On the dynamics of Armbruster Guckenheimer Kim galactic potential in a rotating reference frame

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

“…Llibre and Roberto [12] used classical averaging theory of first order to examine the C 1 nonintegrability in the sense of Liouville and Arnold, and also studied the existence of periodic orbits. In [7], Elmandou considered some dynamical aspects for the AGK potential in a rotating reference frame and examined the nonintegrability by means of Painlevé analysis. Additionally, he employed the Lyapunov method to seek periodic solutions near the equilibrium positions.…”

Painlevé analysis and integrability of three-dimensional Armbruster Guckenheimer Kim galactic potential

“…This Hamiltonian is known in literature as Hamiltonian of Armbruster-Guckenheimer-Kim (AGK). This potential characterizes the local motion of an almost axisymmetric galaxy that rotates with a constant angular velocity ω around a fixed axis, for this reason (Elmandouh [16]) adds the term ωp θ which can also appear in several physical problems, but most work has studied the AGK Hamiltonian for ω = 0 (as a non-rotating system), see for instance (Acosta-Humanez [24], Llibre and Roberto [6]).…”

“…To study the integrability of dynamical systems, that is to say the distinction between integrable and non-integrable systems, different methods and approaches exist such as the Painlevé analysis [15,16], the Ziglin theorem [17], the Liouville theorem [18], Lie algebra [19], separability study [20,21], the Smaller Alignment index (SALI) method for chaos detection [22], and the Poincaré sections [23].…”

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).…”