Bifurcations of Liouville tori of a two fixed center problem

El-Sabaa, F. M.; Hosny, Mona; Zakria, S. K.

doi:10.1007/s10509-018-3297-y

Cited by 2 publications

(1 citation statement)

References 24 publications

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“…In the current section, we examine the topological analysis of the real invariant manifold utilizing the theory of Fomenko. 33 The Fomenko classification had been investigated in several areas in mechanics: in Celestial Mechanics, El-Sabaa et al 35 studied the bifurcation of the problem of two fixed centers. The bifurcation set was constructed by Vozmischeva 36–38 in which the domain of possible motion on the configurational space was classified, while Waalkens et al 39 studied the bifurcation in the topology of energy surfaces for the motion in three dimensions.…”

Section: The First Case Of Integrabilitymentioning

confidence: 99%

Bifurcations of Liouville tori of coupled sextic anharmonic oscillators

El-Sabaa

Amer

Gad

et al. 2023

Journal of Low Frequency Noise, Vibration and Active Control

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In the current paper, the problem of sextic anharmonic oscillators is investigated. There are three integrable cases of this problem. Emphasis is placed on two integrable cases, and a full description of each one is provided. The separated functions of the first and second integrability cases are transformed from a higher degree to the third and fourth degrees. Respectively, the periodic solution is obtained using Jacobi’s elliptic functions. The topology of phase space and Liouville tori’s bifurcations are discussed. The phase portrait is studied to determine singular points and classify their types in addition to the graphic representation for each of them. Finally, the numerical illustrations are introduced using the Poincaré surface section to emphasize the problem’s integrability.

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