ON NONSMOOTH PERTURBATIONS OF NONDEGENERATE PLANAR CENTERS

We provide sufficient conditions for the existence of limit cycles of nonsmooth perturbed planar centers when the discontinuity set is a union of regular curves. We introduce a mechanism which allows us to deal with such systems. The main tool used in this paper is the averaging method. Some applications are explained with careful details. 2010 Mathematics Subject Classification: 34A36, 34C29, 37G15, 37C27, 37C30.


Introduction
The theory of discontinuous systems has been developing at a very fast pace in recent years and it has become certainly an important common frontier between Mathematics, Physics, Engineering and other fields of science.The study of this kind of systems is motivated by various applications.For instance, we may cite some problems in control theory [3], nonlinear oscillations [1,20], nonsmooth mechanics [6], economics [11,15], biology [4], and others.
On the other hand, the knowledge of the existence or not of periodic solutions is very important for understanding the dynamics of differential systems.One of the useful tools to detect such objects is the averaging theory, which is a classical and mature tool that provides techniques to study the behavior of nonlinear smooth dynamical systems.We refer to the books of Sanders and Verhulst [21] and Verhulst [22] for a general introduction about this subject.
In [7], Buicȃ and Llibre generalized the averaging theory for studying periodic solutions of continuous differential systems mainly using the Brouwer degree.More recently in [18], Llibre, Novaes, and Teixeira extended the averaging theory for studying periodic solutions of a class of piecewise continuous differential systems with two zones.
In what follows, we introduce the class of piecewise continuous systems with two zones.
Let D be an open subset of R n .Let X, Y : D → R n be two continuous vector fields and let h : D → R be a C 1 function.The discontinuity set h −1 (0) is denoted by M .So we define a piecewise continuous differential system with two zones as (1) x which we denote concisely by Z = (X, Y ) h .It is worth to say that this definition can be easily extended to non-autonomous systems.
Let the sign function be defined in R \ {0} as The piecewise continuous differential system (1) can be conveniently written as (2) x ′ (t) = Z(x) = Z 1 (x) + sign(h(x))Z 2 (x), where In [18], conditions for the existence of periodic solutions when the discontinuity set M is a regular manifold are exhibited.However, many applications deal with discontinuous systems having the discontinuity set as an algebraic variety, see for instance the book of Andronov, Vitt, and Khaikin [1] and the book of Barbashin [3].In fact, some problems contained in [3] were the main source of motivation of the present work.
In few words, our main result deals with discontinuous perturbation of nondegenerate planar centers.The discontinuity set M is supposed to be a union of regular curves, which includes, particularly, the case when M is an algebraic variety.Moreover, conditions for the existence of periodic solutions of such perturbed systems are presented, via averaging theory.
We also provide two applications with careful details.The first one generalizes the problem of an m-piecewise discontinuous Liénard polynomial differential equation of degree n proposed by Llibre and Teixeira [19]; the second application deals with a plane divided in a mesh, where each piece admits one of the two vector fields.For these systems, the existence of periodic solutions is studied.
The paper is organized as follows.In Section 2 the main result is stated.In Section 3 we present some useful elements of the averaging theory and Brouwer degree theory.In Section 4 we prove the results presented in Section 2, and in Section 5 applications of the results are discussed.

Statement of the main result
Let D be an open subset of R 2 .We consider the following planar discontinuous differential system with where X, Y, F i,j : D → R 2 for i, j = 1, 2 are continuous functions being F i,j for i, j = 1, 2 locally Lipschitz, and h : R 2 → R is a a C 1 function.Furthermore, we shall consider that the origin of the unperturbed system (3) (ε = 0) is a nondegenerate global center in D.
Usually, 0 is assumed to be a regular value of the function h which implies that M = h −1 (0) is a regular manifold, see for instance Theorem 3 of this paper.Here, we assume that (H1) The set of nonregular points in M = h −1 (0) is bounded.In other words, for is the open ball with radius δ centered at (0, 0).
We denote M = M \B δ (0, 0), which is an embedded submanifold in D ⊂ R 2 .In addition for the set M we assume that (H2) ∇h(x, y), (−y, x) = 0 for all (x, y) ∈ M. Remark 1. Geometrically, Hypothesis (H2) is equivalent to (−y, x) / ∈ T (x,y) M because T (x,y) M is the kernel of the operation inner product by ∇h(x, y).
The idea of the proof of our main result (see Theorem A) consists in defining a convenient change of variables which drives some restrictions of system (3) to a system whose discontinuity set is a regular manifold.To do this we define the function Ψ δ : where δ > 0 is chosen in (H1).Clearly, this function is a diffeomorphism onto its image.Furthermore, B δ (0, 0) ∩ Ψ δ (S 1 × R + ) = ∅.Now, let ρ > 0 be a real number such that Ψ δ (S 1 × (0, ρ)) ⊂ D, and denote D = S 1 × (0, ρ).
For simplicity, given a function Example 1.To illustrate Hypothesis (H1) and the change of variables (4), we consider the function h(x, y) = (x 2 − 1)(y 2 − 1).The set M = h −1 (0) is represented by the bold lines in Figure 1.Observe that M is not a regular manifold since it has self-intersections at the points N = {(1, 1), (−1, 1), (1, −1), (−1, −1)} ⊂ B 1 (0, 0).So, choosing δ = √ 2 and proceeding with the change of variables defined above, the set M = (δ * h) −1 (0), represented by the bold lines in Figure 2, becomes a regular submanifold of D (we shall use this example in Application 2 of Section 5).This procedure of finding a conveniently change of variables to remove undesirable regions can be replied for other systems, even in higher dimension.For the functions X and Y from (3) we assume that (H3) For each (θ, r) ∈ D the following relations hold: S(θ, r) = cos(θ)δ * X(θ, r) + sin(θ)δ * Y (θ, r) = 0, and The next proposition gives a class of nondegenerate planar centers for which Condition (H3) is verified, assuring then its non-emptiness.
with (n, j) ∈ N 2 and n + j ≤ µ−1 2 ; and b m,i = 0 otherwise.Then, (H3) holds for the functions X and Y .Remark 2. The Hypothesis (H3) implies the existence of ρ ′ > 0 such that the restriction of the unperturbed system (3) (i.e.ε = 0) to the ball B ρ ′ (0, 0) is conjugated to the linear center.In other words, the unperturbed system (3) is locally conjugated to the linear center at the origin.From our assumptions, it follows that D ⊂ B ρ ′ (0, 0).Now, we define the averaged function f : (0, ρ − δ) → R as The function ( 5) is a suitable modification via the change of variables defined in (4) for system (3), of the averaged function (12) of Theorem 2 (see Section 3).In Example 2 we can see how useful is this function.
In what follows we state a hypothesis for the function f .It uses the concept of Brouwer degree d B which is defined in Section 3. (H4) For some a ∈ (δ, ρ) with f (a − δ) = 0, there exists a neighborhood Remark 3. When f (defined in ( 5)) is a C 1 function, Hypothesis (H4) becomes: (H4') For some a ∈ (δ, ρ) with f (a − δ) = 0 we have f ′ (a − δ) = 0.
Our main result, which provides conditions for the existence of periodic solutions of the nonsmooth perturbed system (3), is the following.
Theorem A. If (H1)-(H4) hold, then for |ε| > 0 sufficiently small there exists a periodic solution (x(t, ε), y(t, ε) The following corollary deals with the perturbations of the linear planar center.

Basic results on averaging theory and Brouwer degree theory
In Theorem A the function d B (f, V, 0) denotes the Brouwer degree, which is uniquely determined by the conditions of the next theorem (for a proof see [3]).
Theorem 2. Let P = R n = Q for a given positive integer n.For bounded open subsets V of P , consider continuous mappings f : V → Q, and points y 0 in Q such that y 0 does not lie in f (∂V ) where ∂V denotes the boundary of V .Then to each such triple (f, V, y 0 ), there corresponds an integer d B (f, V, y 0 ) having the following three properties: (1) If d B (f, V, y 0 ) = 0, then y 0 ∈ f (V ).If f 0 is the identity map of P onto Q, then for every bounded open set V and y 0 ∈ V , we have and consider a continuous homotopy In [18] the methods of averaging theory for studying crossing periodic solutions were extended to a class of discontinuous differential systems.It has been established the following result: Theorem 3. We consider the following discontinuous differential system (11) x where We also suppose that h is a C 1 function having 0 as a regular value.We denote M = h −1 (0).
The averaged function f : D → R n is defined as We also assume that the following conditions hold: (ii) ∂h/∂t = 0, for all p ∈ M ; (iii) for some a ∈ D with f (a) = 0, there exists a neighbourhood V of a such that f (z) = 0 for all z ∈ V \{a} and d B (f, V, 0) = 0.
To prove Claim A.1 we must show that for some decomposition (18) the involved functions are continuous, 2π-periodic in the variable θ and locally Lipschitz with respect to r.
We note that where Applying the Binomial Formula, expression (19) becomes Again, by applying the Binomial Formula to (δ * F i ) a with i = 1, 2 and a ∈ N, we obtain where, as usual, ⌊u⌋ denotes the greatest integer less than or equal to u and ⌈u⌉ denotes the smallest integer greater than or equal to u.

Since
and Now, it is easy to see that the functions R 1 and R 2 are continuous, 2π-periodic in the variable θ and locally Lipschitz with respect to r.So Claim A.1 is verified.
Rewriting system ( 17) by making explicit the sign function, we obtain where and The functions G 1 and G 2 are also continuous, 2π-periodic in the variable θ and locally Lipschitz with respect to r.

Applications
where The zero set of the function sgn(g m (x, y)) with m = 2, 4, 6, . . . is the product of m/2 straight lines passing through the origin of coordinates dividing the plane in sectors of angle 2π/m.
Here, we shall consider a generalization of this problem.Let T m denote the m-Torus.
Thus the discontinuity set M = h α −1 (0) is represented, partially, by the bold lines in Figure 3.We stress that only the behavior of the set M outside the ball B δ (0, 0) is considered, because the part of the discontinuity set M contained in B δ (0, 0) is not important for our arguments.A system of the form will be called an α-piecewise discontinuous Liénard polynomial differential system of degree n.Also, H(m, n) will denote the maximum number of limit cycles that system (23) can have for any α = (α 1 , α 2 , . . ., α m ) ∈ T m , with m = 2, 4, 6, . . ., such that 0 We shall use the theory developed in Section 2 to obtain estimates of H(m, n).
Clearly, Theorem 4 is valid for m = 2 provided that Proposition 6 holds.
To prove Propositions 5 and 6 and Theorem 4 we need a technical lemma about the number of zeros of a real polynomial.Lemma 7. Consider the real polynomial p(x) = a i1 x i1 + a i2 x i2 + • • • + a ir x ir with 0 ≤ i 1 < i 2 < • • • < i r and a ij = 0 real constants for j ∈ {1, 2, . . ., r}.Then p(x) has at most r − 1 positive real roots.Moreover, given δ > 0 it is always possible to choose the coefficients of p(x) in such a way that p(x) has exactly r − 1 distinct real roots greater than δ.
Proof: The proof of the lemma follows immediately by observing that the set of functions {x i1 , x i2 , . . ., x ir } is an Extended Complete Chebyshev System (or just ECT-system) on the interval (δ, ∞).For more details, see the book of Berezin and Shidkov [5], and the book of Karlin and Studden [16].
We start by proving Proposition 6 since it will be used to prove Proposition 5 and Theorem 4.

Proof of Proposition 6:
To prove that n is a lower bound for the maximum number of limit cycles of system (23) we shall find a polynomial function F n (x) of degree n such that the differential system (23) has n limit cycles.Thus, taking F n (x) = εP n (x), with In order to prove the proposition we have to identify in system (23) the elements of Corollary B. Thus, Computing the averaged function (6), for system (24), we have Proceeding by induction on l, we have that, for l ≥ 0, for p, q ∈ Z, where, as usual, n!! denotes the Double Factorial: Following Arfken and Weber [2], these are related to the regular factorial function by (26) (2n)!! = 2 n n! and (2n + 1)!! = (2n + 1)! 2 n n! .
Proof: First we have to identify the elements of Corollary B in system (28): Thus, (H4) holds.Clearly, (H1), (H2) and (H3) also hold in this case.Hence, the proposition follows from Corollary B.
In Figure 4 we can see a numeric approximation of the periodic solution given by Proposition 8.

Proposition 1 .
Consider the functions X(x, y) = µ m=1 f m (x, y) and Y (x, y) = µ m=1 g m (x, y), where f m (x, y) = m i=0 a m,i x m−i y i and g m (x, y) = m i=0 b m,i x m−i y i ; and assume the following conditions are satisfied: (a) a m,0 = b m,m = 0 and b m,i = −a m,i+1 for i = 0, 2, . . ., m − 1 and for m = 1, 2, . . ., µ; (b) b

Figure 4 .
Figure 4. Numerical simulation of the periodic solution of (28).The dashed lines indicate the solutions for ε = −1; −0.7; −0.4; −0.1; the non dashed bold line indicates the solution for ε = 0 which is a sphere centered at the origin (0, 0) with radius equal 4 + 2 √ 2; and the dashed bold line indicates the discontinuity set.