Strong Dynamical Trappings Originating Ergodicity Breaking in Coupled Hamiltonian Systems

Ergodicity breaking has a profound effect on the transport of particles in typical nonlinear Hamiltonian systems. In this paper, we analyze the survival and extinction of ergodicity in weakly chaotic Hamiltonian coupled maps. The key feature for the ergodicity breaking is the existence of sporadic strong dynamical trappings (or strong quasi-invariant sets in higher dimensions). Such trappings eventually occur when zero Kolmogorov-Sinai-entropy (KSE) is observed along a chaotic trajectory. The finite-time KSE ^ω is obtained from the spectrum of finite-time Lyapunov exponents (FTLEs ^ω ) calculated during an arbitrary time window of size ω along the chaotic trajectory. Zero KSE ^ω occurs when all FTLEs ^ω are sufficiently close to zero, and positive KSE ^ω when the sum of the FTLEs ^ω is larger than a predefined threshold. The associated points in phase space belong, respectively, to strong and weak quasi-invariant structures. The key observations are that (i) for zero KSE ^ω , solely power-law decays are observed in the cumulative recurrence distribution (characterizing the strong quasi-invariant sets), and (ii) for positive KSE ^ω values we obtain asymptotically only exponential decays (characterizing the chaotic motion and the weak quasi-invariant sets). For N=1,… ,5 coupled Hamiltonian maps with mixed dynamics, we obtain the power-law decay exponent μ∼ 1.20 , which corroborates with former investigations. This result also persists for arbitrarily small coupling strength between the maps. Both outcomes (i) and (ii) are valid for asymptotic times so that our analysis precisely confirms the concept of ergodicity extinction and survival in weakly chaotic systems with a moderate number of dimensions.