Scaling for a Periodic Forcing of a Period-Doubling System

Abstract

The effects of a periodic excitation on the transition to chaos via the period-doubling cascade are considered. Numerical experiments on discrete systems suggest that the Lyapunov characteristic exponent satisfies scaling in the vicinity of the transition. A renormalization-group approach is used to predict the dependence of the critical exponent on the ratio of the driving to the internal frequencies. When this ratio is equal to the reciprocal of the golden mean, it is shown that this critical exponent is the same as for an external random noise.