The Effects of Noise on Nonlinear Systems Near Crisis

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 noise-induced crisis). Our main result is that for systems whose characteristic times scale as above in the zero-noise limit, the characteristic time in the noisy case scales as tau ~ sigma ^{-gamma}g[ (p_ c-p)/sigma] , where sigma is the characteristic strength of the noise, g( cdot) is a non-universal function depending on the system and noise, and gamma is the critical exponent of the corresponding deterministic crisis. Illustrative numerical examples are given for two-dimensional maps and a three-dimensional 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, noise-induced bursts for a chaotic mechanical system near a crisis. The system studied is a periodically driven, variable-noise, magnetoelastic ribbon. We determine gamma for the low-noise ("deterministic") system, then add noise and observe that the scaling for tau is as predicted.