Characteristic times in the standard map.

Phys Rev E 2019 Mar;99(3-1):032203

Research Center for Astronomy, Academy of Athens Soranou Efesiou 4, GR-115 27 Athens, Greece.

We study and compare three characteristic times of the standard map: the Lyapunov time t_{L}, the Poincaré recurrence time t_{r}, and the stickiness (or escape) time t_{st}. The Lyapunov time is the inverse of the Lyapunov characteristic number (L) and in general is quite small. We find empirical relations for the L as a function of the nonlinearity parameter K and of the chaotic area A. We also find empirical relations for the Poincaré recurrence time t_{r} as a function of the nonlinearity parameter K, of the chaotic area A, and of the size of the box of initial conditions ε. As a consequence, we find relations between t_{r} and L. We compare the distributions of the stickiness time and the Poincaré recurrence time. The stickiness time inside the sticky regions at the boundary of the islands of stability is orders of magnitude smaller than the Poincaré recurrence time t_{r} and this affects the diffusion exponent μ, which converges always to the value μ=1. This is shown in an extreme stickiness case. The diffusion is anomalous (ballistic motion) inside the accelerator mode islands of stability with μ=2 but it is normal everywhere outside the islands with μ=1. In a particular case of extreme stickiness, we find the hierarchy of islands around islands.

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March 2019
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References

(Supplied by CrossRef)

A. S. Monin et al.
1971

G. Contopoulos et al.
2002

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