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.

中文翻译：

---

Painlevé通过三维Armbruster Guckenheimer Kim银河势的SALI和mLE方法分析，可积性并检测混沌-有序-混沌转变

考虑了一般形式的三维（3D）Armbruster Guckenheimer Kim银河势。通过使用Painlevé分析，可以执行此问题的可积性。我们报告了三种情况的可积性，并为每种情况明确获得了运动的确切积分。为了研究经典的相空间并探索混沌-有序-混沌过渡，进行了一些数值模拟。同样，最大李雅普诺夫指数（mLE）和较小对齐指数（SALI）方法用于快速有序地区分混沌运动和混沌运动。