Homoclinic dynamics in a restricted four-body problem: transverse connections for the saddle-focus equilibrium solution set

Kepley, Shane; James, J. D. Mireles

doi:10.1007/s10569-019-9890-8

Cited by 13 publications

(33 citation statements)

References 51 publications

(74 reference statements)

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“…Far from being competitors, the various techniques complement one another. See [71,72,73] for examples of calculations which combine the parameterization method in Step 1 with adaptive advection schemes in Step 2.…”

Section: Numerical Approximation Methods For Stable/unstable Manifoldsmentioning

confidence: 99%

Resonant tori, transport barriers, and chaos in a vector field with a Neimark–Sacker bifurcation

Fleurantin

James

2020

Communications in Nonlinear Science and Numerical Simulation

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We make a detailed numerical study of a three dimensional dissipative vector field derived from the normal form for a cusp-Hopf bifurcation. The vector field exhibits a Neimark-Sacker bifurcation giving rise to an attracting invariant torus. Our main goals are to (A) follow the torus via parameter continuation from its appearance to its disappearance, studying its dynamics between these events, and to (B) study the embeddings of the stable/unstable manifolds of the hyperbolic equilibrium solutions over this parameter range, focusing on their role as transport barriers and their participation in global bifurcations. Taken together the results highlight the main features of the global dynamics of the system.

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Section: Numerical Approximation Methods For Stable/unstable Manifoldsmentioning

confidence: 99%

Resonant tori, transport barriers, and chaos in a vector field with a Neimark–Sacker bifurcation

Fleurantin

James

2020

Communications in Nonlinear Science and Numerical Simulation

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“…Actually, there has been a considerable progress in understanding the basic but important aspects of this problem. We refer to the interested reader to the works [6], [15], [7], [5], [16], [17] and references therein for a deeper discussion in the dynamics of this fascinating system.…”

Section: The Hill Approximation In the Four Body Problemmentioning

confidence: 99%

“…It is worth mentioning that nowadays it is possible to develop numerical studies with high order precision to be used for a posteriori analysis in order to provide computer assisted proofs of the existence of equilibrium points, periodic orbits and invariant manifolds. The interested reader can consult [29], [17], [7] and references therein for more details. Nevertheless, the H4BP allows us to obtain general formulas for the equilibrium points and their stability could be analized by pen and paper techniques, as a consequence, we will privilege this analysis in the following subsection.…”

Section: Some Invariant Sets and Dynamical Propertiesmentioning

confidence: 99%

The spatial Hill four-body problem I -- An exploration of basic invariant sets

Burgos-Garcia,

Bengochea,

Franco-Perez

2021

Preprint

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In this work we perform a first study of basic invariant sets of the spatial Hill's four-body problem, where we have used both analytical and numerical approaches. This system depends on a mass parameter µ in such a way that the classical Hill's problem is recovered when µ = 0. Regarding the numerical work, we perform a numerical continuation, for the Jacobi constant C and several values of the mass parameter µ by applying a classical predictor-corrector method, together with a high-order Taylor method considering variable step and order and automatic differentiation techniques, to specific boundary value problems related with the reversing symmetries of the system. The solution of these boundary value problems defines initial conditions of symmetric periodic orbits. Some of the results were obtained departing from periodic orbits within Hill's three-body problem. The numerical explorations reveal that a second distant disturbing body has a relevant effect on the stability of the orbits and bifurcations among these families. We have also found some new families of periodic orbits that do not exist in the classical Hill's three-body problem; these families have some desirable properties from a practical point of view.

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“…(In addition to the centrifugal effect, the dynamical problem must take into account the Coriolis effect, but at points of equilibrium the Coriolis term vanishes.) Equilibrium solutions in the co-rotating frame of reference play a fundamental role in understanding the dynamical problem [KMJ19a], [KKMJ18], [KMJ19b]. Such equilibria are also important for classifying central configurations with an inferior mass [Are82].…”

Section: Introductionmentioning

confidence: 99%

On the Number of Equilibria Balancing Newtonian Point Masses with a Central Force

Arustamyan,

Cox,

Lundberg

et al. 2021

Preprint

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We consider the critical points (equilibria) of a planar potential generated by n Newtonian point masses augmented with a quadratic term (such as arises from a centrifugal effect). Particular cases of this problem have been considered previously in studies of the circular restricted n-body problem. We show that the number of equilibria is finite for a generic set of parameters, and we establish estimates for the number of equilibria. We prove that the number of equilibria is bounded below by n + 1, and we provide examples to show that this lower bound is sharp. We prove an upper bound on the number of equilibria that grows exponentially in n. In order to establish a lower bound on the maximum number of equilibria, we analyze a class of examples, referred to as "ring configurations", consisting of n − 1 equal masses positioned at vertices of a regular polygon with an additional mass located at the center. Previous numerical observations indicate that these configurations can produce as many as 5n−5 equilibria. We verify analytically that the ring configuration has at least 5n−5 equilibria when the central mass is sufficiently small. We conjecture that the maximum number of equilibria grows linearly with the number of point masses. We also discuss some mathematical similarities to other equilibrium problems in mathematical physics, namely, Maxwell's problem from electrostatics and the image counting problem from gravitational lensing.

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Homoclinic dynamics in a restricted four-body problem: transverse connections for the saddle-focus equilibrium solution set

Cited by 13 publications

References 51 publications

Resonant tori, transport barriers, and chaos in a vector field with a Neimark–Sacker bifurcation

Resonant tori, transport barriers, and chaos in a vector field with a Neimark–Sacker bifurcation

The spatial Hill four-body problem I -- An exploration of basic invariant sets

On the Number of Equilibria Balancing Newtonian Point Masses with a Central Force

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