The Effects of Noise on Nonlinear Systems Near Crisis
Abstract
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 t. For an important class of deterministic systems, the dependence of t on p is t^(p-Pc)-gamma for p slightly greater than Pc. When noise is added to a system with p<Pc, orbits can exhibit the same sorts of characteristic temporal behavior as in the deterministic case for p>Pc (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 t^sigma(-gamma) g(Pc - p)/sigma, 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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1991
- Accession Number
- ADA357013
Entities
People
- John C. Sommerer
Organizations
- University of Maryland