Periodic and Nonperiodic Responses of Shape-Memory Oscillators

Nonlinear responses of shape-memory oscillators are investigated systematically using a numerical procedure and a modified Ivshin-Pence model for the restoring force. Due to the discontinuities in the tangent stiffness, classical gradient-based shooting techniques for determining periodic responses are not applicable. Herein the implemented algorithm searches the periodic solutions as fixed points of the Poincar´e map. The Jacobian of the map is calculated via a central finite-difference scheme and its eigenvalues are computed to ascertain the stability of the solutions and the associated codimension-one bifurcations. A number of frequency-response curves are constructed for some meaningful shape-memory oscillators (characterized by different hysteresis loops) and for various excitation levels in the primary and superharmonic frequency ranges. The investigations are conducted both in isothermal and non-isothermal conditions. A rich class of solutions and bifurcations - including jump phenomena, pitchfork, period doubling, complete or incomplete bubble structures with a variety of nonperiodic responses - is found and discussed.