This article is cited in 1 paper

Nonlinear Systems

On border-collision bifurcations in a pulse system

Zh. T. Zhusubaliev^a, D. V. Titov^a, O. O. Yanochkina^a, U. A. Sopuev<sup>b

a</sup> Southwest State University, Kursk
^b Osh State University

Abstract: Considering a piecewise smooth map describing the behavior of a pulse-modulated control system, we discuss border-collision related phenomena. We show that in the parameter space which corresponds to the domain of oscillatory mode a mapping is piecewise linear continuous. It is well known that in piecewise linear maps, classical bifurcations, for example, period doubling, tangent, fold bifurcations become degenerate (“degenerate bifurcations”), combining the properties of both smooth and border-collision bifurcations. We found unusual properties of this map, that consist in the fact that border-collision bifurcations of codimension one, including degenerate ones, occur when a pair of points of a periodic orbit simultaneously collides with two switching manifolds. This paper also discuss bifurcations of chaotic attractors such as merging and expansion (“interior”) crises, associated with homoclinic bifurcations of unstable periodic orbits.

Keywords: pulse system, differential equations with a discontinuous right-hand side, bimodal piecewise linear mapping, border-collision bifurcations, degenerate bifurcations, merging and expansion bifurcations, chaotic oscillations.

Presented by the member of Editorial Board: A. L. Fradkov

Received: 26.09.2022
Revised: 07.12.2023
Accepted: 30.12.2023

DOI: 10.31857/S0005231024020027

 English version: Automation and Remote Control, 2024, 85:2, 103–122