Dynamical indicators for chaotic systems: Lyapunov exponents, entropies and beyond

At any time there is only a thin layer separating what is trivial from what is impossibly difficult. It is in that layer that discoveries are made … Andrei N. Kolmogorov An important aspect of the theory of dynamical systems is the formalization and quantitative characterization of the sensitivity to initial conditions. The Lyapunov exponents {λ i } are the indicators used to measure the average rate of exponential error growth in a system. Starting from the idea of Kolmogorov of characterizing dynamical systems by means of entropy-like quantities, following the work by Shannon in information theory, another approach to dynamical systems has been developed in the context of information theory, data compression and algorithmic complexity theory. In particular, the Kolmogorov–Sinai entropy, h ks , can be defined and interpreted as a measure of the rate of information production of a system. Since the ability to produce information is tightly linked to the exponential diversification of trajectories, it is not a surprise that a relation exists between h ks and {λ i }, the Pesin relation. One has to note that quantities such as {λ i } and h ks are properly defined only in specific asymptotic limits, that is, very long times and arbitrary accuracy. Since in realistic situations one has to deal with finite accuracy and finite time – as Keynes said, in the long run we shall all be dead – it is important to take into account these limitations. Relaxing the requirement of infinite time, one can investigate the relevance of finite time fluctuations of the “effective” Lyapunov exponent.