1999/09/01 by Awadhesh Prasad, Ramakrishna Ramaswamy · 5 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Chaos control and synchronization #Complex Systems and Time Series Analysis #Mathematical Dynamics and Fractals
paper · doi:10.1103/physreve.60.2761
openalex publication_date 1999/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We study the probability densities of finite-time or local Lyapunov exponents 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-->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 nonuniform spatial organization on chaotic attractors, are robust to noise and can, therefore, be measured from experimental data.