https://doi.org/10.1140/epjp/s13360-022-03445-5
Regular Article
Order and chaos in Hamiltonian systems with quartic coupling
1
Laboratoire de Physique du Solide, Faculté des Sciences Dhar El Mahraz, Université Sidi Mohamed Ben Abdellah, B.P. 1796, 30003, Fès-Atlas, Morocco
2
Laboratoire Systèmes et Environnements Durables, Université Privée de Fès, Lot. Quaraouiyine Route Ain Chkef, 30000, Fès, Morocco
Received:
25
April
2022
Accepted:
26
October
2022
Published online:
10
November
2022
In this study, a variety of analytical approaches and numerical methods are carried out to explore the complex phenomena associated with nonlinear Hamiltonian systems with quartic coupling through the generalized three-dimensional (3D) Yang–Mills Hamiltonian system depending on four control parameters. We provide sufficient conditions on the four control parameters of the system which guarantee the 3D integrability in the Liouvillian sense. The new integrable cases of the 3D Yang–Mills Hamiltonian system are identified, and the 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 evolution of deviation vectors related to the studied orbit. This set of methods includes the Poincaré surface of section, the maximum Lyapunov exponent, the Smaller Alignment Index, the Generalized Alignment Index. In this view, the chaotic behavior will be explored and the order–chaos transition could be evaluated in both 2D and 3D, when any control parameters on which the system depends are slightly changed. 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.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
