Validated Numerics for Continuation and Bifurcation of Connecting Orbits of Maps

Adams, Ronald; James, J. D. Mireles

doi:10.1007/s12346-018-0279-y

Cited by 3 publications

(2 citation statements)

References 36 publications

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“…While the transversality results of the present work imply that the homoclinic connections persist for small enough changes in the masses of the primaries, numerical simulations suggest that the homoclinic connections are typical. Recent work in [71,72] develop methods for mathematically rigorous computer assisted existence proofs of continuation and bifurcation of connecting orbits. These techniques could be used to study global branches of homoclinic connections for the CRTBP over a large range of parameters.…”

Section: Discussionmentioning

confidence: 99%

Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem

Kepley

James

2019

Journal of Differential Equations

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We prove the existence of chaotic motions in a planar restricted four body problem, establishing that the system is not integrable. The idea of the proof is to verify the hypotheses of a topological forcing theorem. The forcing theorem applies to two freedom Hamiltonian systems where the stable and unstable manifolds of a saddle-focus equilibrium intersect transversally in the energy level set of the equilibrium. We develop a mathematically rigorous computer assisted argument which verifies the hypotheses of the forcing theorem, and we implement our approach for the restricted four body problem. The method is constructive and works far from any perturbative regime and for non-equal masses. Being constructive, our argument results in useful byproducts like information about the locations of transverse connecting orbits, quantitative information about the invariant manifolds, and upper/lower bounds on transport times.

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Section: Discussionmentioning

confidence: 99%

Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem

Kepley

James

2019

Journal of Differential Equations

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“…We remark that one parameter families of connecting orbits have been studied using computer assisted methods of proof, see for example [48]. Computer assisted proofs of bifurcations of connecting orbits for maps were studied in [49], and there is reason to believe that these techniques could be extended to differential equations. We further mention that computer assisted methods of proof have been devised for studying center manifolds [50] in celestial mechanics problems.…”

Section: Conjecturementioning

confidence: 99%

Critical homoclinics in a restricted four body problem: numerical continuation and center manifold computations

Hetebrij,

James

2020

Preprint

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The present work studies the robustness of certain basic homoclinic motions in an equilateral restricted four body problem. The problem can be viewed as a two parameter family of conservative autonomous vector fields. The main tools are numerical continuation techniques for homoclinic and periodic orbits, as well as formal series methods for computing normal forms and center stable/unstable manifold parameterizations. After careful numerical study of a number of special cases we formulate several conjectures about the global bifurcations of the homoclinic families.Keywords 4-body problem • homoclinic dynamics • critical equilibria • center manifold parameterization

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Critical homoclinics in a restricted four-body problem: numerical continuation and center manifold computations

Hetebrij

James

2021

Celest Mech Dyn Astr

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Validated Numerics for Continuation and Bifurcation of Connecting Orbits of Maps

Cited by 3 publications

References 36 publications

Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem

Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem

Critical homoclinics in a restricted four body problem: numerical continuation and center manifold computations

Critical homoclinics in a restricted four-body problem: numerical continuation and center manifold computations

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