Geometry of Weak Stability Boundaries

Belbruno, Edward; Gidea, Marian; Topputo, Francesco

doi:10.1007/s12346-012-0069-x

Cited by 17 publications

(16 citation statements)

References 11 publications

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“…When Î P S R 0 2 for = t t 1 , this implies it lies in the three-dimensional region bounded by M u 1 . Moreover, for > t t 1 , due to the separatrix property, P 0 stays within this region inside M u 1 for all time moving forward [21]. This manifold stays within a bounded region, 1 M , bounded by the following: S R 2 , the boundary of H 1 (a zero velocity curve), and a two-dimensional McGehee torus, T M , about P 1 [18,21] 6 .…”

Section: Whenmentioning

confidence: 94%

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Relation between solutions of the Schrödinger equation with transitioning resonance solutions of the gravitational three-body problem

Belbruno

2020

J. Phys. Commun.

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It is shown that a class of approximate resonance solutions in the three-body problem under the Newtonian gravitational force are equivalent to quantized solutions of a modified Schrödinger equation for a wide range of masses that transition between energy states. In the macroscopic scale, the resonance solutions are shown to transition from one resonance type to another through weak capture at one of the bodies, while in the Schrödinger equation, one obtains quantized wave solutions transitioning between different energies. The resonance transition dynamics provides a classical model of a particle moving between different energy states in the Schrödinger equation. This methodology provides a connection between celestial and quantum mechanics.

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Section: Whenmentioning

confidence: 94%

“…A purely analytic proof for the global manifold stricture about P 2 and  k is not available at this time. However, in the case of motion about P 1 , the analogous structure of  k is analytically proven [21].…”

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confidence: 99%

Relation between solutions of the Schrödinger equation with transitioning resonance solutions of the gravitational three-body problem

Belbruno

2020

J. Phys. Commun.

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“…Much work on studying transfer dynamics in celestial mechanics exploits these tools. See for example [2,3,4,8,9,10,11,12,33,40,41]. The study of invariant manifolds for periodic orbits also plays a role in the study of biological and chemical oscillations, and we refer for example to the work of [15,13,17,16,72].…”

Section: Related Workmentioning

confidence: 99%

Parameterization of Invariant Manifolds for Periodic Orbits I: Efficient Numerics via the Floquet Normal Form

Castelli

Lessard

James

2015

SIAM J. Appl. Dyn. Syst.

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We present an efficient numerical method for computing Fourier-Taylor expansions of stable/unstable manifolds associated with hyperbolic periodic orbits. Three features of the method are (1) that we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) that the method admits natural a-posteriori error analysis, and (3) that the method does not require numerical integrating the vector field. Our method is based on the Parameterization Method for invariant manifolds, and studies a certain partial differential equation which characterizes a chart map of the (un)stable manifold. The method requires only that some mild non-resonance conditions hold between the Floquet multipliers of the periodic orbit. The novelty of the the present work is that we exploit the Floquet normal form in order to efficiently compute the Fourier-Taylor expansion. We present a number of example computations, including stable/unstable manifolds in phase space dimension as hight as ten, computation of manifolds which are two and three dimensional, and computation of some homoclinic connecting orbits.

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“…Such symbolic dynamics has been proved using rigorous computer assisted computations by Wilczak and Zgliczyński [23,24]. In recent papers of Belbruno, Gidea, and Topputo [4,5] it is shown that the weak stability boundary method and the invariant manifold method coincide.…”

Section: Introductionmentioning

confidence: 97%

Computer Assisted Existence Proofs of Lyapunov Orbits at $L_2$ and Transversal Intersections of Invariant Manifolds in the Jupiter--Sun PCR3BP

Capiński¹

2012

SIAM J. Appl. Dyn. Syst.

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We present a computer assisted proof of existence of a family of Lyapunov orbits which stretches from L2 (the collinear libration point between the primaries) up to half the distance to the smaller primary in the Jupiter-Sun planar circular restricted three body problem. We then focus on a small family of Lyapunov orbits with energies close to comet Oterma and show that their associated invariant manifolds intersect transversally. Our computer assisted proof provides explicit bounds on the location and on the angle of intersection.

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Geometry of Weak Stability Boundaries

Cited by 17 publications

References 11 publications

Relation between solutions of the Schrödinger equation with transitioning resonance solutions of the gravitational three-body problem

Relation between solutions of the Schrödinger equation with transitioning resonance solutions of the gravitational three-body problem

Parameterization of Invariant Manifolds for Periodic Orbits I: Efficient Numerics via the Floquet Normal Form

Computer Assisted Existence Proofs of Lyapunov Orbits at $L_2$ and Transversal Intersections of Invariant Manifolds in the Jupiter--Sun PCR3BP

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