Diffusion in the standard map

doi:10.1016/0167-2789(85)90134-4

Physica D: Nonlinear Phenomena

Volume 17, Issue 1, August 1985, Pages 63-74

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Abstract

Diffusion in the standard map is studied numerically. The stochasticity parameter K is near the critical value K_c, and the diffusion coefficient D is calculated. It is found to satisfy to a good approximation the scaling relation D ∝ (K − K_c)^η, with η in good agreement with the value predicted by the scaling theory of the disappearance of the last bounding KAM torus. The critical region where this scaling relation holds is surprisingly large, i.e. K ≤ 2.5. The mechanism of transport from the chaotic region to the remnants of the last KAM torus is investigated. Evidence for the existence of a narrow stochastic channel mediating this transport is presented and its origin is discussed. Although the scaling of D agrees with the predictions of the scaling theory the transport mechanism is different from the one assumed in this theory.

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Regarding the problem of transport in Hamiltonian systems, the standard map has become over the years a classical case study. One of its advantages is that it depends on just one control parameter K and many attempts were made to find the link between K and a diffusion coefficient [1,5,10,11]. Depending on the values of K, we can get a system which is very close to an integrable one or one that is fully chaotic, with, in between, the picture of a mixed phase space with a chaotic sea and regular islands.
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Transport times for a chaotic system are highly sensitive to initial conditions and parameter values. We present a technique to find rough orbits (epsilon chains) that achieve a desired transport rapidly and which can be stabilized with small parameter perturbations. The strategy is to build the epsilon chain from segments of a long orbit — the point is that long orbits have recurrences in neighborhoods where faster orbits must also pass. The recurrences are used as the switching points between segments. The resulting epsilon chain can be refined by gluing orbit segments over the switching points, providing that a local hyperbolicity condition is satisfied. As an example, we show that transport times for the standard map can be reduced by factors of 10⁴. The techniques presented here can be easily generalized to higher dimensions and to systems where the dynamics is known only as a time-series.
Critical function for the standard map
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^†: Address after September 1984, Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104, USA.

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Diffusion in the standard map

Abstract

Phys. Rep.

Topology

Physica

Phys. Rev.

Regular and Stochastic Motion

J. Math. Phys.

Phys. Rev. Lett.

J. Stat. Phys.