The Effects of Noise on Nonlinear Systems Near Crisis
MARYLAND UNIV COLLEGE PARK
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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 pPc by convention, with a characteristic time scale t. For an important class of deterministic systems, the dependence of t on p is tp-Pc-gamma for p slightly greater than Pc. When noise is added to a system with pPc, orbits can exhibit the same sorts of characteristic temporal behavior as in the deterministic case for pPc 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 tsigma-gamma gPc - psigma, where sigma is the characteristic strength of the noise, g. 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.
- Operations Research
- Numerical Mathematics