Existence of a Center Manifold in a Practical Domain around $L_1$ in the Restricted Three-Body Problem

Capiński, Maciej J.; Roldán, Pablo

doi:10.1137/100810381

Cited by 37 publications

(31 citation statements)

References 11 publications

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“…These computations can be made mathematically rigorous and play a critical role in the computer assisted proof of the existence of the Lorenz attractor [77,78,79]. See also [86] for an application of mathematically rigorous computation for invariant manifolds in celestial mechanics based on normal forms. Normal forms are also useful for computing invariant tori, and are used in the numerical study of many problems coming from celestial mechanics [28,5,38,84,85].…”

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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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 refer to the work of [18,20,21,19,36,69] for a general theory of validated computation for stable/unstable (and other types of normally hyperbolic) invariant manifolds based on the topological notion of covering relations and cone conditions. The computer assisted topological arguments are carried out in phase space using polygonal elements.…”

Section: Computing Invariant Manifolds: a Brief Overviewmentioning

confidence: 99%

Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds

James

2015

Indagationes Mathematicae

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This work describes a method for approximating a branch of stable or unstable manifolds associated with a branch of hyperbolic fixed points or equilibria in a one parameter family of analytic dynamical systems. We approximate the branch of invariant manifolds by polynomials and develop a-posteriori theorems which provide mathematically rigorous bounds on the truncation error. The hypotheses of these theorems are formulated in terms of certain inequalities which are checked via a finite number of calculations on a digital computer. By exploiting the analytic category we are able to obtain mathematically rigorous bounds on the jets of the manifolds, as well as on the derivatives of the manifolds with respect to the parameter. A number of example computations are given.

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“…Using the above method it is impossible to continue with the orbits to L 2 . At the fixed point one would need to apply alternative methods, such as the method of majorants [21], the Lyapunov theorem by tracing the radius of convergence of the normal form [20], or topological computer assisted tools such as those in [8,10].…”

Section: Notationmentioning

confidence: 99%

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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Existence of a Center Manifold in a Practical Domain around $L_1$ in the Restricted Three-Body Problem

Cited by 37 publications

References 11 publications

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

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

Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds

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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