Crossing periodic orbits of nonsmooth Liénard systems and applications

Abstract

Continuing the investigation for the number of crossing periodic orbits of nonsmooth Liénard systems in (2008 Nonlinearity 21 2121-42) for the case of a unique equilibrium, in this paper we allow the considered system to have one or multiple equilibria. By constructing two control functions that are decreasing in a much narrower interval than the one used in the above work in the estimate of divergence integrals, we overcome the difficulty of comparing the heights of orbital arcs caused by the multiplicity of equilibria and give results about the existence and uniqueness of crossing periodic orbits, which hold not only for a unique equilibrium but also for multiple equilibria. Moreover, we find a sufficient condition for the existence of periodic annuli formed by crossing periodic orbits. Applying our results to planar piecewise linear systems with a line of discontinuity and without sliding sets, we prove the uniqueness of crossing limit cycles and hence give positive answers to conjectures 1 and 2 of Freire et al's work (2013 Planar Filippov Systems with Maximal Crossing Set and Piecewise Linear Focus Dynamics (Progress and Challenges in Dynamical Systems vol 54) (Heidelberg: Springer) pp 221-32).

Limit cycles in piecewise polynomial systems allowing a non-regular switching boundary

“…Among discontinuous differential systems, one of the most studied classes is the one formed by the discontinuous piecewise Liénard systems, which is widely used to model or analyze many real problems, as for instance the mechanical engineering with dry frictions [8], the integrate-and-fire neurons [32], the discontinuous control in the buck electronic converter [3,11], ... Of course, the study of the limit cycles for the discontinuous piecewise Liénard systems is also of fundamental importance and many researchers are devoted to the study of this subject, see the papers [5,7,19,23,27,30,31] for one switching boundary and [9,16,25,28] for multiple ones.…”

“…for i = 0, 1, ..., n and either i = 1 if i is odd, or i = 0 if i is even. For any given n there exists a choice of the parameters a i , b i , d i and e i such that the second order averaged function associated to (19) has exactly 2n − 1 simple positive zeros.…”

“…The procedure of choosing parameters in the proof of Proposition 8 relies on statements (a)-(d). As an example we give the explicit expression of F 2 (r) for system (19) with n = 5,…”