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 {dollar}psb c{dollar}. For each type of change, there is a characteristic temporal behavior of orbits after the crisis (p {dollar}>{dollar} {dollar}psb c{dollar} by convention), with a characteristic time scale {dollar}tau{dollar}. For an important class of deterministic systems, the dependence of {dollar}tau{dollar} on p is {dollar}tau sim(p-psb c)sp{lcub}-gamma{rcub}{dollar} for p slightly greater than {dollar}psb c{dollar}. When noise is added to a system with p {dollar}<{dollar} {dollar}psb c{dollar}, orbits can exhibit the same sorts of characteristic temporal behavior as in the deterministic case for p {dollar}>{dollar} {dollar}psb c{dollar} (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 {dollar}tau sim sigmasp{lcub}-gamma{rcub}glbrack (psb c-p)/sigmarbrack {dollar}, where {dollar}sigma{dollar} is the characteristic strength of the noise, {dollar}g(cdot){dollar} is a non-universal function depending on the system and noise, and {dollar}gamma{dollar} 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 {dollar}tau{dollar} 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 {dollar}gamma{dollar} for the low-noise ("deterministic") system, then add noise and observe that the scaling for {dollar}tau{dollar} is as predicted.