A method for calculating the spectrum of Lyapunov exponents by local maps in non-smooth impact-vibrating systems

Abstract

A general calculation method of calculating the spectrum of Lyapunov exponents is presented for n-dimensional nonlinear non-smooth systems by using the Poincaré map method. The Poincaré map is constructed by means of local maps to avoid calculating the Jacobian matrices at non-smooth points. The calculation of the spectrum of Lyapunov exponents for impact-vibrating systems with rigid constraints is given in detail. This method can be generally applied to systems with rigid or flexible (that is, perfectly elastic) constraints. In order to show the validity of this method, the spectrums of Lyapunov exponents are calculated in a large range of parameters for two given dynamical systems with rigid constraints and used to predict the dynamic behaviour.