Collapse of hierarchical phase space and mixing rates in Hamiltonian systems

Abstract

We employ statistical properties of Poincaré recurrences to investigate dynamical behaviors of coupled Hamiltonian maps with a mixed phase space, where sticking to regular (or resonant) islands degrades chaotic properties of the system. In particular we investigate two and three coupled Chirikov–Taylor standard mappings (SM) and choose nonlinearity parameters corresponding to a mixed phase space. Our first set of findings are: (i) the collapse of hierarchical phase space depends of the coupling intensity, (ii) the penetration of resonance islands generates a trapping regime responsible for a long intermediate algebraic decay which is suppressed as coupling increases; and (iii) the projection of recurrence-time statistics (RTS) onto two-dimensional planes of the high-dimensional phase space of coupled maps is useful to understand different dynamical features, when coupling strengths are varied. We also estimate the asymptotic polynomial decay exponent of RTS γ for both cases, and elaborate an indirect way to estimate the decay exponent of correlations χ by using large deviations theory applied to the probability distributions of finite-time largest Lyapunov exponents (FTLLEs). In the asymptotic regime both methods yield γ∼1.20 (2 SM) and γ∼1.10 (3 SM): the corresponding power-law exponents for time correlation decay χ is given by the known relationship χ=γ−1. As higher-dimensional Hamiltonian systems (with mixed phase space) share the crucial property that resonance islands are no more forbidden domains in phase space, the results obtained here for two and three coupled maps should be extended to even higher-dimensional systems.