Stagnant Motions in Hamiltonian Systems

Abstract

Hamiltonian systems often reveal the chaotic phenomena with long time memory, and then the time average of a dynamical variable does not seem to converge toward a certain constant monotonically. Even when the time average looks like a constant in the limit under a given initial condition, for many cases the limiting value often depends sensitively on the initial data. Chaos in the dynamical system is usually discussed in the framework of the ergodic theory which guarantees the weak law of the large number and the unique existence of the time average except for the measure zero set. However, the hamiltonian chaos seems to be difficult to understand on the same line straightforwardly. The essence of the hamiltonian chaos seems to be more complex (or robust) than that of the purely ergodic ones. In this paper we devote ourselves to the research of the origin of such wild long time tails. A result of our phenomenological approaches is that the hamiltonian chaos is nonstationary and multi-ergodic. The various effects of the long time tails such as the f-2 spectrum and the anomalous large deviation will be explained from a universality law in the transition regime between chaos and torus.