Effective intermittency and cross-correlations in the Standard Map

Abstract

We define auto- and cross-correlation functions capable to capture dynamical characteristics induced by local phase space structures in a general dynamical system. These correlation functions are calculated in the Standard Map for a range of values of the non-linearity parameter $k$. Using a model of non-interacting particles, each evolving according to the same Standard Map dynamics and located initially at specific phase space regions, we show that for $0.6 < k \leq 1.2$ long-range cross-correlations emerge. They occur as an ensemble property of particle trajectories by an appropriate choice of the phase space cells used in the statistical averaging. In this region of $k$-values the single particle phase space is either dominated by local chaos ($k \leq k_c$ with $k_c \approx 0.97$) or it is characterized by the transition from local to global chaos ($k_c < k \leq 1.2$). Introducing suitable symbolic dynamics we demonstrate that the emergence of long-range cross-correlations can be attributed to the existence of an effective intermittent dynamics in specific regions of the phase space. Our findings further support the recently found relation of intermittent dynamics with the occurrence of cross-correlations (F.K. Diakonos, A.K. Karlis, and P. Schmelcher, Europhys. Lett. {\bf 105}, 26004 (2014)) in simple one-dimensional intermittent maps, suggesting its validity for two-dimensional Hamiltonian systems.