Resonances and transport in the sawtooth map

Q. Chen; I. Dana; J. D. Meiss; N. W. Murray; I. C. Percival

doi:10.1016/0167-2789(90)90037-p

Resonances and transport in the sawtooth map

Q. Chen, I. Dana, J. D. Meiss, N. W. Murray, I. C. Percival

Department of Physics - at Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

49 Scopus citations

Abstract

We study transport in a completely chaotic Hamiltonian system, the hyperbolic sawtooth map. Analytical expressions are obtained for its cantori and resonances. We show that resonances give a complete partition of phase space. The flux leaking out of a resonance is given by its turnstiles, whose form and areas are obtained analytically. When the total flux out of a resonance becomes one third the area of an island, the topology of the turnstiles changes. At the same parameter value, a horseshoe is formed corresponding to the orbits trapped within the resonance. Based on this, a coding scheme for the trapped orbits is introduced and expressions for trapped ordered orbits are obtained. The partial flux transferred from one resonance to another is determined by the degree of overlap of their turnstiles. We calculate the survival probability within a resonance using the Markov model; the results are compared with results obtained numerically and from periodic-orbit theory.

Original language	English
Pages (from-to)	217-240
Number of pages	24
Journal	Physica D: Nonlinear Phenomena
Volume	46
Issue number	2
DOIs	https://doi.org/10.1016/0167-2789(90)90037-p
State	Published - Nov 1990

Bibliographical note

Funding Information:
Support for this work was obtained from US Department of Energy grant DE-FG05-80ET-53088, the UK Science and Engineering Research Council, and a NATO grant for international collaboration. This collaboration was begun during a visit by Q.C. and J.D.M. to Queen Mary College, we would like to thank them for their hospitality.

Funding

Support for this work was obtained from US Department of Energy grant DE-FG05-80ET-53088, the UK Science and Engineering Research Council, and a NATO grant for international collaboration. This collaboration was begun during a visit by Q.C. and J.D.M. to Queen Mary College, we would like to thank them for their hospitality.

Funders	Funder number
UK Science and Engineering Research Council
US Department of Energy	DE-FG05-80ET-53088
North Atlantic Treaty Organization

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title = "Resonances and transport in the sawtooth map",

abstract = "We study transport in a completely chaotic Hamiltonian system, the hyperbolic sawtooth map. Analytical expressions are obtained for its cantori and resonances. We show that resonances give a complete partition of phase space. The flux leaking out of a resonance is given by its turnstiles, whose form and areas are obtained analytically. When the total flux out of a resonance becomes one third the area of an island, the topology of the turnstiles changes. At the same parameter value, a horseshoe is formed corresponding to the orbits trapped within the resonance. Based on this, a coding scheme for the trapped orbits is introduced and expressions for trapped ordered orbits are obtained. The partial flux transferred from one resonance to another is determined by the degree of overlap of their turnstiles. We calculate the survival probability within a resonance using the Markov model; the results are compared with results obtained numerically and from periodic-orbit theory.",

author = "Q. Chen and I. Dana and Meiss, {J. D.} and Murray, {N. W.} and Percival, {I. C.}",

note = "Funding Information: Support for this work was obtained from US Department of Energy grant DE-FG05-80ET-53088, the UK Science and Engineering Research Council, and a NATO grant for international collaboration. This collaboration was begun during a visit by Q.C. and J.D.M. to Queen Mary College, we would like to thank them for their hospitality.",

year = "1990",

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doi = "10.1016/0167-2789(90)90037-p",

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AU - Dana, I.

AU - Meiss, J. D.

AU - Murray, N. W.

AU - Percival, I. C.

N1 - Funding Information: Support for this work was obtained from US Department of Energy grant DE-FG05-80ET-53088, the UK Science and Engineering Research Council, and a NATO grant for international collaboration. This collaboration was begun during a visit by Q.C. and J.D.M. to Queen Mary College, we would like to thank them for their hospitality.

PY - 1990/11

Y1 - 1990/11

N2 - We study transport in a completely chaotic Hamiltonian system, the hyperbolic sawtooth map. Analytical expressions are obtained for its cantori and resonances. We show that resonances give a complete partition of phase space. The flux leaking out of a resonance is given by its turnstiles, whose form and areas are obtained analytically. When the total flux out of a resonance becomes one third the area of an island, the topology of the turnstiles changes. At the same parameter value, a horseshoe is formed corresponding to the orbits trapped within the resonance. Based on this, a coding scheme for the trapped orbits is introduced and expressions for trapped ordered orbits are obtained. The partial flux transferred from one resonance to another is determined by the degree of overlap of their turnstiles. We calculate the survival probability within a resonance using the Markov model; the results are compared with results obtained numerically and from periodic-orbit theory.

AB - We study transport in a completely chaotic Hamiltonian system, the hyperbolic sawtooth map. Analytical expressions are obtained for its cantori and resonances. We show that resonances give a complete partition of phase space. The flux leaking out of a resonance is given by its turnstiles, whose form and areas are obtained analytically. When the total flux out of a resonance becomes one third the area of an island, the topology of the turnstiles changes. At the same parameter value, a horseshoe is formed corresponding to the orbits trapped within the resonance. Based on this, a coding scheme for the trapped orbits is introduced and expressions for trapped ordered orbits are obtained. The partial flux transferred from one resonance to another is determined by the degree of overlap of their turnstiles. We calculate the survival probability within a resonance using the Markov model; the results are compared with results obtained numerically and from periodic-orbit theory.

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Resonances and transport in the sawtooth map

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