Leading Pollicott-Ruelle resonances for chaotic area-preserving maps

Venegeroles, Roberto

doi:10.1103/physreve.77.027201

Cited by 11 publications

(5 citation statements)

References 27 publications

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“…Since numerical experiments estimate that γ ≈ 1.5, then (35) implies that β ≈ 1.5 as well. This has been confirmed by Venegeroles using the Perron-Frobenius operator to analytically estimate diffusion due to accelerator modes [Ven08,Ven09]. Stickiness can also be studied using finite time Lyapunov exponents: the distribution of exponents is bimodal due to orbits sticking near elliptic regions [SLV05].…”

Section: B Stickiness and Anomalous Diffusionmentioning

confidence: 78%

Thirty years of turnstiles and transport

Meiss

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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To characterize transport in a deterministic dynamical system is to compute exit time distributions from regions or transition time distributions between regions in phase space. This paper surveys the considerable progress on this problem over the past thirty years. Primary measures of transport for volume-preserving maps include the exiting and incoming fluxes to a region. For area-preserving maps, transport is impeded by curves formed from invariant manifolds that form partial barriers, e.g., stable and unstable manifolds bounding a resonance zone or cantori, the remnants of destroyed invariant tori. When the map is exact volume preserving, a Lagrangian differential form can be used to reduce the computation of fluxes to finding a difference between the action of certain key orbits, such as homoclinic orbits to a saddle or to a cantorus. Given a partition of phase space into regions bounded by partial barriers, a Markov tree model of transport explains key observations, such as the algebraic decay of exit and recurrence distributions. The problem of transport in dynamical systems is to quantify the motion of collections of trajectories between regions of phase space with physical significance, for example, to determine chemical reaction rates, mixing rates in a fluid, or particle confinement times in an accelerator or fusion plasma device. For Hamiltonian systems with a phase space containing regular (elliptic islands and tori) and irregular (chaotic) components, transport is impeded by partial barriers bounding resonance zones or by remnants of invariant tori. In the former case the barriers are formed from the broken separatrices of an unstable periodic orbit, and in the latter, they are formed from similar stable and unstable manifolds of a remnant torus-a cantorus. Thirty years ago, MacKay, Meiss, and Percival discovered that cantori form robust partial barriers for area-preserving maps: in any region of phase space the most resistant barriers are the cantori. In this review, I discuss this work and survey subsequent developments. While much has been done, there are still many open questions.

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Section: B Stickiness and Anomalous Diffusionmentioning

confidence: 78%

Thirty years of turnstiles and transport

Meiss

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Moreover, those orbits that either originate from the neighborhood of the accelerator modes, or come close to them in the course of time, becoming trapped there for a while due to the stickiness of the neighborhood (containing cantori), get "dragged", or accelerated by them, and therefore display anomalous diffusion with µ > 1 (superdiffusion). Furthermore, apart from the accelerator modes of period 1, there exist also accelerator modes of higher periods, 2,3,4..., which we also observe in this work, but their role becomes less important with increasing K much faster than for period 1. There have been many attempts to account for the anomalous diffusion in the area preserving maps, most notably by Venegeroles [20,21,22], but it seems still largely impossible to predict the diffusion exponent µ (see also [23]) for a set of initial conditions at a given K.…”

Section: Introductionmentioning

confidence: 99%

“…There have been many attempts to account for the anomalous diffusion in the area preserving maps, most notably by Venegeroles [20,21,22], but it seems still largely impossible to predict the diffusion exponent µ (see also [23]) for a set of initial conditions at a given K.…”

Section: Introductionmentioning

confidence: 99%

Survey on the role of accelerator modes for anomalous diffusion: The case of the standard map

Manos

Robnik

2014

Phys. Rev. E

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We perform an extensive and detailed analysis of the generalized diffusion processes in deterministic area preserving maps with noncompact phase space, exemplified by the standard map, with the special emphasis on understanding the anomalous diffusion arising due to the accelerator modes. The accelerator modes and their immediate neighborhood undergo ballistic transport in phase space, and also the greater vicinity of them is still much affected ("dragged") by them, giving rise to the non-Gaussian (accelerated) diffusion. The systematic approach rests upon the following applications: the GALI method to detect the regular and chaotic regions and thus to describe in detail the structure of the phase space, the description of the momentum distribution in terms of the Lévy stable distributions, and the numerical calculation of the diffusion exponent and of the corresponding diffusion constant. We use this approach to analyze in detail and systematically the standard map at all values of the kick parameter K, up to K = 70. All complex features of the anomalous diffusion are well understood in terms of the role of the accelerator modes, mainly of period 1 at large K ≥ 2π, but also of higher periods (2,3,4,...) at smaller values of K ≤ 2π.

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“…Here we are interested mostly in dynamical systems confined to a compact phase space in which the RP resonances are believed to depend on the fine structure of the mixing process as opposed to the diffusion in phase space. Namely, for a large area preserving maps on the cylinder (an infinite phase-space) it was shown in [6,7] that there exists a strict connection between the resonances and the diffusion process.…”

Section: Preliminariesmentioning

confidence: 99%