A class of discontinuous systems exhibit perturbed period doubling bifurcation

Abstract

This research considers discontinuous dynamical systems, which have related vector fields that shift over a discontinuity surface. These systems appear in a variety of applications, including ecology, medicine, neuroscience, and nonsmooth mechanics. The purpose of this paper is to develop a perturbation technique that measures the effect of a nonsmooth perturbation on the period doubling bifurcation of an unperturbed system. The unperturbed system is assumed to be close to a period doubling orbit, such that when the bifurcation parameter varies, the response changes from a period one to a period two limit cycle. The generalized determination of the Poincaré map associated with perturbed systems subjected to nonsmooth transitions is derived. The main techniques used in the proof of the results are normal forms and Melnikov functions, which are defined in two zones. Various examples are presented to show that non-smoothness is responsible for period doubling. To illustrate the interesting period doubling phenomenon that emerges from an existing flat periodic orbit via the non-smooth perturbation, a simple and novel discontinuous system is provided. An additional example is provided to show the emergence of a perturbed period doubling orbit near an unperturbed one.

Keywords

• nonsmooth perturbation
• melnikov function
• discontinuous systems
• perturbed period doubling
• stability