On the maximum number of limit cycles for a piecewise smooth differential system

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Abstract

In this paper, we consider an upper bound for the maximum number of limit cycles bifurcating from the periodic solutions of x¨+x=0 under piecewise smooth perturbation. By using the first order average method, we prove that the maximum number is at most n and n can be reached.

Introduction

In recent years, piecewise smooth differential systems, as an interesting class of nonlinear differential systems, have been studied extensively. Due to the rich dynamics, they appear in a wide range of natural science and play a vital role, for instance, in control systems [1], Biology [18], economy [17], mechanical systems with impact [3], nonlinear oscillations [22] and neuroscience [9], [15], [23]. Therefore, many researchers have always been interested in trying to understand their dynamic richness, especially the study of the bifurcation of limit cycles.

There have been some tools and theories for studying the dynamics of piecewise smooth differential systems. One of the main tools for studying the bifurcation of limit cycles is the averaging theory. Averaging theory has a long history, it began with classical works of Lagrange and Laplace, who provided an intuitive proof of the process. The first formalization of that for smooth differential systems was provided by Fatou [10] in 1928. Its important practical was made by Bogoliubov and Krylov [5] in 1930s, and by Bogoliubov [4] in 1945. For smooth differential systems the theory has been greatly developed, and plenty of excellent results have been obtained, see [6], [7], [10], [11], [12], [20] and the references therein. It has been developed for studying the bifurcation of limit cycles of the piecewise smooth differential systems in recent years. Llibre et al. [21] developed a variation of the classical averaging theory, which can be used to detect the limit cycles of some piecewise continuous dynamic systems. Llibre et al. [19] further extended the averaging theory to the first and second order for analyzing periodic solutions of any finite dimensional piecewise smooth differential systems using techniques of regularization. Itikawa et al. further extended the averaging theory to arbitrary order for studying the periodic solutions of piecewise smooth differential systems in [16]. These results can be applied to obtain a lower bound for the maximum number of limit cycles for some piecewise smooth differential systems by the first order average function. Han [13] studied a class of piecewise smooth periodic differential systems and obtained a sufficient condition for the maximum number of periodic solutions. Then Han et al. [14] developed the average method for estimating the maximum number of periodic solutions of multi-dimensional periodic systems.

Recently, Chen and Llibre [8] studied the number of limit cycles of the following systemx˙=y,y˙=x+ε(1+cosmθ)Q(x,y), where θ=arctan(y/x), Q(x,y) is a polynomial of degree n and m0 a non-negative integer. They obtained the following results by using the first order averaging method.

(a) If m is odd, the maximum number of limit cycles of Eq. (1.1) bifurcating from the periodic solutions of the linear center x¨+x=0 is at most n1.

(b) If m is even, the maximum number of limit cycles of Eq. (1.1) bifurcating from the periodic solutions of the linear center x¨+x=0 is at most [n12].

(c) These upper bounds can be reached.

Observe that (1.1) is in fact a piecewise smooth when m is odd. Inspired by the above discussions, in this paper we will consider the number of limit cycles of a piecewise smooth differential system as followsx˙=y,y˙=x+ε(1+cosmθ)k=12XSk(x,y)Qk(x,y), where ε>0 is a small parameter, m is a non-negative integer, θ=arctan(y/x), Qk(x,y)=i+j=0naijkxiyj is a real polynomial of degree n, S1={(x,y)|y0}, S2={(x,y)|y0}. Here, XS denotes the characteristic function of a set SR2 which is defined byXS(x,y)={1,if(x,y)S,0,if(x,y)S. In this paper, we study the maximum number of limit cycles bifurcating from the periodic orbits of the linear system (1.2)|ε=0.

The structure of this paper is as follows. In Section 2, we present some known results of the averaging theory for piecewise smooth differential systems and our main result. Section 3 is dedicated to prove the main result.

Section snippets

Preliminaries and the main result

At first, we present preliminaries about the averaging theory for calculating periodic solutions of piecewise smooth vector fields which are used to prove our main result. Consider the periodic equationdxdt=εF(t,x,ε,δ),tR,xJ. We give our assumptions as follows.

(H1) There exist an open interval J, a positive constant T and k1 Cr functions h1(x), …, hk1(x) defined on J, satisfying0<h1(x)<<hk1(x)<T,xJ,k2,r1. (H2) Set h0(x)=0 and hk(x)=T. Introduce k regions as followsDj={(t,x)|hj1(x)t<hj

Proof of the main result

In the calculation of the average function, we need to use the following formulas (3.1) and (3.4), for more details of these integrals one can see pages 152-153 in [24]. The first formula iscospxsin2nxdx=cosp+1x2n+p[sin2n1x+k=1n1(2n1)(2n3)(2n2k+1)sin2n2k1x(2n+p2)(2n+p4)(2n+p2k)]+(2n1)!!(2n+p)(2n+p2)(p+2)cospxdx. This formula is applicable for any real p and any positive integer n, except for the following negative even integers p=2,4,...,2q (q is a positive integer).

Declaration of Competing Interest

There is no competing interest.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 11771296 and 11931016).

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