Painlevé's analysis, integrability and detecting chaos-order-chaos transition by the SALI and mLE methods of three-dimensional Armbruster Guckenheimer Kim galactic potential

Highlights: • We study the integrability of three-dimensional Armbruster Guckenheimer Kim potential. • We determine three integrable cases by using the Painlevé analysis. • We build the integrals of motion for all cases obtained. • Numerical investigations are performed to observe the chaos-order-chaos phenomenon of each case. The three-dimensional (3D) Armbruster Guckenheimer Kim galactic potential in the general form is considered. The integrability of this problem is performed by using the Painlevé analysis. We report three cases of integrability, and the exact integrals of motion are obtained explicitly for each case. Several numerical simulations were performed in order to investigate the classical phase space and explore chaos-order-chaos transition. Also, the maximum Lyapunov exponent (mLE) and the smaller alignment index (SALI) methods are used to distinguish rapidly and with certainty between ordered and chaotic motion.