The Effects of Noise on Nonlinear Systems Near Crisis
Abstract
We consider the influence of random noise on low dimensional, nonlinear dynamical systems with parameters near values leading to a crisis in the absence of noise. In a crisis, one of several characteristic changes in a chaotic attractor takes place as a system parameter p passes through its crisis value p_ c. For each type of change, there is a characteristic temporal behavior of orbits after the crisis (p > p _ c by convention), with a characteristic time scale tau. For an important class of deterministic systems, the dependence of tau on p is tau ~(p p_ c)^{gamma} for p slightly greater than p_ c. When noise is added to a system with p < p _ c, orbits can exhibit the same sorts of characteristic temporal behavior as in the deterministic case for p > p_ c (a noiseinduced crisis). Our main result is that for systems whose characteristic times scale as above in the zeronoise limit, the characteristic time in the noisy case scales as tau ~ sigma ^{gamma}g[ (p_ cp)/sigma] , where sigma is the characteristic strength of the noise, g( cdot) is a nonuniversal function depending on the system and noise, and gamma is the critical exponent of the corresponding deterministic crisis. Illustrative numerical examples are given for twodimensional maps and a threedimensional flow. In addition, the relevance of the noise scaling law to experimental situations is discussed. We investigate experimentally the scaling of the average time tau between intermittent, noiseinduced bursts for a chaotic mechanical system near a crisis. The system studied is a periodically driven, variablenoise, magnetoelastic ribbon. We determine gamma for the lownoise ("deterministic") system, then add noise and observe that the scaling for tau is as predicted.
 Publication:

Ph.D. Thesis
 Pub Date:
 1991
 Bibcode:
 1991PhDT.......183S
 Keywords:

 CHAOS;
 Physics: General