Applying PSS, mLE, SALI and GALI Methods for Exploring Order and Chaos in Two and Three Dimensional Hamiltonian Systems with Quartic Coupling

Abstract In this article, the complex nonlinear dynamics and chaos control have been examined in Hamiltonians systems with quartic coupling through the generalized three-dimensional (3D) Yang-Mills Hamiltonian system with four control parameters. We provide sufficient conditions on the four control parameters of the system which guarantee the 3D integrability in the Liouvillian sense. Therefore, we get a classification of the 3D Yang-Mills Hamiltonian system in sense of integrability and non-integrability. The integrable cases are identified and the detailed calculations of their associated first integrals of motion are given. The nature of the behavior orbits could be distinguished in a fast and efficient way by using a set of reliable methods based on the so-called the evolution of deviation vectors related to the studied orbit. This set of methods includes the Poincaré surface of section (PSS), the maximum Lyapunov exponent (mLE), the Smaller Alignment Index (SALI), the Generalized Alignment Index (GALI). In this view, the chaotic behavior will be explored and the order-chaos transition could be evaluated both in 2D and 3D, when any control parameters on which the system depends vary. Finally, the efficiency and rapidity of these proposed methods are proven by using several numerical illustrative paradigms for identifying whether the system is in chaos or order state.