Chaos and Information

Chaotic systems are characterised by their sensitivity to small disturbances in initial position, which results in diverging trajectories, with the exponential rate of divergence given by the Lyapunov exponent. This amplification of the details in the initial position can be seen as a flow of information from smaller to larger length scales in state space, and it can be quantified using information theory. This flow can also be quantified as an entropy that is generated by the chaotic system, despite the fact that the dynamics is deterministic. In order to make an information-theoretic analysis of chaotic dynamical systems, we describe the system trajectory as a symbol sequence, which is referred to as symbolic dynamics. If the description of the chaotic dynamics in terms of symbol sequences is properly done, the full characteristics of the dynamical system can be derived from the symbol sequence description. In chemical self-organizing systems one can identify a tendency of information to flow from larger to smaller length scales, as we have discussed in the previous chapter. Still, it is clear that noise or fluctuations are important to break symmetries and to initiate the formation of spatial structure. Quantitatively this information flow from the noise, or from the microscales of the system, is very small and is hidden by the much larger thermodynamically related flow towards the microscales that is present in chemical dynamics. In chaotic systems, noise is of an even higher importance. Chaos may even be characterised by the extent to which a system is sensitive to noise. The Lyapunov exponents of a dynamical system quantifies how noise is amplified in the dynamics. In this chapter, we shall make an information-theoretic interpretation of these exponents and relate them to an entropy concept in dynamical systems, the measure entropy.