Chaos in the Gyldén problem

Diacu, Florin; Şelaru, Dan

doi:10.1063/1.532663

Cited by 13 publications

(18 citation statements)

References 14 publications

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“…where p ≡ (p 1 , p 2 ), i.e a standard (planar) Kepler problem plus a smooth perturbation ǫW (r, t). This problem can be easily reduced to a 2-dimensional dynamical system for the two variables r,ṙ, and for this reason we shall treat first this case, not only for better illustrating our procedure, but also in view of some direct and interesting applications, which include the classical Gyldén problem 13 . Consider the "parabolic" solution of the unperturbed (ǫ = 0) problem (1), which plays here the role of the homoclinic solution corresponding to the critical point at the infinity, i.e.…”

Section: The Gyldén-like Problemsmentioning

confidence: 99%

An approach to Mel’nikov theory in celestial mechanics

Cicogna

Santoprete

2000

Journal of Mathematical Physics

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Using a completely analytic procedure -based on a suitable extension of a classical method -we discuss an approach to the Poincaré-Mel'nikov theory, which can be conveniently applied also to the case of non-hyperbolic critical points, and even if the critical point is located at the infinity. In this paper, we concentrate our attention on the latter case, and precisely on problems described by Kepler-like potentials in one or two degrees of freedom, in the presence of general time-dependent perturbations. We show that the appearance of chaos (possibly including Arnol'd diffusion) can be proved quite easily and in a direct way, without resorting to singular coordinate transformations, such as the McGehee or blowing-up transformations. Natural examples are provided by the classical Gyldén problem, originally proposed in celestial mechanics, but also of interest in different fields, and by the general 3-body problem in classical mechanics. PACS(1999): 05.45.Ac, 45.05.+x, 95.10.CeIn a previous paper 1 , we discussed a completely analytic procedurebased on a suitable extension of a classical method 2 -to introduce the Poincaré-Melnikov theory 3−5 concerning the appearance of chaotic behaviour, which can be applied even to the case where the critical point is not hyperbolic. The case of nonhyperbolic stationary points has been already considered by several authors, although with quite different methods or in different contexts: we mention here Ref.s 6-11; some other references, more strictly related to our arguments, will be given in the following. In Ref. 1, we have also shown that our procedure may work even in some problems where the critical point is located at the infinity of the real line, as occurs in the case of the classical Sitnikov problem 12 .In this paper, we want first of all to reconsider and refine this approach, concentrating our attention precisely on problems described by Kepler-like potentials, in the presence of time-dependent perturbations of very general form. An example is provided by the classical Gyldén problem originally proposed in celestial mechanics, but also of interest in different fields (see Ref. 13). We shall then extend this procedure to problems with 2 degrees of freedom and in the presence of rotational symmetry, and show that the appearance of chaos (possibly including Arnol'd diffusion 14 ) can be proved quite easily and in a direct way, without resorting to singular coordinate transformations, such as the McGehee transformations 15 . Natural examples are provided by the general planar 3-body problem in classical mechanics 16−18 .Let us remark that, although our procedure is quite general, we shall consider here -for the sake of definiteness and clarity -only the case of the Kepler potential V (r) = 1/r: from the following discussion it will appear completely clear that the method can be easily applied -with suitable minor adjustments -to different classes of problems, for instance to problems where the Kepler potential is replaced by another "long range" potential, as e.g. V ∼ 1...

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Section: The Gyldén-like Problemsmentioning

confidence: 99%

An approach to Mel’nikov theory in celestial mechanics

Cicogna

Santoprete

2000

Journal of Mathematical Physics

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“…Within this framework, the relative motion of one particle with respect to the other (fixed at the origin of the coordinates) is confined to a plane. The corresponding Hamiltonian, in its most general form and in conveniently chosen units, reads (e.g., Diacu & Ş elaru 1998):…”

Section: Basic Equationsmentioning

confidence: 99%

“…Mathematically speaking, the Gyldén-type problem with periodically changing ν was approached by us from many standpoints: first-order analytical solutions in the case of a periodic variation of the equivalent gravitational parameter (Ş elaru et al 1992), the slowly-changing equivalent gravitational parameter (Cucu-Dumitrescu & Ş elaru 1997), KAM theory applied to this problem (Ş elaru & Mioc 1997), periodic orbits at infinity and chaotic behavior (Diacu & Ş elaru 1998), etc.…”

Section: Introductionmentioning

confidence: 99%

The Gyldén-type problem revisited: More refined analytical solutions

et al. 2006

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We resume and consistently extend our previous researches concerning the Gyldén-type problem (a two-body problem with time-dependent equivalent gravitational parameter). To approach most of the concrete astronomical situations to be modelled in this way, we consider a periodic small perturbation. For the nonresonant case, we present a second-order analytical solution. For the resonant case, we adopt the most realistic astronomical situation: only one dominant term of the Hamiltonian. In this case we point out a fundamental model of resonance, common to every resonant situation, and, moreover, identical to the first fundamental model of resonance. Considering the simplest model of periodic change of the equivalent gravitational parameter, we find that all possible resonances are confined to the first fundamental model.

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“…Consider the Gyldén problem, a time dependent perturbation of the Kepler problem, whose Hamiltonian is of the form where r = r , p = p and G(r, t) vanishes at infinity and is 2π periodic in t. This problem was considered in [5] and in [7], where sufficient conditions are given for the existence of transversal homoclinic points and hence of chaotic behaviour for the perturbed system. While the methods in [5] are self-contained, the aproach in [7], based on the change to McGehee's variables and on the usual geometric construction of the Melnikov function, requires justification as given by Theorem 2.1.…”

Section: Theorem 21mentioning

confidence: 99%

“…While the methods in [5] are self-contained, the aproach in [7], based on the change to McGehee's variables and on the usual geometric construction of the Melnikov function, requires justification as given by Theorem 2.1.…”

Section: Theorem 21mentioning

confidence: 99%

Melnikov method for parabolic orbits

Casasayas

Faísca

Nunes

2003

NoDEA : Nonlinear Differential Equations and Applications

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The present work completes the study of the conditions under which Melnikov method can be used when the unperturbed system has a parabolic periodic orbit with a homoclinic loop, by considering the case of orbits whose associated Poicaré map has linear part equal to the identity. The result is that the conditions for the persistence under perturbation of the invariant manifolds also ensure the convergence of the Melnikov integral and hence the applicability of the method.

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Chaos in the Gyldén problem

Cited by 13 publications

References 14 publications

An approach to Mel’nikov theory in celestial mechanics

An approach to Mel’nikov theory in celestial mechanics

The Gyldén-type problem revisited: More refined analytical solutions

Melnikov method for parabolic orbits

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