Nonlinear Sciences – Chaotic Dynamics
Scientific paper
1999-06-14
Nonlinear Sciences
Chaotic Dynamics
To be appear in Phys. Rev E
Scientific paper
10.1103/PhysRevE.60.2761
We study the probability densities of finite-time or \local Lyapunov exponents (LLEs) in low-dimensional chaotic systems. While the multifractal formalism describes how these densities behave in the asymptotic or long-time limit, there are significant finite-size corrections which are coordinate dependent. Depending on the nature of the dynamical state, the distribution of local Lyapunov exponents has a characteristic shape. For intermittent dynamics, and at crises, dynamical correlations lead to distributions with stretched exponential tails, while for fully-developed chaos the probability density has a cusp. Exact results are presented for the logistic map, $x \to 4x(1-x)$. At intermittency the density is markedly asymmetric, while for `typical' chaos, it is known that the central limit theorem obtains and a Gaussian density results. Local analysis provides information on the variation of predictability on dynamical attractors. These densities, which are used to characterize the {\sl nonuniform} spatial organization on chaotic attractors are robust to noise and can therefore be measured from experimental data.
Prasad Awadhesh
Ramaswamy Ramakrishna
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