Quasi-periodic route to chaos in a near-integrable PDE: Homoclinic crossings

A. R. Bishop; D. W. McLaughlin; M. G. Forest; E. A. Overman

doi:10.1016/0375-9601(88)90580-4

Quasi-periodic route to chaos in a near-integrable PDE: Homoclinic crossings

A. R. Bishop, D. W. McLaughlin, M. G. Forest, E. A. Overman

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Abstract

A new numerical experiment is discussed which shows a quasi-periodic route to intermittent chaos - typical for near-conservative, dispersive waves of small amplitude (nonlinear Schrödinger regime) in one spatial dimension. This route has: temporally one frequency, then two, then chaos; associated spatial symmetry changes; a low dimensional intermittent strange attractor. A nonlinear spectral transform has been used to show: a small number of nonlinear modes in the chaotic state; interaction of coherent modes with radiation modes: and, most importantly, that (unperturbed) homoclinic states are crossed repeatedly in these regimes. These homoclinic states: (a) separate spatially localized modes from radiation modes, and (b) act as sources of extreme sensitivity which can produce temporal chaos: for the first time homoclinic states have been simultaneously associated with both spatial patterns and temporal chaos.

Original language	English (US)
Pages (from-to)	335-340
Number of pages	6
Journal	Physics Letters A
Volume	127
Issue number	6-7
DOIs	https://doi.org/10.1016/0375-9601(88)90580-4
State	Published - Mar 7 1988

ASJC Scopus subject areas

General Physics and Astronomy

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10.1016/0375-9601(88)90580-4

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title = "Quasi-periodic route to chaos in a near-integrable PDE: Homoclinic crossings",

abstract = "A new numerical experiment is discussed which shows a quasi-periodic route to intermittent chaos - typical for near-conservative, dispersive waves of small amplitude (nonlinear Schr{\"o}dinger regime) in one spatial dimension. This route has: temporally one frequency, then two, then chaos; associated spatial symmetry changes; a low dimensional intermittent strange attractor. A nonlinear spectral transform has been used to show: a small number of nonlinear modes in the chaotic state; interaction of coherent modes with radiation modes: and, most importantly, that (unperturbed) homoclinic states are crossed repeatedly in these regimes. These homoclinic states: (a) separate spatially localized modes from radiation modes, and (b) act as sources of extreme sensitivity which can produce temporal chaos: for the first time homoclinic states have been simultaneously associated with both spatial patterns and temporal chaos.",

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T1 - Quasi-periodic route to chaos in a near-integrable PDE

T2 - Homoclinic crossings

AU - Bishop, A. R.

AU - McLaughlin, D. W.

AU - Forest, M. G.

AU - Overman, E. A.

PY - 1988/3/7

Y1 - 1988/3/7

N2 - A new numerical experiment is discussed which shows a quasi-periodic route to intermittent chaos - typical for near-conservative, dispersive waves of small amplitude (nonlinear Schrödinger regime) in one spatial dimension. This route has: temporally one frequency, then two, then chaos; associated spatial symmetry changes; a low dimensional intermittent strange attractor. A nonlinear spectral transform has been used to show: a small number of nonlinear modes in the chaotic state; interaction of coherent modes with radiation modes: and, most importantly, that (unperturbed) homoclinic states are crossed repeatedly in these regimes. These homoclinic states: (a) separate spatially localized modes from radiation modes, and (b) act as sources of extreme sensitivity which can produce temporal chaos: for the first time homoclinic states have been simultaneously associated with both spatial patterns and temporal chaos.

AB - A new numerical experiment is discussed which shows a quasi-periodic route to intermittent chaos - typical for near-conservative, dispersive waves of small amplitude (nonlinear Schrödinger regime) in one spatial dimension. This route has: temporally one frequency, then two, then chaos; associated spatial symmetry changes; a low dimensional intermittent strange attractor. A nonlinear spectral transform has been used to show: a small number of nonlinear modes in the chaotic state; interaction of coherent modes with radiation modes: and, most importantly, that (unperturbed) homoclinic states are crossed repeatedly in these regimes. These homoclinic states: (a) separate spatially localized modes from radiation modes, and (b) act as sources of extreme sensitivity which can produce temporal chaos: for the first time homoclinic states have been simultaneously associated with both spatial patterns and temporal chaos.

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Quasi-periodic route to chaos in a near-integrable PDE: Homoclinic crossings

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