Poisson–Lie groups, bi-Hamiltonian systems and integrable deformations

Ballesteros, Ángel; Marrero, Juan Carlos; Ravanpak, Zohreh

doi:10.1088/1751-8121/aa617b

Cited by 5 publications

(7 citation statements)

References 26 publications

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“…Finally, the Hamiltonian structure here presented allows us to apply the well-known machinery of integrable deformations of Hamiltonian systems in order to define new compartmental models as integrable deformations of the dynamical systems here studied, including coupled ones. In particular, deformations based on Poisson coalgebras [32] provide a suitable arena in order to face this problem by following the same techniques that have been previously used, for instance, in order to get integrable deformations and coupled versions of Lotka–Volterra, Lorenz, Rössler and Euler top dynamical systems [33] , [34] , [35] . Work on this line is in progress and will be presented elsewhere.…”

Section: Discussionmentioning

confidence: 99%

Hamiltonian structure of compartmental epidemiological models

Ballesteros

Blasco

Gutiérrez-Sagredo

2020

Physica D: Nonlinear Phenomena

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Any epidemiological compartmental model with constant population is shown to be a Hamiltonian dynamical system in which the total population plays the role of the Hamiltonian function. Moreover, some particular cases within this large class of models are shown to be bi-Hamiltonian. New interacting compartmental models among different populations, which are endowed with a Hamiltonian structure, are introduced. The Poisson structures underlying the Hamiltonian description of all these dynamical systems are explicitly presented, and their associated Casimir functions are shown to provide an efficient tool in order to find exact analytical solutions for epidemiological models, such as the ones describing the dynamics of the COVID-19 pandemic.

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Section: Discussionmentioning

confidence: 99%

Hamiltonian structure of compartmental epidemiological models

Ballesteros

Blasco

Gutiérrez-Sagredo

2020

Physica D: Nonlinear Phenomena

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“…Under the same hypotheses as in Corollary 3.3, we deduce that (note that [ ← − r k , − → r l ] = 0). Therefore, Π k and Π l are compatible multiplicative Poisson structures on G. A generalization of the Poisson coalgebra approach to bi-Hamiltonian systems was shown in [4], and a construction of integrable deformations of bi-Hamiltonian systems, based on the theory of multiplicative Poisson structures on the Lie groups, was presented (for more details, see [3], [4], [5]).…”

Section: Discussionmentioning

confidence: 99%

“…for the solutions r (1) , r (2) , r (3) and r (4) , respectively; their corresponding constants of motion are…”

Section: Physical Applicationmentioning

confidence: 99%

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Invariant Poisson–Nijenhuis structures on Lie groups and classification

Ravanpak

Rezaei-Aghdam

Haghighatdoost

2018

Int. J. Geom. Methods Mod. Phys.

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Abstract. We study right-invariant (resp., left-invariant) Poisson-Nijenhuis structures on a Lie group G and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra g. We show that r-n structures can be used to find compatible solutions of the classical Yang-Baxter equation. Conversely, two compatible r-matrices from which one is invertible determine an r-n structure. We classify, up to a natural equivalence, all r-matrices and all r-n structures with invertible r on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by r-matrices on a phase space whose symmetry group is Lie group G, can be specifically determined.

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“…In [4], the authors presented a method for obtaining integrable deformations of Lie-Poisson bi-Hamiltonian systems. The initial data is a dynamical system D on R n , which is bi-Hamiltonian with respect to two compatible linear Poisson structures.…”

Section: A Deformation Of a Conservative Lorenz Systemmentioning

confidence: 99%

Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson–Lie groups

Gutiérrez-Sagredo¹,

Ponte²,

Marrero³

et al. 2023

J. Phys. A: Math. Theor.

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In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson-Lie groups and the unimodularity of the Poisson-Lie structure. In particular, we prove that Hamiltonian vector fields on a Lie group endowed with a unimodular Poisson-Lie structure preserve a multiple of any left-invariant volume on the group. Conversely, we also prove that if there exists a Hamiltonian function such that the identity element of the Lie group is a nondegenerate singularity and the associated Hamiltonian vector field preserves a volume form, then the Poisson-Lie structure is necessarily unimodular. Furthermore, we illustrate our theory with different interesting examples, both on semisimple and unimodular Poisson-Lie groups.

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Poisson–Lie groups, bi-Hamiltonian systems and integrable deformations

Cited by 5 publications

References 26 publications

Hamiltonian structure of compartmental epidemiological models

Hamiltonian structure of compartmental epidemiological models

Invariant Poisson–Nijenhuis structures on Lie groups and classification

Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson–Lie groups

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