Relaxation and mixed mode oscillations in a shape memory alloy oscillator driven by parametric and external excitations

This paper performs analytical investigations on symmetric jump phenomena reflecting multi-timescale dynamics in a nonlinear shape memory alloy oscillator with parametric and external cosinoidal excitations by means of geometrical singular perturbation theory (GSPT). The conditions concerning the existence of homoclinic and heteroclinic chaos subjected to periodic perturbations are studied in mathematical models by applying Melnikov method. Then we treat the excitation item as the slow variable acting on the bifurcation structure of the fast subsystem and present explicit algebraic expressions of the critical manifold of this subsystem for the investigation of fast-slow motions in the whole system. The study reveals the dynamical features of the relaxation oscillations formed by the alternative stable states on the different branches. In addition, the range of parametric excitation amplitude and stiffness values of the system plays an important role in the singularities and shaping the oscillation patterns. We determine numerically how the parameter values affect the manifold of the fast subsystem during the active fast transitions. Using the fast-slow analysis, one can provide the correct analytical predictions of the region of parameter space where the periodic orbits that alternate between epochs of fast and slow motions occur. Numerical simulations are also carried out to illustrate the validity of our study.