Discontinuous dynamics with grazing points

TL;DR

This paper analyzes the complex behavior of discontinuous dynamical systems with grazing points, focusing on stability, bifurcations, and solution properties, extending classical methods to these non-smooth systems.

Contribution

It provides a comprehensive analysis of grazing solutions, including stability, bifurcation, and solution continuation, with new methods for neighborhoods of grazing orbits.

Findings

Grazing cycles can undergo bifurcations, leading to complex dynamics.

Linearization techniques help analyze stability near grazing points.

The extended small parameter method reveals new bifurcation phenomena.

Abstract

Discontinuous dynamical systems with grazing solutions are discussed. The group property, continuation of solutions, continuity and smoothness of motions are thoroughly analyzed. A variational system around a grazing solution which depends on near solutions is constructed. Orbital stability of grazing cycles is examined by linearization. Small parameter method is extended for analysis of neighborhoods of grazing orbits, and grazing bifurcation of cycles is observed in an example. Linearization around an equilibrium grazing point is discussed. The mathematical background of the study relies on the theory of discontinuous dynamical systems [1]. Our approach is analogous to that one of the continuous dynamics analysis and results can be extended on functional differential, partial differential equations and others. Appropriate illustrations with grazing limit cycles and bifurcations are depicted to support the theoretical results.