Spectral analysis of non-smooth dynamical systems

In this paper, we consider non-smooth dynamical systems driven by broad-banded excitations. Typical examples of. non-smooth characteristics of interest are backlash or bang-bang restorings in first order systems, respectively, piecewise linear dampings or dissipation by dry friction in oscillatory systems. The analysis applied is based on the invariant measures of systems, known by strong solutions of associated Fokker-Planck equations. The densities of these invariant measures are utilized to calculate adapted orthogonal polynomials, by which weak solutions of transient densities or time-dependent moments are expanded. Power spectra and correlations are calculated, correspondingly. Existence and uniqueness are proofed by discrete eigenvalue spectra of the expansion matrices. Evaluations of Monte-Carlo simulations show high agreements with calculated results.