Mixing and Diffusion in Chaos of Conservative Systems

Abstract

The characteristic feature of the chaotic sea in a conservative dynamical system is the presence of a multitude of tori of different sizes, forming an ‘islands around islands’ self-similar hierarchical structure. Chaotic orbits are often trapped for long times within such structure, and as a result the longtime correlation $C_t^\lambda \alpha {t^{ - \left( {\beta - 1} \right)}}\left( {2 > \beta > 1} \right)$ appears. In this situation, for Λ > 0 the probability distribution of mixing P(Λ;n) obeys the anomalous scaling relation P(Λ; n) = n ^δ p(n ^δ(Λ – Λ ^∞)), where ^δ = (β-1)/β < 1/2, and p(x) is a universal function determined by β. As determined by the folding of the unstable manifold induced by the tangency structure, for 0 > Λ > Λ _min we have ψ(Λ) = -2Λ. In the case in which islands of accelerator mode tori exist, the diffusion coefficient for chaotic orbits (or fluid particles) diverges, and the corresponding probability distribution of the coarse-grained velocity of such orbits (or particles) obeys an anomalous scaling relation similar to that given above. In this chapter we investigate the universality displayed by the statistical structure of this kind for chaos in conservative systems and the self-similar time series of the last KAM torus.