Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory

Elskens, Yves; Escande, Dominique

doi:10.1088/0951-7715/4/3/002

Cited by 89 publications

(93 citation statements)

References 21 publications

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“…Conversely, if the surface of section in (Υ, υ) plane reveals an area-filling manifold, conservation of J is broken as the orbit repeatedly encounters a separatrix in the (Ξ, ξ) plane (Henrard 1982a;Morbidelli 2002). Indeed the situation is quite analogous to the well-studied problem of a modulated pendulum (Elskens & Escande 1991;Bruhwiler & Cary 1989). A surface of section of the (Correia et al 2009) orbital solution is shown as a thick purple line in Fig.…”

Section: Adiabatic Evolutionmentioning

confidence: 91%

Analytical treatment of planetary resonances

Batygin

Morbidelli

2013

A&A

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An ever-growing observational aggregate of extrasolar planets has revealed that systems of planets that reside in or near mean-motion resonances are relatively common. While the origin of such systems is attributed to protoplanetary disk-driven migration, a qualitative description of the dynamical evolution of resonant planets remains largely elusive. Aided by the pioneering works of the last century, we formulate an approximate, integrable theory for first-order resonant motion. We utilize the developed theory to construct an intuitive, geometrical representation of resonances within the context of the unrestricted three-body problem. Moreover, we derive a simple analytical criterion for the appearance of secondary resonances between resonant and secular motion. Subsequently, we demonstrate the onset of rapid chaotic motion as a result of overlap among neighboring first-order mean-motion resonances, as well as the appearance of slow chaos as a result of secular modulation of the planetary orbits. Finally, we take advantage of the integrable theory to analytically show that, in the adiabatic regime, divergent encounters with first-order mean-motion resonances always lead to persistent apsidal anti-alignment.

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Section: Adiabatic Evolutionmentioning

confidence: 91%

Analytical treatment of planetary resonances

Batygin

Morbidelli

2013

A&A

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“…Statistical independence follows from the divergence of phases along trajectories. For separatrix crossings consecutive crossings for some initial conditions are statistically dependent (Cary and Skodje, 1989) and islands of stability, albeit being of a small measure, do exist (Elskens and Escande, 1991;Neishtadt et al, 1997) inside large chaotic see. Consecutive resonance crossings should be treated as independent as shows the following reasoning from (Neishtadt, 1999).…”

Section: The Long-time Behaviour Of the Particlesmentioning

confidence: 99%

Resonances and Particle Stochastization in Nonhomogeneous Electromagnetic Fields

Vainchtein

Rovinsky

Zelenyǐ

et al. 2004

Journal of Nonlinear Science

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In the present paper we investigate the resonant interaction between monochromatic electromagnetic waves and charged particles in configurations with magnetic field reversals (e.g. in the earth magnetotail). The smallness of certain physical parameters allows us to solve this problem using perturbation theory reducing the problem of resonant wave-particle interaction to the analysis of slow passages of a particle through a resonance.We discuss in details two of the most important resonant phenomena: capture into resonance and scattering on resonance. We show that these processes result in destruction of the adiabatic invariants, chaotization of particles, may lead to significant (almost free) acceleration of particles and govern transport in the phase space.We calculate the characteristic times of mixing due to resonant effects and separatrix crossings and discuss the relative importance of these phenomena.

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“…when τ discr ≫ τ spread , numerical calculations revealed [30] that after a time τ s ∼ τ spread , ⟨∆v 2 (t)⟩ grows with a slope in between the quasilinear one and 2.3 times this value 42 (in the range 40 In the opposite limit when τ spread /τac is small, the time evolution of the waves is slow with respect to the trapping motion in the instantaneous wave potential. Then chaotic dynamics may be described in an adiabatic way with the picture of a slowly pulsating separatrix [46,47] (see also section 5.5 of [48] and 14.5.2 of [54]). In this limit, for the case of the motion in two waves, the resonance overlap defined hereafter is large.…”

Section: Diffusion In a Given Spectrum Of Wavesmentioning

confidence: 99%

How to Face the Complexity of Plasmas?

Escande

2013

Nonlinear Systems and Complexity

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This paper has two main parts. The first part is subjective and aims at favoring a brainstorming in the plasma community. It discusses the present theoretical description of plasmas, with a focus on hot weakly collisional plasmas. It comprises two sub-parts. The first one deals with the present status of this description. In particular, most models used in plasma physics are shown to have feet of clay, there is no strict hierarchy between them, and a principle of simplicity dominates the modeling activity. At any moment the description of plasma complexity is provisional and results from a collective and somewhat unconscious process. The second sub-part considers possible methodological improvements, some of them specific to plasma physics, some others of possible interest for other fields of science. The proposals for improving the present situation go along the following lines: improving the way papers are structured and the way scientific quality is assessed in the referral process, developing new data bases, stimulating the scientific discussion of published results, diversifying the way results are made available, assessing more quality than quantity, making available an incompressible time for creative thinking and non purpose-oriented research. Some possible improvements for teaching are also indicated. The suggested improvement of the structure of papers would be for each paper to have a "claim section" summarizing the main results and their most relevant connection to previous literature. One of the ideas put forward is that modern nonlinear dynamics and chaos might help revisiting and unifying the overall presentation of plasma physics.The second part of this chapter is devoted to one instance where this idea has been developed for three decades: the description of Langmuir wave-electron interaction in one-dimensional plasmas by a finite dimensional Hamiltonian. This part is more specialized, and is written like a classical scientific paper. This Hamiltonian approach enables recovering Vlasovian linear theory with a mechanical understanding. The quasilinear description of the weak warm beam is discussed, and it is shown that self-consistency vanishes when the plateau forms in the tail distribution function. This leads to consider the various diffusive regimes of the dynamics of particles in a frozen spectrum of waves with random phases. A recent numerical simulation showed that diffusion is quasilinear when the plateau sets in, and that the variation of the phase of a given wave with time is almost non fluctuating for random realizations of the initial wave phases. This led to new analytical calculations of the average behavior of the self-consistent dynamics when the initial wave phases are random. Using Picard iteration technique, they confirm numerical results, and exhibit a spontaneous emission of spatial inhomogeneities.Non quia difficilia sunt, non audemus, sed quia non audemus, difficilia sunt. It is not because things are difficult that we do not dare, it is because we do not dare that things a...

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Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory

Cited by 89 publications

References 21 publications

Analytical treatment of planetary resonances

Analytical treatment of planetary resonances

Resonances and Particle Stochastization in Nonhomogeneous Electromagnetic Fields

How to Face the Complexity of Plasmas?

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