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 pc. For each type of change, there is a characteristic temporal behavior of orbits after the crisis (p>pc by convention), with a characteristic time scale {τ}. For an important class of deterministic systems, the dependence of {τ} on p is {τ}{~}(p-pc{)}^{{{-}}{γ}}$ 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 {τ}{~}σ^-γg((pc-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.