On the Number of Limit Cycles for Discontinuous Piecewise Linear Differential Systems in R with Two Zones

We study the number of limit cycles of the discontinuous piecewise linear differential systems in R 2n with two zones separated by a hyperplane. Our main result shows that at most (8n − 6) n−1 limit cycles can bifurcate up to first-order expansion of the displacement function with respect to a small parameter. For proving this result, we use the averaging theory in a form where the differentiability of the system is not necessary.


Introduction and statement of the main result
For a given differential system a limit cycle is a periodic orbit isolated in the set of all its periodic orbits.Inside the qualitative theory of differential systems the study of their limit cycles is one of the main topics.
Many questions are considered on the limit cycles of the differential systems in R 2 .Thus, one of the main lines of research for such systems is the study of how many limit cycles emerge from the periodic orbits of a centre when we perturb it inside a given class of differential equations, see for example the book [6] and the references there in.More precisely, the problem of considering the planar linear differential centre ẋ = −y, ẏ = x and perturbing it ẋ = −y + εf (x, y), ẏ = x + εg(x, y), inside a given class of differential equations for studying the limit cycles which bifurcate from the periodic orbits of the linear centre.Of course, ε is a small parameter.Here our main concerning is to bring this problem to higher dimension when the perturbation is discontinuous and piecewise linear.
In [16] Lum and Chua conjectured that a continuous piecewise linear differential system in the plane with two zones separated by an straight line has at most one limit cycle.This conjecture was proved by Freire, Ponce, Rodrigo and Torres [9].The number of limit cycles for continuous piecewise linear differential system with three zones separated by two parallel hyperplanes in R 2n has been studied in [5] by Cardin, De Carvalho and Llibre.
The purpose of this paper is to study the problem of Lum and Chua extended to discontinuous piecewise linear differential systems in R 2n with two zones separated by a hyperplane.More precisely, we want to study the number of limit cycles of the discontinuous piecewise linear vector fields with two zones in R 2n separated by a hyperplane.This problem in R 2 has been studied by several authors, see for instance [10,12,13,14,15].The same problem in R 4 has been studied in [4].
Here we shall study this problem in R 2n for n ≥ 3.
The study of continuous and discontinuous piecewise linear differential systems goes back to Andronov and coworkers [1], and nowadays they are also studied by many researchers.Moreover, these systems can exhibit complicated dynamical phenomena such as those exhibited by general nonlinear differential systems.Thus, these last years a big interest from the mathematical community takes place trying to understand their dynamical richness, because such systems are widely used to model many real processes and different modern devices, see for more details the book [8] and the references therein.More recently, these systems become also important as idealized models of cell activity, see [7,18,19].
We shall study the number of limit cycles of the discontinuous piecewise linear differential systems in R 2n of the form where ε ̸ = 0 is a small parameter, Our main result is the following generic upper bound for the maximum number of limit cycles of the discontinuous piecewise differential linear systems (1) in R 2n , which can bifurcate up to first-order expansion of the displacement function, from the periodic orbits of the 2n-dimensional center (1) with ε = 0.Here by "generic" we mean that the first-order expansion of the displacement function of (1) is not identically zero.
We recall the basic results from averaging theory that we shall need for proving Theorem 1 in section 2, and we shall prove Theorem 1 in section 3.
We note that the bound obtained in Theorem 1 for the discontinuous piecewise differential linear systems (1) in R 2n with two zones separated by a hyperplane, coincides with the bound obtained in [5] for the continuous piecewise differential linear systems in R 2n with three pieces separated by two parallel hyperplanes.

Averaging method
The averaging method is a powerful tool in the qualitative theory of differential equations.We present below the first-order averaging method as obtained in [3].For a systematic treatment of the averaging theory, see [17].
Consider the following differential system where For each z ∈ D, we denote by x(•, z, ε) : [0, t z ) → R m the solution of (3) with x(0, z, ε) = z.We assume that t z > T for all z ∈ D. Define the displacement map g : Then, the zeros of the displacement map g correspond to periodic solutions of the system (3), while isolated zeros of g correspond to limit cycles of (3).
The first-order averaging method for studying the periodic orbits of the differential system (3) is stated in the next result.Assume that for a ∈ D with h(a) = 0, there exists a neighborhood V of a such that h(z) ̸ = 0 for all z ∈ V \{a}, and the Brouwder degree d B (h, V, a) ̸ = 0.Then, for |ε| > 0 sufficiently small, there exists an isolated T -periodic solution ψ(•, ε) of system (3) such that ψ(0, ε) → a as ε → 0.
In this paper, we want to use the averaging method to study discontinuous differential systems.Let (6) ẋ = G 0 (t, x, ε) = εH 0 (t, x) + ε 2 R 0 (t, x, ε) be the differential system (3) satisfying the same assumptions with the exception that G 0 is discontinuous in the second variable at x = 0. Clearly, our discontinuous piecewise differential linear systems (1) satisfy the assumptions of the differential systems (6).Moreover, the displacement function of systems (1) are smooth.We have the following result.
Proposition 3. Assume that the displacement map g 0 associated to system (6) is smooth.If there exists a family of T -periodic smooth maps G δ (t, x, ε) such that lim δ→0 G δ (t, x, ε) = G 0 (t, x, ε), then we can apply the first-order averaging method directly to system (6).
Proof.Denote by g δ the displacement map associated to Then g δ is smooth and lim δ→0 g δ = g 0 .Write Then since g 0 and g δ are smooth, we have ).Therefore, we have lim δ→0 h δ = h 0 .
Since we can apply the first-order averaging method to systems (7), by passing to the limit, we see that it is equivalent to applying the first-order averaging method directly to system (6).

On the number of limit cycles
For applying the averaging method of section 2 to our discontinuous piecewise differential linear systems (1), we must write system (1) into the "normal" form of the averaging method, i.e. into the form (3), for doing this we need the following lemma of [5].
Proof.Under the change of variables ( 9), we have The lemma then follows from a straightforward computation.
Note that the displacement map for system (10) is smooth because the Poincaré map is a composition of two smooth maps.It is also easy to find smooth functions approximating the function φ in (2).Therefore, by Proposition 3, we can apply the averaging method directly to the discontinuous system (10).Lemma 6.We have I 2 (r) = 0 for all r > 0 and Proof.For r < a, we have φ(r cos θ) ≡ 0. Thus assume r ≥ a, and let θ * be such that cos θ * = a/r.Then The lemma then follows from a straightforward computation.
Lemma 7. We have continuous functions, T -periodic in the first variable, locally Lipschitz in the second variable and D is an open subset of R m .