Bifurcations of Armbruster Guckenheimer Kim galactic potential

Bifurcations of Liouville tori of coupled sextic anharmonic oscillators

“…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. In El-Sabaa et al, 40 the bifurcation of AGK galactic potential was analyzed. In Refs.…”

Armbruster – Guckenheimer – Kim Hamiltonian System in $\bs 1$:$\bs 1$ Resonance

“…In fact, these are the only Liouville integrable cases. El-Sabaa et al [8] made a topological study of the integral manifolds for the integrable cases b = 2a and b = −a, finding the structure of the bifurcation of the level sets. As expected, if bifurcations of Liouville tori take place, the level set of the first integrals becomes degenerate, and the existence of families of periodic solutions is possible, and they were given in terms of Jacobi's elliptic functions.…”

Global dynamics of the integrable Armbruster-Guckenheimer-Kim galactic potential

“…The existence of such ω denotes that the rotation of the galaxy must be taken into account when we study the stellar orbits (see [8]). Many studies concerning the integrability and non-integrability of such systems have been done (see for instance [1,4,5]) using different techniques such as the Painlevé analysis and the Morales-Ramis theory as well as the study of the existence of periodic orbits which was done in [7]. In particular, it was proved in [5] that if b = 2a or b = −a the system is completely integrable but the authors do not describe completely the dynamics of the integrable systems form the point of view of the Liouville-Arnold theorem (see section 2).…”