ANALYSIS OF A NONLINEAR-SYSTEM EXHIBITING CHAOTIC, NOISY CHAOTIC, ANDRANDOM BEHAVIORS

This study presents a stochastic approach for the analysis of nonchaot ic, chaotic, random and nonchaotic, random and chaotic, and random dyn amics of a nonlinear system. The analysis utilizes a Markov process ap proximation, direct numerical simulations, and a generalized stochasti c Melnikov process. The Fokker-Planck equation along with a path integ ral solution procedure are developed and implemented to illustrate the evolution of probability density functions. Numerical integration is employed to simulate the noise effects on nonlinear responses. In rega rd to the presence of additive ideal white noise, the generalized stoc hastic Melnikov process is developed to identify the boundary for nois y chaos. A mathematical representation encompassing all possible dynam ical responses is provided. Numerical results indicate that noisy chao s is a possible intermediate state between deterministic and random dy namics. A global picture of the system behavior is demonstrated via th e transition of probability density function over its entire evolution . It is observed that the presence of external noise has significant e ffects over the transition between different response slates and betwe en co-existing attractors.