Non-integrability criterium for normal variational equations around an integrable subsystem and an example: The Wilberforce spring-pendulum

Acosta-Humánez, Primitivo B.; Álvarez-Ramírez, Martha; Blázquez-Sanz, David; Delgado, Joaquín

doi:10.3934/dcds.2013.33.965

Cited by 10 publications

(10 citation statements)

References 11 publications

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“…Algebraic Methods. Concerning algebraic aspects considered in this paper, we follow the references [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] and also [1,2,3,4,5,6].…”

Section: Saddle-focus-saddle Bifurcationsmentioning

confidence: 99%

Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families

Contreras¹,

Linero²,

Montes³

et al. 2021

Preprint

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This article reveals an analysis of the quadratic systems that hold multiparametric families therefore, in the first instance the quadratic systems are identified and classified in order to facilitate their study and then the stability of the critical points in the finite plane, its bifurcations, stable manifold and lastly, the stability of the critical points in the infinite plane, afterwards the phase portraits resulting from the analysis of these families are graphed. To properly perform this study it was necessary to use some results of the non-linear systems theory, for this reason vital definitions and theorems were included because of their importance during the study of the multiparametric families. Algebraic aspects are also included.

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“…Algebraic Methods. Concerning algebraic aspects considered in this paper, we follow the references [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] and also [1,2,3,4,5,6].…”

Section: Saddle-focus-saddle Bifurcationsmentioning

confidence: 99%

Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families

Contreras¹,

Linero²,

Montes³

et al. 2021

Preprint

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“…Poincaré sections were applied in problems related with the integrability and non-integrability of hamiltonian dynamical systems, see [4,5,6,7,8] Since Poincaré maps are simpler to evaluate, even by computer, they were used to study complex trajectories, such as the ones determined by Lorenz system in Figure 3: Edward Lorenz as in [14] ( [19]). Once the discrete dynamics is created, our path follows in two branches: Lorenz (and Hénon) and May.…”

Section: Poincaré Sectionsmentioning

confidence: 99%

“…Dynamical systems are very important in the development of different areas, see for example [1] for applications in Quantum Mechanics and see also [5,6,7,8] for applications in Classical Mechanics. In particular, discrete dynamical systems play an important role due to they are an useful tool to understand continuous systems by decreasing its complexity, either by diminish dimensions or passing from continuous to discrete time.…”

Section: Final Remarksmentioning

confidence: 99%

From Poincaré to May: The Genesis of Discrete Dynamics

Martínez-Castiblanco,

Acosta-Humánez

2021

Preprint

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In this paper, the origin of discrete dynamics is stated from a historical point of view, as well as its main ideas: fixed and periodic points, chaotic behaviour, bifurcations. This travel will begin with Poincaré's work and will finish with May's work, one of the most important scientific papers of 20th century.This paper is based on the M.Sc. thesis "Simple Permutations, Pasting and Reversing" ([22]), written by the first author under the guidance of the second author.

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“…On the other hand, the differential Galois group is an algebraic group, in particular, a Lie group, and can be calculated by infinitesimal methods. Since then, Morales-Ramis theory has been applied successfully for studying the nonintegrability of large numbers of physical models, such as the planar three-body problem [13][14][15][16][17], Hill's problem [18], generalized Yang-Mills Hamiltonian [19], Wilberforce spring-pendulum problem [20], and double pendula problem [21]. It should be pointed out that the differential Galois group can also be used to investigate the nonintegrability of general dynamical systems which may be non-Hamiltonian [22][23][24].…”

Section: Remarkmentioning

confidence: 99%

Complex Dynamics of Some Hamiltonian Systems: Nonintegrability of Equations of Motion

2019

Advances in Mathematical Physics

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The main purpose of this paper is to study the complexity of some Hamiltonian systems from the view of nonintegrability, including the planar Hamiltonian with Nelson potential, double-well potential, and the perturbed elliptic oscillators Hamiltonian. Some numerical analyses show that the dynamic behavior of these systems is very complex and in fact chaotic in a large range of their parameter. I prove that these Hamiltonian systems are nonintegrable in the sense of Liouville. My proof is based on the analysis of normal variational equations along some particular solutions and the investigation of their differential Galois group.

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Non-integrability criterium for normal variational equations around an integrable subsystem and an example: The Wilberforce spring-pendulum

Cited by 10 publications

References 11 publications

Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families

Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families

From Poincaré to May: The Genesis of Discrete Dynamics

Complex Dynamics of Some Hamiltonian Systems: Nonintegrability of Equations of Motion

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