H. Lin et Scs. Yim, ANALYSIS OF A NONLINEAR-SYSTEM EXHIBITING CHAOTIC, NOISY CHAOTIC, ANDRANDOM BEHAVIORS, Journal of applied mechanics, 63(2), 1996, pp. 509-516
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.