On Hopf bifurcation in non-smooth planar systems

Abstract

MSC: 34C05 34C07 37G15Keywords: Hopf bifurcation Non-smooth dynamical system Limit cycle Piecewise linear system As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincaré map with its analytical property and the problem of Hopf bifurcation have been studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles can appear near a focus of either FF, FP or PP type (Theorem 3.3).

Piecewise linear differential systems without equilibria produce limit cycles?

“…Up to now we know that there are discontinuous systems with at least three limit cycles, see for instance [2,4,3,6,8,9,10,11,12,13,14,15,22,17,18,19,20,22,24].…”

Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

“…The proof of Theorem B follows the same ideas of [14] for proving their Theorem 3.1. Indeed our proof could be obtained by applying their theorem to the restriction of system (9) to Ω.…”

The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems

“…In smooth systems there is a well known mechanism to search for the occurrence of limit cycles, the Hopf bifurcation theorem, see [13,19]. There are analogous results for piecewise smooth systems, for the case of continuous systems see for example [6,7,26,27], and for the case of discontinuous systems see [1,8,11,12,14,18]. In the discontinuous ones we can have more than one limit cycle, either all crossing cycles or including one sliding cycle, and in fact, the determination of the number of limit cycle has been the subject of several recent papers, see [2,3,4,10,15,16,17,20,22,23,24].…”