SCALING LAW FOR CHARACTERISTIC TIMES OF NOISE-INDUCED CRISES

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 approximately (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 approximately sigma-gamma-g((p(c)-p)/sigma), where sigma is the characteristic strength of the noise, g(.) is a nonuniversal function depending on the system and noise, and gamma is the critical exponent of the corresponding deterministic crisis. A previous analysis of the noisy logistic map [F. T. Arecchi, R. Badii, and A. Politi, Phys. Lett. 103A, 3 (1984)] is shown to be consistent with our general result. 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.