Bifurcation analysis of piecewise-linear systems with hysteresis nonlinearity

Abstract

A perturbation-incremental (PI) method is presented for the computation, continuation, bifurcation analysis of periodic solutions and homoclinic and heteroclinic orbits of piecewise-linear systems and systems with hysteresis nonlinearity. We first study the computation of homoclinic and heteroclinic orbits of piecewise-linear system. From the continuation in a bifurcation parameter using the PI method, high codimension points can be obtained, which give useful information of how the dynamics of the system is organized. The Chua’s circuit is used as an illustration. Next, we study a two-degree-of-freedom aeroelastic system with hysteresis nonlinearity. Explicit form of a limit cycle with arbitrary parameter values can be obtained, which enables the phase portraits to be constructed. Periodic solutions can be calculated and their stabilities are determined by means of the Poincar´e map. As the parameter varies, bifurcations such as period-doubling, symmetric-breaking, saddle-node and Hopf-like bifurcations can be identified. In particular, we find two narrow intervals in which period-doubling sequence to chaos is observed. From the above investigation, the advantage of the PI method lies in its simplicity and ease of implementation. It is well known that chaos does not occur in a two-dimensional continuous autonomous system. For a two-dimensional piecewise-linear system with hysteresis nonlinearity, the flow at a point in the phase plane may not be unique. Therefore, a trajectory may traverse itself. We investigate the properties of such a system and obtain a chaos attractor using an one-dimensional map.