Scaling law for characteristic times of noise-induced crises
- Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742, (USA) Applied Physics Laboratory, The Johns Hopkins University, Laurel, Maryland 20723 (USA)
- Laboratory for Plasma Research and Departments of Electrical Engineering and Physics, University of Maryland, College Park, Maryland 20742 (USA)
- Laboratory for Plasma Research, Institute for Physical Science Technology, and Department of Mathematics, University of Maryland, College Park, Maryland 20742 (USA)
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 {ital p} passes through its crisis value {ital p}{sub {ital c}}. For each type of change, there is a characteristic temporal behavior of orbits after the crisis ({ital p}{gt}{ital p}{sub {ital c}} by convention), with a characteristic time scale {tau}. For an important class of deterministic systems, the dependence of {tau} on {ital p} is {tau}{similar to}({ital p}{minus}{ital p}{sub {ital c}}){sup {minus}{gamma}} for {ital p} slightly greater than {ital p}{sub {ital c}}. When noise is added to a system with {ital p}{lt}{ital p}{sub {ital c}}, orbits can exhibit the same sorts of characteristic temporal behavior as in the deterministic case for {ital p}{gt}{ital p}{sub {ital 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}{similar to}{sigma}{sup {minus}{gamma}}{ital g}(({ital p}{sub {ital c}}{minus}{ital p})/{sigma}), where {sigma} is the characteristic strength of the noise, {ital g}({center dot}) 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.
- OSTI ID:
- 5838160
- Journal Information:
- Physical Review, A; (USA), Vol. 43:4; ISSN 1050-2947
- Country of Publication:
- United States
- Language:
- English
Similar Records
Effect of Noise on Nonhyperbolic Chaotic Attractors
Barium and xenon isotope yields in photopion reactions of sup 133 Cs