Geometric properties of the scattering map of a normally hyperbolic invariant manifold

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

doi:10.1016/j.aim.2007.08.014

Cited by 102 publications

(295 citation statements)

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“…In Proposition 9.2 in [DLS06a] it is proved that if hypothesis H2' in Theorem 2.1 is satisfied, then the stable and unstable manifolds W s Λ ε and W u Λ ε of the NHIM intersect transversally along a homoclinic manifold Γ ε , which is also called a homoclinic channel (see [DLS08] for more details, in particular for the definition of the wave operators, needed for the construction of the scattering map). So, we will be able to locally define the scattering map associated to Γ ε and compute it in first order perturbation theory using the results in [DLS08].…”

Section: Second Part: Outer Dynamicsmentioning

confidence: 99%

“…The first two parts follow readily from [DLS06a] and Theorems stated in [DLS06a] apply straightforwardly because hypotheses H1 and H2' required for the proof of the mentioned results are the same as in our case. Moreover, for the second part we use the symplectic properties developed in [DLS08] to generalize the computation of the scattering map using its Hamiltonian function. So, for these parts, we only refer in Section 2.3.1 and 2.3.2, to the results in [DLS06a] and [DLS08] that we are using.…”

Section: Partmentioning

confidence: 99%

“…Moreover, for the second part we use the symplectic properties developed in [DLS08] to generalize the computation of the scattering map using its Hamiltonian function. So, for these parts, we only refer in Section 2.3.1 and 2.3.2, to the results in [DLS06a] and [DLS08] that we are using.…”

Section: Partmentioning

confidence: 99%

“…We would like to remark that, in contrast to [DLS06a], and thanks to the new results about the scattering map obtained in [DLS08], we use the Hamiltonian function generating the deformation of the scattering map instead of the scattering map itself, in order to compute the images of the leaves of a certain foliation under the scattering map.…”

Section: Introductionmentioning

confidence: 99%

“…The strategy in the mentioned papers is based on the incorporation of new invariant objects, created by the resonances, like secondary KAM tori and the stable and unstable manifolds of lower dimensional tori in the transition chain, together with the primary KAM tori. The scattering map, introduced by the same authors (see [DLS08] for a geometric study) is the essential tool for the heteroclinic connections between invariant objects of different topology.…”

Section: Introductionmentioning

confidence: 99%

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Geography of resonances and Arnold diffusion ina prioriunstable Hamiltonian systems

Delshams

Huguet

2009

Nonlinearity

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Abstract. In the present paper we consider the case of a general C r+2 perturbation, for r large enough, of an a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom, and we provide explicit conditions on it, which turn out to be C 2 generic and are verifiable in concrete examples, which guarantee the existence of Arnold diffusion. This is a generalization of the result in Delshams et al., Mem. Amer. Math. Soc., 2006, where it was considered the case of a perturbation with a finite number of harmonics in the angular variables.The method of proof is based on a careful analysis of the geography of resonances created by a generic perturbation and it contains a deep quantitative description of the invariant objects generated by the resonances therein. The scattering map is used as an essential tool to construct transition chains of objects of different topology. The combination of quantitative expressions for both the geography of resonances and the scattering map provides, in a natural way, explicit computable conditions for instability.

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Section: Second Part: Outer Dynamicsmentioning

confidence: 99%

Section: Partmentioning

confidence: 99%

Section: Partmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Geography of resonances and Arnold diffusion ina prioriunstable Hamiltonian systems

Delshams

Huguet

2009

Nonlinearity

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A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

Gidea

Llave

Seara

2019

Comm Pure Appl Math

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We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach is based on following the “outer dynamics” along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined “scattering map.” We show that for every finite sequence of successive iterations of the scattering map, there exists a true orbit that follows that sequence, provided that the inner dynamics is recurrent. We apply this result to prove the existence of diffusing orbits that cross large gaps in a priori unstable models of arbitrary degrees of freedom, when the unperturbed Hamiltonian is not necessarily convex and the induced inner dynamics is not necessarily a twist map, and the perturbation satisfies explicit conditions that are generic. We also mention several other applications where this mechanism is easy to verify (analytically or numerically), such as the planar elliptic restricted three‐body problem and the spatial circular restricted three‐body problem. Our method differs, in several crucial aspects, from earlier works. Unlike the well‐known “two‐dynamics” approach, the method we present here relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, such as on existence of its invariant objects (e.g., primary and secondary tori, lower‐dimensional hyperbolic tori, and their stable/unstable manifolds, Aubry‐Mather sets), which are not used at all. © 2019 Wiley Periodicals, Inc.

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Arnold Diffusion, Quantitative Estimates, and Stochastic Behavior in the Three‐Body Problem

Capiński

Gidea

2021

Comm Pure Appl Math

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We consider a class of autonomous Hamiltonian systems subject to small, time‐periodic perturbations. When the perturbation parameter is set to zero, the energy of the system is preserved. This is no longer the case when the perturbation parameter is non‐zero. We describe a topological method to establish orbits which diffuse in energy for every suitably small perturbation parameter ε>0. The method yields quantitative estimates: (i)the existence of orbits along which the energy drifts by an amount independent of ε; the time required by such orbits to drift is O()1/ε; (ii)the existence of orbits along which the energy makes chaotic excursions; (iii)explicit estimates for the Hausdorff dimension of the set of such chaotic orbits; (iv)the existence of orbits along which the time evolution of energy approaches a stopped diffusion process (Brownian motion with drift), as ε tends to 0. For each ε fixed, the set of initial conditions of the orbits that yield the diffusion process has positive Lebesgue measure, and in the limit the measure of these sets approaches 0. Moreover, we can obtain any desired values of the drift and variance for the limiting Brownian motion for appropriate sets of initial conditions. A key feature of our topological method is that it can be implemented in computer‐assisted proofs. We give an application to a concrete model of the planar elliptic restricted three‐body problem, on the motion of an infinitesimal body relative to the Neptune‐Triton system. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

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Geometric properties of the scattering map of a normally hyperbolic invariant manifold

Cited by 102 publications

References 50 publications

Geography of resonances and Arnold diffusion ina prioriunstable Hamiltonian systems

Geography of resonances and Arnold diffusion ina prioriunstable Hamiltonian systems

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

Arnold Diffusion, Quantitative Estimates, and Stochastic Behavior in the Three‐Body Problem

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