Abstract
We develop an analogy between the transitions to chaos observed in dissipative dynamical systems and critical phenomena. We mainly focus on the most popular scenarios to weak turbulence, namely the cascade of period-doubling bifurcations and intermittency. We consider the (largest) Lyapunov characteristic exponent as a (dis)order parameter which displays scaling behavior in the vicinity of the onset of chaos. We use renormalization group techniques to calculate the corresponding critical exponents. In the presence of an external perturbation, one can carry on the analogy with second-order phase transitions and define cross-over exponents relating the effect of the external field on the Lyapunov characteristic exponent. We compare the effects of both a random noise and a periodic excitation. We report numerical results on discrete systems which corroborate the renormalization group predictions.
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