Spatial periodic orbits in the equilateral circular restricted four-body problem: computer-assisted proofs of existence

Burgos-Garcia, Jaime; Lessard, Jean‐Philippe; James, J. D. Mireles

doi:10.1007/s10569-018-9879-8

Cited by 23 publications

(31 citation statements)

References 61 publications

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“…In this section, we turn the non-polynomial DDE (13) into a higher dimensional DDE with polynomial nonlinearities, using the automatic differentiation technique as in [31,32,33].…”

Section: Automatic Differentiation: Obtaining a Polynomial Problemmentioning

confidence: 99%

“…This reduction greatly simplifies the analysis of the delay differential equation in Fourier space, but requires adding an unfolding parameter to balance the system. In addition we utilize automatic differentiation as in [31,32,33], and reformulate (5) as a problem with polynomial nonlinearities. The polynomial problem is amenable to straight forward analysis exploiting the Banach algebra properties of the solution space and the use of the FFT algorithm as in [34].…”

Section: Introductionmentioning

confidence: 99%

“…The relevant literature is rich and we direct the interested reader to the works of [44,45,16,46,47] for a much more complete view of the literature. Other authors have studied center manifolds [48], transverse intersections of stable/unstable manifolds [31,49], Melnikov theory [50], Arnold diffusion and transport [51,52,53], and existence/continuation/bifurcation of Halo orbits [32,54] -all in gravitational n-body problems and all using computer assisted arguments. Especially relevant to the present work are the computer assisted existence and KAM stability proofs for n-body choreographies in [13,14,15,16].…”

Section: Introductionmentioning

confidence: 99%

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Torus knot choreographies in the $n$-body problem

Calleja¹,

García-Azpeitia²,

Lessard³

et al. 2019

Preprint

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We develop a systematic approach for proving the existence of spatial choreographies in the gravitational n body problem. After changing to rotating coordinates and exploiting symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. We study periodic solutions of this DDE in a Banach space of rapidly decaying Fourier coefficients. Imposing appropriate constraint equations lets us isolate choreographies having prescribed symmetries and topological properties. Our argument is constructive and makes extensive use of the digital computer. We provide all the necessary analytic estimates as well as a working implementation for any number of bodies. We illustrate the utility of the approach by proving the existence of some spatial torus knot choreographies for n = 4, 5, 7, and 9 bodies.

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“…In this section, we turn the non-polynomial DDE (13) into a higher dimensional DDE with polynomial nonlinearities, using the automatic differentiation technique as in [31,32,33].…”

Section: Automatic Differentiation: Obtaining a Polynomial Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Torus knot choreographies in the $n$-body problem

Calleja¹,

García-Azpeitia²,

Lessard³

et al. 2019

Preprint

Self Cite

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“…The problem is the subject of many studies beginning with the work of [Simó, 1978]. Detailed analysis of the equilibrium solutions and their stability are found in [Barros and Leandro, 2014;Leandro, 2006;Baltagiannis and Papadakis, 2011b;Rusu and Santoprete, 2016;Alvarez-Ramírez and Delgado, 2003], while periodic orbits are studied in [Papadakis, 2016a;Baltagiannis and Papadakis, 2011a;Papadakis, 2016b;Burgos-García and Delgado, 2013;Burgos-García et al, 2017]. The earlier study of [Blazevski and Ocampo, 2012] considers stable/unstable manifolds attached to periodic orbits (using the linear approximation given by the stable/unstable normal bundles combined with numerical integration).…”

Section: A Circular Restricted Four Body Problemmentioning

confidence: 99%

Chebyshev–Taylor Parameterization of Stable/Unstable Manifolds for Periodic Orbits: Implementation and Applications

James

Murray

2017

Int. J. Bifurcation Chaos

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This paper develops Chebyshev-Taylor spectral methods for studying stable/unstable manifolds attached to periodic solutions of differential equations. The work exploits the parameterization method -a general functional analytic framework for studying invariant manifolds. Useful features of the parameterization method include the fact that it can follow folds in the embedding, recovers the dynamics on the manifold through a simple conjugacy, and admits a natural notion of a-posteriori error analysis. Our approach begins by deriving a recursive system of linear differential equations describing the Taylor coefficients of the invariant manifold. We represent periodic solutions of these equations as solutions of coupled systems of boundary value problems. We discuss the implementation and performance of the method for the Lorenz system, and for the planar circular restricted three and four body problems. We also illustrate the use of the method as a tool for computing cycle-to-cycle connecting orbits.

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“…For more "advanced" applications the effect of domain decomposition for Chebyshev series in computerassisted proofs easily gets convoluted with other technical aspects of the problem. However, we point out that our approach has been applied already to other problems, such as for heteroclinic travelling wave profiles in Lotka-Volterra systems and fourth order equations (originating from the Swift-Hohenberg PDE) [41], as well as for periodic orbits in the equilateral circular restricted four body problem [13].…”

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confidence: 99%

Rigorous numerics for ODEs using Chebyshev series and domain decomposition

Berg¹,

Sheombarsing²

2021

JCD

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<p style='text-indent:20px;'>In this paper we present a rigorous numerical method for validating analytic solutions of nonlinear ODEs by using Chebyshev-series and domain decomposition. The idea is to define a Newton-like operator, whose fixed points correspond to solutions of the ODE, on the space of geometrically decaying Chebyshev coefficients, and to use the so-called radii-polynomial approach to prove that the operator has an isolated fixed point in a small neighborhood of a numerical approximation. The novelty of the proposed method is the use of Chebyshev series in combination with domain decomposition. In particular, a heuristic procedure based on the theory of Chebyshev approximations for analytic functions is presented to construct efficient grids for validating solutions of boundary value problems. The effectiveness of the proposed method is demonstrated by validating long periodic and connecting orbits in the Lorenz system for which validation without domain decomposition is not feasible.</p>

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Spatial periodic orbits in the equilateral circular restricted four-body problem: computer-assisted proofs of existence

Cited by 23 publications

References 61 publications

Torus knot choreographies in the $n$-body problem

Torus knot choreographies in the $n$-body problem

Chebyshev–Taylor Parameterization of Stable/Unstable Manifolds for Periodic Orbits: Implementation and Applications

Rigorous numerics for ODEs using Chebyshev series and domain decomposition

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