Darboux integrability of generalized Yang–Mills Hamiltonian system

Llibre, Jaume; Valls, Clàudìa

doi:10.1080/14029251.2016.1175820

Cited by 13 publications

(4 citation statements)

References 19 publications

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“…The integrability of the 2D Yang-Mills Hamiltonian has been studied by several authors, see for example [28,31,[36][37][38]. We would like to emphasize that research works in this direction are interested in the integrability of this system only in two dimensions.…”

Section: Basic Conceptsmentioning

confidence: 99%

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

Ghamari¹,

Chatar²,

Kharbach³

et al. 2021

Preprint

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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.

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Section: Basic Conceptsmentioning

confidence: 99%

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

Ghamari¹,

Chatar²,

Kharbach³

et al. 2021

Preprint

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show abstract

“…Therefore, the authors worked on both topics separately for the same system. For example, the authors studied meromorphic first integral in Maciejewski and Przybylska [11] and Shi and Li [12], while in Llibre and Valls [13, 14] respectively, they studied the Darboux integrability. The authors Maciejewski and Szumiski [9] used differential Galois theory to investigate the integrability of system () for both cases

c&amp;#x0003D;0

and

c&amp;#x0003D;1

under the condition

b\ne 0

.…”

Section: Introductionmentioning

confidence: 99%

Darboux and analytic integrability of five‐dimensional semiclassical Jaynes–Cummings system

Taha

Amen

Hussein

2023

Math Methods in App Sciences

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We study the dynamics of a version of the Jaynes-Cummings system without the rotating wave approximation, which essentially consists of the interaction of two systems, n two-level atoms and a single system of electromagnetic field.In particular, we investigate some types of first integrals such as Darboux, analytic, and time-dependent first integrals of this system. We prove that only two Darboux first integrals and two analytic first integrals exist for this system. Moreover, we determine all exponential factors and invariant algebraic hypersurfaces of this system.

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“…For this, the invariant algebraic surfaces need to be fully characterized based on their parameters. To achieve invariant algebraic surfaces as such, the theory of Darboux integrability needs to be utilized, for further details on this theory see [20][21][22][23][24][25][26][27] .…”

Section: Introductionmentioning

confidence: 99%

Darboux Integrability of a Generalized 3D Chaotic Sprott ET9 System

2022

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Darboux integrability of generalized Yang–Mills Hamiltonian system

Abstract: We show that the generalized Yang-Mills system with Hamiltonian H = (p2 )/4 is not completely integrable with Darboux first integrals.

Cited by 13 publications

References 19 publications

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

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

Darboux and analytic integrability of five‐dimensional semiclassical Jaynes–Cummings system

Darboux Integrability of a Generalized 3D Chaotic Sprott ET9 System

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