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

Delshams, Amadeu; Gidea, Marian; Roldán, Pablo

doi:10.1016/j.physd.2016.06.005

Cited by 22 publications

(30 citation statements)

References 51 publications

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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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“…To the best of our knowledge, this is the first regime of motions of the problem of three bodies naturally modelling a region in the Solar system, where nonlocal transition chains are established 2 . Previous results showing transition chains of tori in the problem of three bodies naturally modelling a region in the Solar system are confined to small neighborhoods of the Lagrangian Equilibrium points [CZ11,DGR11], and therefore, are local in the Configuration and Phase space.…”

Section: The Problem Of the Stability Of Gravitating Bodiesmentioning

confidence: 95%

“…A somewhat opposite strategy was developed by Bolotin and McKay, using the Poincaré orbits of the second species to show the existence of symbolic dynamics in the threebody problem, hence of chaotic orbits, but considering far from integrable, non-planetary conditions; see for example [Bol06]. Also, Delshams, Gidea and Roldán have shown an instability mechanism in the spatial restricted three-body problem, but only locally around the equilibrium point L 1 (see [DGR11]). …”

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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Global Instability in the Restricted Planar Elliptic Three Body Problem

Delshams

Калошин

Rosa

et al. 2018

Commun. Math. Phys.

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The restricted planar elliptic three body problem (RPETBP) describes the motion of a massless particle (a comet or an asteroid) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter) revolving around their center of mass on elliptic orbits with some positive eccentricity. The aim of this paper is to show the existence of orbits whose angular momentum performs arbitrary excursions in a large region. In particular, there exist diffusive orbits, that is, with a large variation of angular momentum.The leading idea of the proof consists in analyzing parabolic motions of the comet. By a well-known result of McGehee, the union of future (resp. past) parabolic orbits is an analytic manifold P + (resp. P − ). In a properly chosen coordinate system these manifolds are stable (resp. unstable) manifolds of a manifold at infinity P∞, which we call the manifold at parabolic infinity.On P∞ it is possible to define two scattering maps, which contain the map structure of the homoclinic trajectories to it, i.e. orbits parabolic both in the future and the past. Since the inner dynamics inside P∞ is trivial, two different scattering maps are used. The combination of these two scattering maps permits the design of the desired diffusive pseudo-orbits. Using shadowing techniques and these pseudo orbits we show the existence of true trajectories of the RPETBP whose angular momentum varies in any predetermined fashion.2000 Mathematics Subject Classification: Primary 37J40, 70F15. Keywords: Elliptic Restricted Three Body problem, parabolic motions, Manifold of parabolic motions at infinity, Arnold diffusion, splitting of separatrices, Melnikov integral. Main result and methodologyThe restricted planar elliptic three body problem (RPETBP) describes the motion q of a massless particle (a comet) under the gravitational field of two massive bodies (the primaries, say the Sun and Jupiter ) with mass ratio µ revolving around their center of mass on elliptic orbits with eccentricity J . In this paper we search for trajectories of motion which show a large variation of the angular momentum G = q ×q. In other words, we search for global instability ("diffusion"This is one of the reasons why we are going to restrict ourselves to the region G ≥ C large enough and J G ≤ c small enough along this paper to get the diffusive orbits.Among the harmonics L 0, of 0 order in s, by (42), the harmonic L 0,0 appears to be the dominant one, but we will also estimate L 0,1 to get information about the variable α, and bound the rest of harmonics L 0, for ≥ 2. Among the harmonics of first order L 1,k , again by (42), the and the error functions satisfyTo bound the integral (81) for m ≥ 0 we will consider two different cases: 0 ≤ q ≤ m and 0 ≤ m < q. Let us first consider the case 0 ≤ q ≤ m. By the analyticity and periodicity of the integral we change the path of integration from (E) = 0 to E = ln(2a 2 / J )so that e iE = e iu−ln(2a 2 / J) = J 2a 2 e iu and then, by (78), (79) and (83a), (83b), 23Therefore along the complex path...

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

Cited by 22 publications

References 51 publications

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

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

Global Instability in the Restricted Planar Elliptic Three Body Problem

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