Detecting Order and Chaos by the mLE, SALI and GALI Methods in Three-Dimensional Nonlinear Yang–Mills System

Abstract

The main objective of this study is to explore the nonlinear dynamics and chaos detecting in the three-dimensional (3D) generalized Hamiltonian Yang–Mills system as coupled quartic oscillators. We investigate its Liouvillian integrability. The integrable system is obtained and its associated first integral of motion is explicitly given. Also, a variety of nonlinear phenomena could be expressed using various numerical analysis, such as the construction of the system’s Poincaré Surface of Section (PSS), the maximum Lyapunov exponent (mLE), and the modern numerical methods like the Smaller (SALI) and the Generalized (GALI) Alignment Indexes. In this context, a series of numerical simulations are appropriate in order to explore ordered and chaotic motions of the system both in two (2D) and three dimensions, and one can observe chaos–order–chaos transition of the system when any parameters on which the system depends vary. Finally, several numerical schemes are shown to demonstrate the effectiveness and rapidity of these proposed methods for distinguishing chaos and order states of the system.