On perturbed oscillators in 1–1–1 resonance: the case of axially symmetric cubic potentials

Ferrer, S.; Hanßmann, Heinz; Palacián, Jesús F.; Yanguas, Patricia

doi:10.1016/s0393-0440(01)00041-9

Cited by 36 publications

(58 citation statements)

References 44 publications

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“…We can say more, e.g., that the points (iii) are also stable for N small, which is in agreement with the stability character of (iii) for N such that N * /L ≤ |N |/L < 3/5. Finally in order to establish the existence of KAM 3-tori when N ≈ 0 for the full system we can use the result of Proposition 4.3 after undoing the scaling N = ε 4 N and the transformation to normal form that led to (21), we go back to the Hamiltonian in normal form given by (15). Thus repeating the steps of Subsection 4.3 and using Proposition 4.3, we end up with the existence of KAM 3-tori surrounding the periodic solutions that are near rectilinear and whose projections in the coordinate space indicate that they move close to the axis x 3 up and down.…”

Section: Invariant 3-tori Reconstructed From the Points (Iii)mentioning

confidence: 99%

Invariant tori in the lunar problem

Meyer¹,

Palacián²,

Yanguas³

2014

Publicacions Matemàtiques

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Theorems on the existence of invariant KAM tori are established for perturbations of Hamiltonian systems which are circle bundle flows. By averaging the perturbation over the bundle flow one obtains a Hamiltonian system on the orbit (quotient) space by a classical theorem of Reeb. A non-degenerate critical point of the system on the orbit space gives rise to a family of periodic solutions of the perturbed system. Conditions on the critical points are given which insure KAM tori for the perturbed flow.These general theorems are used to show that the near circular periodic solutions of the planar lunar problem are orbitally stable and are surrounded by KAM 2-tori.For the spatial case it is shown that there are periodic solutions of two types, either near circular equatorial, that is, the infinitesimal particle moves close to the plane of the primaries following near circular trajectories and the other family where the infinitesimal particle moves along the axis perpendicular to the plane of the primaries following near rectilinear trajectories. We prove that the two solutions are elliptic and are surrounded by invariant 3-tori applying a recent theorem of Han, Li, and Yi.In the spatial case a second averaging is performed, and the corresponding orbit space (called the twice-reduced space) is constructed. The flow of the averaged Hamiltonian on it is given and several families of invariant 3-tori are determined using Han, Li, and Yi Theorem.

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Section: Invariant 3-tori Reconstructed From the Points (Iii)mentioning

confidence: 99%

Invariant tori in the lunar problem

Meyer¹,

Palacián²,

Yanguas³

2014

Publicacions Matemàtiques

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“…However, in our example, we use the exterior product to verify whether the transformations are canonical or not by using its properties [10,11]. In detail, if A(v, w) denotes the area of the parallelogram determined by the pair of vectors v and w then A has the following properties:…”

Section: Methodsmentioning

confidence: 99%

Analytical Solution to Normal Forms of Hamiltonian Systems

Allahem

2017

MCA

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Abstract:The idea of the normalisation of the Hamiltonian system is to simplify the system by transforming Hamiltonian canonically to an easy system. It is under symplectic conditions that the Hamiltonian is preserved under a specific transformation-the so-called Lie transformation. In this review, we will show how to compute the normal form for the Hamiltonian, including computing the general function analytically. A clear example has been studied to illustrate the normal form theory, which can be used as a guide for arbitrary problems.

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“…When using variables M and N the Poisson structure is given by Table 6, where Casimirs are L 1 and the third relation in (17).…”

Section: Reduction To One Degree Of Freedommentioning

confidence: 99%

“…Among them, recently progress has been made, in particular, in the understanding of bifurcations emanating from singular points of the algebraic varieties defining the reduced phase spaces. More precisely we are interested here in enlarging the studies done in relation to the 1:1:1 resonance [7,14,15,16,17,18,19,20,21], considering now one more resonance with the same frequency; a preliminary report communicating part of the full study presented here was presented recently [12](see also [11]). In [28] it is shown that the reduced phase space of the n-dimensional harmonic oscillator is CP n−1 .…”

Section: Introductionmentioning

confidence: 99%