Stochastic Fluctuations in Deterministic Systems

The unavoidable presence of inhomogeneities in the phase space of a chaotic system induces fluctuations in the degree of stability, even when long trajectories are considered. The characterization of such fluctuations requires to go beyond average indicators: this is achieved with the help of the multifractal formalism which contributes to: (i) establishing a general connection between the positive Lyapunov exponents and the Kolmogorov-Sinai entropy; (ii) identifying and quantifying deviations from a purely hyperbolic dynamics; (iii) characterizing anomalous bifurcations, where the attractor looses progressively its stability. In the context of spatially extended dynamical systems, the study of Lyapunov exponent fluctuations leads to a non conventional assessment of the extensivity of the resulting dynamics. Finally, a careful study of the fluctuations allows clarifying the odd phenomenon of “stable chaos”, where an irregular dynamics is accompanied by a negative (average) Lyapunov exponent.