Lyapunov’s Exponents for Non-Smooth Dynamical Systems: Behavior Identification Across Stability Boundaries With Bifurcations

In this paper some recently developed techniques to evaluate both analytically and numerically stability features of a non-smooth dynamical system, are used to investigate in detail the stability boundaries of regions corresponding to given stable periodic responses. The problem analyzed is that of a rocking block simply supported on a harmonically moving rigid ground; in this case, if the block is assumed to be a rigid body, the strong discontinuities characterizing the dynamic evolution are due to the impacts occurring each time the block crosses the initial equilibrium configuration. Therefore some special tools, specific for non-smooth functions, must be introduced to perform the stability analysis. In the present study, the theory due to Mu¨ller [19] is used to handle the evaluation of Lyapunov’s exponents upon discontinuities, by introducing the treatment, both analytical and numerical, of “saltation matrices”. Such a general theoretical method on one hand has been adapted to the numerical algorithms needed for the solution of the complete, non-linearized, problem and on the other hand, it allowed the development of the closed-form analytical reference solutions, obtained by linearizing assumptions less restrictive than those used by Hogan [8, 9]. The approximated stability boundaries obtained by the linearized closed-form solutions have been the starting point to guide the choice of the system parameters values to locate the responses in regions where bifurcations can arise. Inside these ranges several examples can be presented to illustrate the trends exhibited by the numerically evaluated Lyapunov’s exponents when the values of the forcing amplitude increase over the stability boundary of the symmetric responses while the value of the forcing frequency is fixed. Among these, investigations on sequences of responses composing period doubling cascades toward chaos, can provide a good and interesting test to appreciate the indications offered by the numerically derived Lyapunov’s exponents.