12 th Conference on Nonlinear Vibrations, Dynamics and Multibody Systems, Blacksburg VA USA, June 1-5- 2008 SYSTEMATIC NUMERICAL CHARACTERIZATION OF NON- REGULAR RESPONSES OF SHAPE-MEMORY OSCILLATORS Davide Bernardini Giuseppe Rega Dipartimento di Ingegneria Strutturale e Geotecnica Sapienza Università di Roma Italy davide.bernardini@uniroma1.it giuseppe.rega@uniroma1.it Abstract Previous studies on the nonlinear dynamics of thermomechanically based pseudoelastic oscillators showed the occurrence of chaotic responses in some ranges of the system parameters (Bernardini and Rega 2005, 2007). In order to understand whether nonregular responses only occur in isolated zones or are actually robust outcomes, the analyses need to be carried out through some synthetic measure of nonregularity that allows for systematic investigations in meaningful parameter spaces. Whereas the numerical characterization of chaos in smooth dynamical systems is often carried out via the computation of Lyapunov exponents, in the present case the computation of such exponents, following, for example, (Müller 1995), does not seem to be a convenient strategy. The attention has thus been focused on the simpler direct numerical tool represented by the Method of Wandering Trajectories (MWT) (Awrejcewicz et al. 2004). This method has been successfully applied in the literature to estimate regular and chaotic responses for non-smooth mechanical oscillators with up to two degrees of freedom (Awrejcewicz et al. 2005) and has been validated and calibrated in (Bernardini and Rega 2007). Based on a suitable, numerically oriented, implementation of the underlying thermomechanical model, the purpose of this paper is to present systematic results on the overall characterization of the chaotic response of shape-memory oscillators. The robustness of the chaotic response within the overall behavior of the system is investigated by computing behavior charts in which some control parameters are varied and the MWT is systematically applied to distinguish between regular and nonregular responses. A natural choice for the control parameters is the pair excitation frequency-amplitude at fixed initial conditions and material parameters. The analysis is carried out for a set of reference material parameters (RMP) corresponding to typical loops with medium level of hysteresis and for companion sets accounting for variations of model parameters having a direct mechanical interpretation in terms of pseudoelastic loop shape. In particular, the comparison between RMP and another set (MP1) corresponding to a loop with lower hysteresis provides information about the effect of the hysteresis on the chaotic response. The frequency-amplitude behavior chart for the basic set of parameters RMP is shown in Figure 1 (left) and highlights how chaos is quite a robust outcome even in a case of medium hysteresis level. Two clearly separated regions of non-regular motion are found, a compact one on the right, a more scattered one on the left. They are likely to correspond with the two kinds of chaotic motions highlighted in (Bernardini and Rega 2005) by bifurcation diagrams.