Unpredictability in Hamiltonian systems with a hierarchical phase space

Abstract

One of the main consequences of the complex hierarchical structure of chaotic regions and stability islands in the phase space of a typical nonlinear Hamiltonian system is the phenomenon of stickiness. The chaotic orbits that approach an island are trapped in its neighborhood for arbitrarily long times, in which the orbits behave similarly as quasiperiodic orbits. In this paper, we characterize the boundary between chaos and regular motion in the phase space of the standard map for distinct parameter values. The orbits are distinguished between regular and chaotic employing a recently proposed method of weighted Birkhoff averages. We quantify the dimension of the boundaries of the islands using the uncertainty exponent. In our simulations, we show that the dimension of the island's boundary depends on the scale of the initial condition uncertainty and the level of the hierarchical structure. We also show that the trapping in the vicinity of the islands causes an obstruction in the predictability of the final state of an orbit. We present how this loss of predictability results in larger dimensions at the inner levels of the islands.