Unified Analysis of Complex Nonlinear Motions via Densities

Abstract

In this study analyses of deterministic and randomly perturbed complex nonlinear responses of nonlinear systems using densities are illustrated from an engineering perspective. Motivations to examine deterministic nonlinear dynamical systems via densities are first discussed. Pertinent mathematical backgrounds and techniques common to the analyses of both deterministic chaos and random chaotic processes are reviewed. Probability densities of nonlinear responses are computed by solving the Fokker-Planck equation (FPE) numerically to examine stochastic properties of random chaotic responses. It is shown that, by introducing random perturbations in the otherwise deterministic (periodic) excitation, the boundaries separating co-existing attractors associated with the corresponding deterministic system become blurred, and the existence of the attractors can be depicted by the evolution of a unique density in (physical) phase space. Transient and asymptotic behaviors of the densities reveal the relative stability of the various attractors. Mathematical theories useful for stability analysis of the attractors are discussed. Finally, asymptotically stable co-existing “periodic” and “chaotic” motions of a nonlinear dynamical system subjected to periodic excitation with random perturbations are demonstrated via an engineering example.