Diffusion in the standard map

Itzhack Dana; Shmuel Fishman

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

Diffusion in the standard map

Itzhack Dana, Shmuel Fishman

Technion-Israel Institute of Technology

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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.

Original language	English
Pages (from-to)	63-74
Number of pages	12
Journal	Physica D: Nonlinear Phenomena
Volume	17
Issue number	1
DOIs	https://doi.org/10.1016/0167-2789(85)90134-4
State	Published - Aug 1985
Externally published	Yes

Bibliographical note

Funding Information:
This work was supported in part by the U.S.-Israel Binational Science Foundation (B.S.F.), and by the Bat-Sheva de Rothschild Fund for Advancement of Science and Technology. We thank P. Bak, D. Bensimon, M.V. Berry, S.N. Coppersmith, M. Feingold, L.P. Kadanoff, I.C. Percival, I. Procaccia and A.B. Rechester for useful discussions and correspondence. S.F. thanks R.E. Prange for critical reading of the manuscript and for the hospitality of the University of Maryland where work was completed with support from the NSF through grant No. DMR-79-001172-A02.

Funding

This work was supported in part by the U.S.-Israel Binational Science Foundation (B.S.F.), and by the Bat-Sheva de Rothschild Fund for Advancement of Science and Technology. We thank P. Bak, D. Bensimon, M.V. Berry, S.N. Coppersmith, M. Feingold, L.P. Kadanoff, I.C. Percival, I. Procaccia and A.B. Rechester for useful discussions and correspondence. S.F. thanks R.E. Prange for critical reading of the manuscript and for the hospitality of the University of Maryland where work was completed with support from the NSF through grant No. DMR-79-001172-A02.

Funders	Funder number
Bat-Sheva de Rothschild
National Science Foundation	DMR-79-001172-A02
United States-Israel Binational Science Foundation

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title = "Diffusion in the standard map",

abstract = "Diffusion in the standard map is studied numerically. The stochasticity parameter K is near the critical value Kc, and the diffusion coefficient D is calculated. It is found to satisfy to a good approximation the scaling relation D ∝ (K - Kc)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.",

author = "Itzhack Dana and Shmuel Fishman",

note = "Funding Information: This work was supported in part by the U.S.-Israel Binational Science Foundation (B.S.F.), and by the Bat-Sheva de Rothschild Fund for Advancement of Science and Technology. We thank P. Bak, D. Bensimon, M.V. Berry, S.N. Coppersmith, M. Feingold, L.P. Kadanoff, I.C. Percival, I. Procaccia and A.B. Rechester for useful discussions and correspondence. S.F. thanks R.E. Prange for critical reading of the manuscript and for the hospitality of the University of Maryland where work was completed with support from the NSF through grant No. DMR-79-001172-A02.",

year = "1985",

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N2 - Diffusion in the standard map is studied numerically. The stochasticity parameter K is near the critical value Kc, and the diffusion coefficient D is calculated. It is found to satisfy to a good approximation the scaling relation D ∝ (K - Kc)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.

AB - Diffusion in the standard map is studied numerically. The stochasticity parameter K is near the critical value Kc, and the diffusion coefficient D is calculated. It is found to satisfy to a good approximation the scaling relation D ∝ (K - Kc)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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Diffusion in the standard map

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