Nonlinear Stability in Resonant Cases: A Geometrical Approach

Elipe, A.; Lanchares, Víctor; López-Moratalla, Teodoro; Riaguas, Andrés

doi:10.1007/s00332-001-0001-z

Cited by 16 publications

(8 citation statements)

References 25 publications

(37 reference statements)

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“…The equilibria located in the equatorial plane satisfy x ϭ 0 or y ϭ 0. In the first case, from equations (11) and (12), the y coordinate is a root of the polynomial equation…”

Section: Hamiltonian and Equilibrium Positionsmentioning

confidence: 99%

“…To deal with these higher order resonances and other degenerate cases, we will use a very general result that gives stability criteria in almost all resonant situations [12][13][14]. This result takes advantage of geometrical considerations of phase flow after a normalization procedure and, in some sense, constitutes an adaptation of ideas developed for the cases of third and fourth order resonances in the restricted three-body problem.…”

Section: Introductionmentioning

confidence: 99%

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Resonances and the Stability of Stationary Points Around a Central Body

2012

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The question of Lyapunov stability of stationary points around a central body has been studied in the absence of resonances and in the case of resonances of order 3 and 4, by means of the computation of the normal form up to second order. However, some special degenerate cases are not covered, as it happens for resonances of order 5 and 6, when the free parameters of the problem are chosen properly. In this article, we deal with these resonances and apply appropriate results to establish the stability properties of the stationary points.

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“…The equilibria located in the equatorial plane satisfy x ϭ 0 or y ϭ 0. In the first case, from equations (11) and (12), the y coordinate is a root of the polynomial equation…”

Section: Hamiltonian and Equilibrium Positionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Resonances and the Stability of Stationary Points Around a Central Body

2012

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“…Soon after, it was proven by Elipe and coworkers [8,9] that this result has a nice geometric counterpart giving rise to a geometric criterion of stability a bit more general; it is sufficient to characterize the phase flow of the normalized Hamiltonian system, and roughly speaking, the criterion is based on how two surfaces, related with the normal form, intersect one another.…”

Section: Introductionmentioning

confidence: 95%

Determination of nonlinear stability for low order resonances by a geometric criterion

2012

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We consider the problem of stability of equilibrium points in Hamiltonian systems of two degrees of freedom under low order resonances. For resonances of order bigger than two there are several results giving stability conditions, in particular one based on the geometry of the phase flow and a set of invariants. In this paper we show that this geometric criterion is still valid for low order resonances, that is, resonances of order two and resonances of order one. This approach provides necessary stability conditions for both the semisimple and non-semisimple cases, with an appropriate choice of invariants.MSC2010 numbers: 34D20, 37J40, 70H05

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“…For details, the reader is addressed to the work of Elipe. 20 Noting that the change (Φ 1 , Φ 2 ) → (ω 2 , ω 1 ) made in Arnold's theorem is equivalent to replace M 2 → 0, Elipe and coworkers 10,14 proved that on the manifold M 2 = 0, if the normalized Hamiltonian H(M 1 , C, S; 0) and the surface…”

Section: The Resonant Casesmentioning

confidence: 99%

On the Liapunov Stability of Stationary Points Around a Central Body

Elipe

López-Moratalla

2004

AIAA/AAS Astrodynamics Specialist Conference and Exhibit

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It is well known that an Earth-like planet has four stationary solutions, two stable and the other two unstable in the linear sense, which can be checked by computing the eigenvalues of the linearized system of equations of motion. Determining the orbital stability (or Liapunov stability) is rather more involved, and it requires firstly, the computing of the normal form around the equilibrium; secondly the expression of this normal form in action-and-angle variables in order to apply the so called Arnold's theorem of stability. For some specific values, higher order normal forms are required. However, Arnold's theorem is useless in the presence of resonances, and a new technique (related again with normal forms) must be applied in order to determine orbital stability. In this work, we proceed symbolically, taking the harmonic coefficients as parameters. We find the Liapunov stability diagram in the parametric plane for the stationary points. Resonances 2:1 and 3:1 are also studied, as well as the case at which a higher order normalization is required. The advantage of the symbolic analysis is that it is enough to replace the actual values of a planet or celestial body to have the orbital stability of the stationary points.

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Nonlinear Stability in Resonant Cases: A Geometrical Approach

Cited by 16 publications

References 25 publications

Resonances and the Stability of Stationary Points Around a Central Body

Resonances and the Stability of Stationary Points Around a Central Body

Determination of nonlinear stability for low order resonances by a geometric criterion

On the Liapunov Stability of Stationary Points Around a Central Body

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