Some exact results for the diffusion coefficients of maps with pruning cycles

Tseng, Hsen-Chen; Chen, Hung-Jung; Li, Ping-Cheng; Lai, Wuu-Yih; Chou, Chien Ho; Chen, Hung-Wei

doi:10.1016/0375-9601(94)90429-4

Cited by 10 publications

(1 citation statement)

References 3 publications

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“…Ishii [63] proved the Pruning Front Conjecture for the Lozi family, but his approach is adapted to the piecewise linear nature of this family and is quite different from the one adopted here. There is a vast literature on the Hénon and Lozi families, the transition to chaos in dimension 2, and pruning theory in other contexts (see for example 1 [18,17,60,61,10,9,11,12,14,16,13,15,8,7,78,53,54,52,56,63,64,29,77,65,26,27,25,2,3,46,45]). Most of this work is not discussed here: the topological approach described in this paper has a quite different flavour.…”

Section: Introductionmentioning

confidence: 99%

How to prune a horseshoe

Carvalho

Hall

2002

Nonlinearity

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Let F : R 2 → R 2 be a homeomorphism. An open F-invariant subset U of R 2 is a pruning region for F if it is possible to deform F continuously to a homeomorphism F U for which every point of U is wandering, but which has the same dynamics as F outside of U. This concept is motivated by the Pruning Front Conjecture introduced by Cvitanović which claims that every Hénon map can be understood as a pruned horseshoe. This paper contains recent results on pruning theory, concentrating on prunings of the horseshoe. We give conditions on a disk D which ensure that the orbit of its interior is a pruning region; explain how prunings of the horseshoe can be understood in terms of underlying tree maps; discuss the connection between pruning and Thurston's classification theorem for surface homeomorphisms; motivate a conjecture describing the forcing relation on horseshoe braid types; and use this theory to give a precise statement of the Pruning Front Conjecture.

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Section: Introductionmentioning

confidence: 99%

How to prune a horseshoe

Carvalho

Hall

2002

Nonlinearity

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Deterministic (Anomalous) Transport

Artuso

Cristadoro

2008

Anomalous Transport

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Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

Korabel

Klages

2004

Physica D: Nonlinear Phenomena

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The nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter. We show that on a suitable coarse scale this map generates an oscillating parameterdependent diffusion coefficient, similarly to hyperbolic maps, whose asymptotic functional form can be understood in terms of simple random walk approximations. On finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. By using a Green-Kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations. Starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized Takagi functions obeying generalized de Rham-type functional recursion relations. We finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive, and that for normal diffusion it increases by increasing the parameter value.

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Some exact results for the diffusion coefficients of maps with pruning cycles

Cited by 10 publications

References 3 publications

How to prune a horseshoe

How to prune a horseshoe

Deterministic (Anomalous) Transport

Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

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