to the proof of Theorem A by applying Lemma 1(c), Proposition 4 and the Implicit Function Theorem for the function f 3 (y) = k 1 g 1 (y) + k 2 g 2 (y) + k 3 g 3 3 (y), where the functions g i for i = 1, 2 and g 3 3 the ones defined in (9).Here, the coefficients k m for m = 1, 2, 3 depend linearly on the parameters b ± i for i = 1, 2, 3, 4 and v ± j for j = 1, 2.

Fig. 1 .
Fig.1.Normal form of system (1) assuming the Hypotheses set (Ha).Here Γ is the homoclinic orbit; the points (0, y u ) and (0, y s ) are the intersections between Γ and Σ; the point p is both a center of the system ( ẋ, ẏ) T = M − (x, y) T + u − and a center of the system (1) with its period annulus ending in Γ; the point s is a saddle of the system ( ẋ, ẏ) T = M + (x, y) T + u + ; and the origin 0 is a fold-fold point of Σ.

Fig. 2 .
Fig.2.Normal form of system (1) assuming the Hypotheses set (Hb).Here Γ is the homoclinic orbit; the points (0, y u ) and (0, y s ) are the intersections between Γ and Σ; the point p = 0 is both a center of the system ( ẋ, ẏ) T = M − (x, y) T + u − and a center of the system (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 system ( ẋ, ẏ) T = M + (x, y) T + u + .

Fig. 3 .
Fig.3.Normal form of system (1) assuming the Hypotheses set (Hc).Here Γ is the homoclinic orbit; the points (0, y u ) and (0, y s ) are the intersections between Γ and Σ; the point p is a center of the system ( ẋ, ẏ) T = M − (x, y) T + u − ; the point s is a saddle of the system ( ẋ, ẏ) T = M + (x, y) T + u + ; and the origin 0 is a fold-fold point of Σ wich is a center of the system (1) with its period annulus ending in Γ.

Lemma 1 .
Assume that a > 0, b > 0, c > 0. (a) If d > 0 then G 1 is a set of functions linearly independent on the interval (0, √ b c/ √ a).(b) If d = 0 then G 2 is a ECT-system on the interval (0, √ b c/ √ a).(c) If d < 0 then G 3 is a set of functions linearly independent on the interval (0, √ b c/ √ a).