Periodic orbits of the generalized Friedmann-Robertson-Walker potential in galactic dynamics in a rotating reference frame

El-Dessoky, M. M.; Elmandouh, A. A.; Hobiny, Aatef

doi:10.1063/1.4979583

Cited by 4 publications

(5 citation statements)

References 19 publications

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“…It is noticeable that in the three situations H 2 is independent of , thus the application of the transformations (11), (12) and (13) to the normal form Hamiltonian (3) produces the effect of averaging H 2 with respect to the angle coordinate . This is an expected fact that puts in emphasis the relationship between averaging and normal form theories.…”

Section: Symplectic Coordinates On the Reduced Space B(h)mentioning

confidence: 99%

“…We want to stress the feasibility of the local symplectic variables Q i /P i introduced through the maps (11), (12) and (13). They are indeed rather convenient coordinates to perform the symplectic transformations towards the determination of the various normal forms needed to establish the different bifurcations occurring in a system with three or more degrees of freedom.…”

Section: Bifurcations Of Some Periodic Solutionsmentioning

confidence: 99%

“…We start with the critical point O − 1 when δ ≈ −1. We consider the coordinates Q/P defined in (12) related to the Hamiltonian of (17) noting that for O − 1 , H 2 is expanded around the origin. Rescaling the whole Hamiltonian by a constant factor that does not vanish for δ ≈ −1 and introducing the symplectic linear change…”

Section: Critical Point Eigenvaluesmentioning

confidence: 99%

“…Planar generalisations of this model have been tackled by several authors, for example [26] uses classical averaging theory to prove the existence of three families of periodic solutions in each nonzero energy level. In [12] the authors introduce the planar FLRW Hamiltonian in rotating coordinates and using averaging theory show the existence of two families of periodic solutions for angular velocity ω = 0 and six families of periodic solutions for ω = 0 in each nonzero energy level. In [45] the authors consider the formal stability of the origin of a Hamiltonian system composed by an oscillator in 1: −1:1 resonance plus a quartic perturbation.…”

Section: Introductionmentioning

confidence: 99%

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Periodic solutions, KAM tori and bifurcations in a cosmology-inspired potential

et al. 2019

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A family of perturbed Hamiltonians Hε = 1 2 (x 2 + X 2) − 1 2 (y 2 + Y 2) + 1 2 (z 2 + Z 2) + ε 2 [α(x 4 + y 4 + z 4) + β(x 2 y 2 + x 2 z 2 + y 2 z 2)] in 1:−1:1 resonance depending on two real parameters is considered. We show the existence and stability of periodic solutions using reduction and averaging. In fact, there are at most thirteen families for every energy level h < 0 and at most twenty six families for every h > 0. The different types of periodic solutions for every nonzero energy level, as well as their bifurcations, are characterised in terms of the parameters. The linear stability of each family of periodic solutions, together with the determination of KAM 3-tori encasing some of the linearly stable periodic solutions is proved. Critical Hamiltonian bifurcations on the reduced space are characterised. We find important differences with respect to the dynamics of the 1:1:1 resonance with the same perturbation as the one given here. We end up with an intuitive interpretation of the results from a cosmological viewpoint.

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Section: Symplectic Coordinates On the Reduced Space B(h)mentioning

confidence: 99%

Section: Bifurcations Of Some Periodic Solutionsmentioning

confidence: 99%

Section: Critical Point Eigenvaluesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Periodic solutions, KAM tori and bifurcations in a cosmology-inspired potential

et al. 2019

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“…Averaging method [32,35] is suitable for this purpose. Indeed, it has been extensively used to determine periodic orbits in both non rotating [1,5,6,[18][19][20] and rotating potentials [8,9] and also to establish periodic orbits for system (1), as it is done in [17]. The method is simple to apply, although some times tricky, but, if the system is Hamiltonian, alternative techniques can be used.…”

Section: Introductionmentioning

confidence: 99%

Reeb’s Theorem and Periodic Orbits for a Rotating Hénon–Heiles Potential

et al. 2019

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We apply Reeb's theorem to prove the existence of periodic orbits in the rotating Hénon-Heiles system. To this end, a sort of detuned normal form is calculated that yields a reduced system with at most four non degenerate equilibrium points. Linear stability and bifurcations of equilibrium solutions mimic those for periodic solutions of the original system. We also determine heteroclinic connections that can account for transport phenomena.

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