Fluctuations and Simple Chaotic Dynamics

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. * Present address: Center for Nonlinear Studies, MS B 258, Los Alamos Laboratories, Los Alamos, New Mexico 87545, U.SA. Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 92, No. 2 (1982) 45-82. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 20.00, postage included. o 370-1573/82/0000—0000/$9.50 © 1982 North-Holland Publishing Company J.P. Crutchfield eta!., Fluctuations and simple chaotic dynamics 47