The increasingly complex structure of the bifurcation tree of a piecewise-smooth system

Abstract

A new approach to the study of the dynamics of a piecewise-smooth system is proposed, which uses the a priori known possible bifurcation structures of the parameter space. In Section 1 the synthesis of the structures of the bifurcation tree of the system is considered, namely, the local structures, bifurcation bands, sources and nodes. It is shown that a node corresponding to a doubling bifurcation with reorientation of the domain of existence can generate a sequence of increasingly complex structures. Then the increasing number of unstable orbits serves as one of the mechanisms giving rise to the chaotic behaviour of the dynamical system. In Section 2 the procedure for synthesizing the structures of the bifurcation tree of a piecewise-smooth system proposed in the first part of the paper is applied to the problem of the forced vibrations of a linear oscillator with impacts against a stopping device. Period-doubling cascades are discovered, which are accompanied by the reorientation of the domain of existence of a solution relative to some bifurcation surface, namely, the trunk of the tree. A set of frequency intervals is distinguished on the bifurcation trunk, each containing an infinite sequence of increasingly complex local structures appearing and disappearing at the nodes. This specific mechanism, giving rise to the chaotic motion of the oscillator, is realized in neighbourhoods of the limiting nodal bifurcation points.