A Geometric Approach to the Existence of Orbits with Unbounded Energy in Generic Periodic Perturbations by a Potential of Generic Geodesic Flows of ?2}

Delshams, Amadeu; Llave, Rafael de la; Seara, Tere M.

doi:10.1007/pl00020961

Cited by 110 publications

(270 citation statements)

References 14 publications

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“…However, if we have an in nite sequence, we can give an argument t o p r o ve that the set is non-void. These arguments are also used in 3,7] to nd orbits with unbounded energy in a class of time dependent analytic Hamiltonian systems. Transition chains connecting two neighborhoods of the phase space, if they can be found, are far from being uniquely determined.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian systems with orbits covering densely submanifolds of small codimension

Fontich

Martín

2003

Nonlinear Analysis: Theory, Methods & Applications

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The existence of a transition chain in a Hamiltonian system leads to the existence of orbits shadowing it, if some lambda lemma can be applied. This fact has been used to prove the existence of di usion in perturbations of integrable a priori stable systems. We prove that, under suitable conditions there are orbits which c o ver densely codimension two submanifolds of the (2n ; 1)-dimensional energy level. We also provide an example where the computations to verify the su cient conditions for the appearance of such phenomenon can be performed. We comment on Arnol'd example, which i s covered by this set of hypotheses.

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Section: Introductionmentioning

confidence: 99%

Hamiltonian systems with orbits covering densely submanifolds of small codimension

Fontich

Martín

2003

Nonlinear Analysis: Theory, Methods & Applications

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“…The criteria for the existence of trajectories of the energy that grows up to infinity are known for sufficiently large initial energies [10,24,29,30,44,45,80,81]. The results of the present paper can be used to establish the generic existence of orbits of unbounded energy for all possible values of initial energy.…”

Section: Introductionmentioning

confidence: 83%

“…Then by differentiating (10), (11), we will obtain the inequalities (29), hence the estimate (36) holds at all sufficiently large k for the newly defined γ i , β i . However, instead of (31) we have now…”

Section: Indeed In This Normmentioning

confidence: 99%

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Arnold Diffusion in A Priori Chaotic Symplectic Maps

Gelfreich

Turaev

2017

Commun. Math. Phys.

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“…The instability mechanism shown in this paper is related to a generalized version of Mather's acceleration problem [Mat96,BT99,DdlLS00,GT08,Kal03,Pif06]. Some parts of the proof rely on numerical computations, but our strategy allows us to keep these computations simple and convincing.…”

Section: The Problem Of the Stability Of Gravitating Bodiesmentioning

confidence: 99%

Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem

Fejoz¹,

Guàrdia²,

Калошин³

et al. 2016

J. Eur. Math. Soc.

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We study the dynamics of the restricted planar three-body problem near mean motion resonances, i.e. a resonance involving the Keplerian periods of the two lighter bodies revolving around the most massive one. This problem is often used to model Sun-Jupiter-asteroid systems. For the primaries (Sun and Jupiter), we pick a realistic mass ratio µ = 10 −3 and a small eccentricity e0 > 0. The main result is a construction of a variety of non local diffusing orbits which show a drastic change of the osculating (instant) eccentricity of the asteroid, while the osculating semi major axis is kept almost constant. The proof relies on the careful analysis of the circular problem, which has a hyperbolic structure, but for which diffusion is prevented by KAM tori. In the proof we verify certain nondegeneracy conditions numerically.Based on the work of Treschev, it is natural to conjecture that the time of diffusion for this problem is ∼ − ln(µe 0 ) µ 3/2 e 0 . We expect our instability mechanism to apply to realistic values of e0 and we give heuristic arguments in its favor. If so, the applicability of Nekhoroshev theory to the three-body problem as well as the long time stability become questionable.It is well known that, in the Asteroid Belt, located between the orbits of Mars and Jupiter, the distribution of asteroids has the so-called Kirkwood gaps exactly at mean motion resonances of low order. Our mechanism gives a possible explanation of their existence. To relate the existence of Kirkwood gaps with Arnol'd diffusion, we also state a conjecture on its existence for a typical ε-perturbation of the product of the pendulum and the rotator. Namely, we predict that a positive conditional measure of initial conditions concentrated in the main resonance exhibits Arnol'd diffusion on time scales − ln ε ε 2 .

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A Geometric Approach to the Existence of Orbits with Unbounded Energy in Generic Periodic Perturbations by a Potential of Generic Geodesic Flows of ?2}

Cited by 110 publications

References 14 publications

Hamiltonian systems with orbits covering densely submanifolds of small codimension

Hamiltonian systems with orbits covering densely submanifolds of small codimension

Arnold Diffusion in A Priori Chaotic Symplectic Maps

Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem

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