Arnold diffusion in the planar elliptic restricted three-body problem: mechanism and numerical verification

Capiński, Maciej J.; Gidea, Marian; Llave, Rafael de la

doi:10.1088/1361-6544/30/1/329

Cited by 26 publications

(33 citation statements)

References 48 publications

(118 reference statements)

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“…(4) Our method can be applied to concrete systems-e.g., the planar elliptic restricted three-body problem, and the spatial circular restricted three-body problem-and, further, can be implemented in computer-assisted proofs. See the related papers [19,40,44]. (5) Although the main application in this paper is on diffusion in a priori unstable systems, we expect that this method can be useful when applied to a priori stable systems, as well as to infinite-dimensional systems, once the existence of suitable normally hyperbolic invariant manifolds (called normally hyperbolic cylinders in [6,68,69,75]) and their homoclinic channels is established.…”

Section: Brief Description Of the Main Resultsmentioning

confidence: 99%

“…In many examples, the scattering map can be computed explicitly via perturbation theory [31,34,35] or numerically [18,19,39,40].…”

Section: Normally Hyperbolic Invariant Manifolds and Scattering Mapsmentioning

confidence: 99%

“…In applications, using several scattering maps rather than a single one can be very advantageous. In astrodynamics, for example, the existence of multiple homoclinic intersections can be exploited to obtain diffusion [43,47] and to increase the versatility of space missions; see, e.g., [19,40].…”

Section: Shadowing Of Pseudo-orbits Of the Scattering Mapmentioning

confidence: 99%

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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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Section: Brief Description Of the Main Resultsmentioning

confidence: 99%

“…In many examples, the scattering map can be computed explicitly via perturbation theory [31,34,35] or numerically [18,19,39,40].…”

Section: Normally Hyperbolic Invariant Manifolds and Scattering Mapsmentioning

confidence: 99%

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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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“…Mechanisms based on NHIM's can be verified in concrete systems via nonperturbative methods -e.g., numerical computations [9,23,8,10,40] -, and can be applied in astrodynamics and space mission design.…”

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confidence: 99%

A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold

Gidea¹,

Llave²,

Seara³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We describe a recent method to show instability in Hamiltonian systems. The main hypothesis of the method is that some explicit transversality conditions-which can be verified in concrete systems by finite calculations-are satisfied. In particular, for several types of perturbations of integrable Hamiltonian systems, the hypothesis can be verified by just checking that some Melnikovtype integrals have non-degenerate zeros. This holds for Baire generic sets of perturbations in the C r-topology, for r ∈ [3, ∞) ∪ {ω}. Our method does not require that the unperturbed Hamiltonian system is convex, or that the perturbation is polynomial, which are non-generic properties. Provided that the transversality conditions are verified, one concludes the existence of orbits which change the action coordinate by a quantity independent of the size of the perturbation. In fact, one can obtain orbits that follow any path in action space, up to an error decreasing with the size of the perturbation.

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“…The proof of Theorem 2.4, given in Section 3, relies on the theory of normal hyperbolicity, on the geometric properties of scattering maps, and on the Poincaré recurrence theorem. Relevant references to our approach include [6,22,21,17,33,7].…”

Section: Methodsmentioning

confidence: 99%

Arnold’s mechanism of diffusion in the spatial circular restricted three-body problem: A semi-analytical argument

Delshams¹,

Gidea

Roldán

2016

Physica D: Nonlinear Phenomena

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We consider the spatial circular restricted three-body problem, on the motion of an infinitesimal body under the gravity of Sun and Earth. This can be described by a 3-degree of freedom Hamiltonian system. We fix an energy level close to that of the collinear libration point L 1 , located between Sun and Earth. Near L 1 there exists a normally hyperbolic invariant manifold, diffeomorphic to a 3-sphere. For an orbit confined to this 3-sphere, the amplitude of the motion relative to the ecliptic (the plane of the orbits of Sun and Earth) can vary only slightly. We show that we can obtain new orbits whose amplitude of motion relative to the ecliptic changes significantly, by following orbits of the flow restricted to the 3-sphere alternatively with homoclinic orbits that turn around the Earth. We provide an abstract theorem for the existence of such 'diffusing' orbits, and numerical evidence that the premises of the theorem are satisfied in the three-body problem considered here. We provide an explicit construction of diffusing orbits. The geometric mechanism underlying this construction is reminiscent of the Arnold diffusion problem for Hamiltonian systems. Our argument, however, does not involve transition chains of tori as in the classical example of Arnold. We exploit mostly the 'outer dynamics' along homoclinic orbits, and use very little information on the 'inner dynamics' restricted to the 3-sphere. As a possible application to astrodynamics, diffusing orbits as above can be used to design low cost maneuvers to change the inclination of an orbit of a satellite near L 1 from a nearly-planar orbit to a tilted orbit with respect to the ecliptic. We explore different energy levels, and estimate the largest orbital inclination that can be achieved through our construction.

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Arnold diffusion in the planar elliptic restricted three-body problem: mechanism and numerical verification

Cited by 26 publications

References 48 publications

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold

Arnold’s mechanism of diffusion in the spatial circular restricted three-body problem: A semi-analytical argument

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