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Hamiltonian systems on a one-dimensional lattice with discrete time are studied. As the coupling constant is increased, they show a sharp transition from regular to random motion. Below the threshold, KAM tori and a stochastic layer coexist. The stochastic motion therein is sticky to KAM tori and has a long time-tail. The motion above the threshold is ergodic, characterized by the power spectra and Lyapunov spectra which are consistent with the results of random matrices.
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Kaneko et al. (1987) studied this question.
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