Regularization of the Boundary-Saddle-Node Bifurcation

Xia Liu

doi:10.1155/2018/5094878

Abstract

In this paper we treat a particular class of planar Filippov systems which consist of two smooth systems that are separated by a discontinuity boundary. In such systems one vector field undergoes a saddle-node bifurcation while the other vector field is transversal to the boundary. The boundary-saddle-node (BSN) bifurcation occurs at a critical value when the saddle-node point is located on the discontinuity boundary. We derive a local topological normal form for the BSN bifurcation and study its local dynamics by applying the classical Filippov’s convex method and a novel regularization approach. In fact, by the regularization approach a given Filippov system is approximated by a piecewise-smooth continuous system. Moreover, the regularization process produces a singular perturbation problem where the original discontinuous set becomes a center manifold. Thus, the regularization enables us to make use of the established theories for continuous systems and slow-fast systems to study the local behavior around the BSN bifurcation.

Highlights

The study of Filippov system was motivated by its considerable applications in mechanical systems exhibiting dry friction [1,2,3,4], biological systems [5,6,7], and control systems [8,9,10]
The consequence of doing so is that the established discontinuous model focuses on the overall dynamics of the entire physical process while slightly ignoring the detailed dynamics in the transition stage
Our aim of this work is to describe the local dynamics in the neighborhood of the codimension-2 boundarysaddle-node (BSN) bifurcation where two equilibria of one smooth vector field go through a saddle-node bifurcation while they lie on the boundary

Summary

Introduction

The study of Filippov system was motivated by its considerable applications in mechanical systems exhibiting dry friction [1,2,3,4], biological systems [5,6,7], and control systems [8,9,10]. Filippov systems, or differential equations with discontinuous right-hand side, model physical processes which experience abrupt transitions between different modes. Our aim of this work is to describe the local dynamics in the neighborhood of the codimension-2 boundarysaddle-node (BSN) bifurcation where two equilibria of one smooth vector field go through a saddle-node bifurcation while they lie on the boundary. We treat this particular class of Filippov systems by applying both Filippov’s convex method and the regularization approach to the BSN point to obtain its local bifurcation diagram, to understand how all the codimension-1 bifurcations interact.

Preliminaries

The Boundary-Saddle-Node Bifurcation

Bifurcation Diagram and Phase Portraits

Regularization of the BSN Bifurcation

Conclusions