Estimating Lyapunov spectrum on shape-memory alloy oscillators considering cloned dynamics and tangent map methods

Abstract

Mechanical devices built with shape-memory alloys become popular due to thermomechanical coupling associated with solid-state phase transformations. The main characteristics stemming from these transformations are the intrinsic dissipation and the change of thermomechanical properties as a consequence of the phase transformations. Shape-memory alloy oscillators have a nonlinear behavior that can reach responses with different natures such as periodic, quasiperiodic, chaotic, or even hyperchaotic. A proper identification of these behaviors requires the use of dynamical tools, and among them, Lyapunov exponents are one of the most relevant. This work deals with the calculation of the Lyapunov spectrum of shape-memory alloy oscillators. One- and two-degree-of-freedom systems are of concern. Two different approaches are compared: the classical algorithm that uses a tangent space approach; and the cloned dynamics algorithm that is a Jacobian-free approach. Results show that both methods have similar results, allowing the use of cloned dynamics as an interesting alternative procedure since it avoids Jacobian calculations.