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.
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.
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