Characteristic distributions of finite-time Lyapunov exponents

We study the probability densities of finite-time or local Lyapunov exponen ts in low-dimensional chaotic systems. While the multifractal formalism des cribes how these densities behave in the asymptotic or long-time limit, the re 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, a nd at crises, dynamical correlations lead to distributions with stretched e xponential tails, while for fully developed chaos the probability density h as a cusp. Exact results are presented for the logistic map, x --> 4x(1-x). At intermittency the density is markedly asymmetric, while for "typical" c haos, it is known that the central limit theorem obtains and a Gaussian:den sity results. Local analysis provides information on the variation of predi ctability on dynamical attractors. These densities, which are used to chara cterize the nonuniform spatial organization on chaotic attractors, are robu st to noise and can, therefore, be measured from experimental data. [S1063- 651X(99)10208-3].