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 ${\mathit{p}}_{\mathit{c}}$. For each type of change, there is a characteristic temporal behavior of orbits after the crisis (p>${\mathit{p}}_{\mathit{c}}$ by convention), with a characteristic time scale τ. For an important class of deterministic systems, the dependence of τ on p is τ∼(p-${\mathit{p}}_{\mathit{c}}$ ${)}^{\mathrm{-}\gamma }$ for p slightly greater than ${\mathit{p}}_{\mathit{c}}$. When noise is added to a system with p${\mathit{p}}_{\mathit{c}}$, orbits can exhibit the same sorts of characteristic temporal behavior as in the deterministic case for p>${\mathit{p}}_{\mathit{c}}$ (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 τ∼${\mathrm{\sigma }}^{\mathrm{-}\gamma }$g((${\mathit{p}}_{\mathit{c}}$-p)/σ), where σ is the characteristic strength of the noise, g(⋅) is a nonuniversal function depending on the system and noise, and γ 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.