Analysis of Class of Nonlinear System under Deterministic and Stochastic Excitations

Abstract

In this paper, some analysis techniques of nonlinear dynamics are applied to physical systems which may be modeled by the Duffing nonlinear differential equation. The response of the Duffing oscillator to both deterministic sinusoidal and stochastic loadings is investigated and distinct regimes of the response motion are discerned and discussed. The stochastic input to the system is low-pass Gaussian white noise. The efficacy of studying the variation in time of the probability density of one or more of the system output states to determine the type of motion of the system is examined. Attractors in phase space are defined via Poincaré mapping and bounds on motion which serve as signatures for particular types of motion (e.g., chaotic, periodic) are identified by a hypervolume measurement technique. An accepted method for adapting one measured output state into a higher dimensional space by using time-delayed coordinates is used in conjunction with correlation dimension calculation to supply a lower-bound estimate of the fractal dimension and insight into the character of the motion of a nonlinear dynamic system.