Generation of hysteresis cycles with two and four jumps in a shape memory oscillator

A numerical investigation of the generation of hysteresis cycles with two and four jumps in a shape memory oscillator is presented. The fast subsystem is examined using local stability and bifurcations. When the upper and lower attractors disappear, the fast subsystem can be stable or bi-stable. This leads to a classification of the bifurcation behaviors of the fast subsystem, i.e., the stable case and the bi-stable case, which are associated with different parameter conditions. For the stable case, the hysteresis cycles with two and four jumps are obtained by properly modulating the slow external forcing, and the associated dynamical mechanisms are analyzed. While for the bi-stable case, the transition mechanisms underlying the appearance of hysteresis cycles with two and four jumps are revealed by computing attraction domains of the attractors.