Effect of Islands in Diffusive Properties of the Standard Map for Large Parameter Values

Miguel, Narcís; Simó, Carles; Vieiro, A.

doi:10.1007/s10208-014-9210-3

Cited by 19 publications

(45 citation statements)

References 45 publications

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“…It can be interesting to make a comment on the accelerator modes of the classical SM κ , i.e., points that in the SM κ considered in T 2 are periodic elliptic, but looking at them in S × R 1 they produce islands whose action variable (w in (15)) increases (or decreases) monotonically. For suitable ranges of κ they play a relevant role in the anomalous diffusion properties of SM κ , see [21]. In the SepM the action variable in one "layer" (see Fig.…”

Section: 2 To Obtainmentioning

confidence: 99%

“…Additional examples can be found in [29,30]. For an study of domains of rather chaotic behaviour and anomalous diffusion properties one can refer to [21].…”

Section: Introductionmentioning

confidence: 98%

“…vertical) variable. Different structures like elliptic and hyperbolic fixed points, islands, IRC and chaotic domains, can be seen Proof To recover the format of (4) one has to replace h in (5) by 2|h|, to multiply the time by 4 because of the denominator in the expression of p in(21) and use the fact that c 1,2,4 = 2 ∞ 0 (1 + η 2 ) −2 dη = π/2. The eccentric anomaly E differs from t by O(e) thanks to Kepler equation.…”

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confidence: 99%

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Return Maps, Dynamical Consequences and Applications

Martínez

Simó

2015

Qual. Theory Dyn. Syst.

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After reviewing some general settings for return maps in problems reducible to 2D symplectic maps, details on the construction of return maps are presented. Different forms of such maps close to splitted separatrices (separatrix maps) are introduced, taking into account the size and shape of the splitting function and also the return time to the domains of interest. Then it is shown how to derive approximations by suitable standard-like maps. Dynamical consequences concerning the existence of invariant rotational curves (IRC) are derived. An application is made to theoretically estimate the location of the outermost IRC in the Sitnikov problem, which is in good agreement with numerical data. To compare with the cases which are approximated by the classical standard map, some details on the properties of the standard-like map with two harmonic terms are included. Finally a method to estimate the amount of chaos depending on the form of the separatrix map is introduced. Except otherwise stated all the systems we consider are assumed to be analytic, despite several of the properties we study are no longer analytic.

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Section: 2 To Obtainmentioning

confidence: 99%

“…Additional examples can be found in [29,30]. For an study of domains of rather chaotic behaviour and anomalous diffusion properties one can refer to [21].…”

Section: Introductionmentioning

confidence: 98%

mentioning

confidence: 99%

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Return Maps, Dynamical Consequences and Applications

Martínez

Simó

2015

Qual. Theory Dyn. Syst.

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“…This well known discrete area preserving dynamical system, introduced by Chirikov ([4], [5]) and largely studied in the last four decades (see for instance [27] and references therein), (I, θ) → (I ′ , θ ′ ), is defined by…”

Section: The Standard Mapmentioning

confidence: 99%

“…Left panel) Evolution of I for the phase τ ∈ (0, 1) corresponding to the values of t when the the pendulum crosses the surface θ 1 = π for np = 100 random initial values on the separatrix of the resonance ω 1 = 0 for the Hamiltonian (21) with ε = 0.25, µ = 0.1ε and seven values of ω 0 2 = κ √ The analytical expected behavior for random and ergodic motion given by(27) are also included for N = βnp × t/δt, with β ≈ 3 × 10 −4 . (Right panel) Similar to the plot at the left but for the phase θ 2 ∈ (0, 1).…”

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confidence: 99%

Phase correlations in chaotic dynamics: a Shannon entropy measure

Cincotta

Giordano

2018

Celest Mech Dyn Astr

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In the present work we investigate phase correlations by recourse to the Shannon entropy. Using theoretical arguments we show that the entropy provides an accurate measure of phase correlations in any dynamical system, in particular when dealing with a chaotic diffusion process. We apply this approach to different low dimensional maps in order to show that indeed the entropy is very sensitive to the presence of correlations among the successive values of angular variables, even when it is weak. Later on, we apply this approach to unveil strong correlations in the time evolution of the phases involved in the Arnold's Hamiltonian that lead to anomalous diffusion, particularly when the perturbation parameters are comparatively large. The obtained results allow us to discuss the validity of several approximations and assumptions usually introduced to derive a local diffusion coefficient in multidimensional near-integrable Hamiltonian systems.

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On the chaotic diffusion in multidimensional Hamiltonian systems

Cincotta

Giordano

Marti

et al. 2018

Celest Mech Dyn Astr

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We present numerical evidence that diffusion in the herein studied multidimensional near-integrable Hamiltonian systems departs from a normal process, at least for realistic timescales. Therefore, the derivation of a diffusion coefficient from a linear fit on the variance evolution of the unperturbed integrals fails. We review some topics on diffusion in the Arnold Hamiltonian and yield numerical and theoretical arguments to show that in the examples we considered, a standard coefficient would not provide a good estimation of the speed of diffusion. However, numerical experiments concerning diffusion would provide reliable information about the stability of the motion within chaotic regions of the phase space. In this direction, we present an extension of previous results concerning the dynamical structure of the Laplace resonance in Gliese-876 planetary system considering variations of the orbital parameters accordingly to the error introduced by the radial velocity determination. We found that a slight variation of the eccentricity of planet c would destabilize the inner region of the resonance that, though chaotic, shows stable when adopting the best fit values for the parameters.

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Effect of Islands in Diffusive Properties of the Standard Map for Large Parameter Values

Cited by 19 publications

References 45 publications

Return Maps, Dynamical Consequences and Applications

Return Maps, Dynamical Consequences and Applications

Phase correlations in chaotic dynamics: a Shannon entropy measure

On the chaotic diffusion in multidimensional Hamiltonian systems

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