Fluctuations and Simple Chaotic Dynamics

J. P. Crutchfield and J. D. Farmer
Physics Board of Studies
University of California
Santa Cruz, CA 95064, USA

B. A Huberman
Xerox Palo Alto Research Center
Palo Alto, CA 94304, USA

Abstract

We describe the effects of fluctuations on the period-doubling bifurcation to chaos. We study the dynamics of maps of the interval in the absence of noise and numerically verify the scaling behavior of the Lyapunov characteristic exponent near the transition to chaos. As previously shown, fluctuations produce a gap in the period-doubling bifurcation sequence. We show that this implies a scaling behavior for the chaotic threshold and determine the associated critical exponent. By considering fluctuations as a disordering field on the deterministic dynamics, we obtain scaling relations between various critical exponents relating the effect of noise on the Lyapunov characteristic exponent. A rule is developed to explain the effects of additive noise at fixed parameter value from the deterministic dynamics at nearby parameter values.

Citation

J. P. Crutchfield, J. D. Farmer and B. A. Huberman. "Fluctuations and Simple Chaotic Dynamics." Physics Reports 92 1982: 45-82.

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