Bifurcations and chaos in a shape memory alloy oscillator driven by parametric and external excitations

This paper investigates the nonlinear dynamic behavior of shape memory alloy (SMA) oscillators subjected to combined parametric and external excitations. We employ the fast–slow decomposition method to elucidate the mechanism of relaxation oscillations, yielding an analytical expression for the quiescent interval which approximates half of the excitation period. To address global instability induced by quintic nonlinearity, we rigorously derive exact analytical expressions for homoclinic and heteroclinic orbits within a fifth-order triple-well potential and establish explicit analytical chaos thresholds using the Melnikov method. Parametric analysis indicates that increasing the damping coefficient effectively suppresses chaotic motion, whereas higher amplitudes of parametric and external excitations promote global instability. Numerical simulations corroborate the analytical predictions and further identify boundary crises as primary routes to chaos. The study reveals underlying physical mechanisms governed by the interaction between energy accumulation and rapid release. These findings provide a theoretical basis for determining stable operation boundaries and engineering design constraints for SMA smart structures.