Bifurcation analysis on delay-induced bursting in a shape memory alloy oscillator with time delay feedback

Abstract

The mechanism for the action of time delay in a non-autonomous system with two time scales is investigated in this paper. The original mathematical model under consideration is a shape memory alloy oscillator with external forcing. The delayed system is obtained by adding both linear and nonlinear time delayed position feedbacks to the original system. Typical bursting patterns can be presented, including symmetric fold/supHopf, double-fold/supHopf and supHopf/supHopf bursting when periodic forcing changes slowly. The time delay is taken as a variable parameter to investigate its effect on the dynamics of the system such as the stability and bifurcation. We calculate the conditions of fold bifurcation and Hopf bifurcation as well as its stability with the aid of the normal form theory and center manifold theorem. Through bifurcation analysis, we can identify that the occurrence and evolution of bursting dynamic depends on the magnitude of the delay itself and the strength of time delayed coupling in the model. Furthermore, we use phase space analysis to explore the associated mechanisms for the oscillator with multiple coexisting attractors. Numerical simulations are also included to illustrate the validity of our study.