Seminar

Geometric Mechanics and Control Seminar

Lie-Poisson (bi)Hamiltonian systems: integrable deformations and clustering

Speaker:   Ángel Ballesteros (Universidad de Burgos)
Date:  Friday, 30 April 2021 - 15:30
Place:  Online - zoom.us/j/96739216821?pwd=UDhPYlNzM0M3dXFHUnJRSW92enl1dz09

Abstract:
A method for the construction of integrable deformations of Lie-Poisson Hamiltonian systems of ODEs by considering (dual) Poisson-Lie groups as deformations of Lie-Poisson algebras g is reviewed. In this approach, for each Lie bialgebra structure on the Lie algebra g that defines the undeformed Hamiltonian structure, an integrable deformation of the Lie-Poisson dynamical system is obtained. This deformation turns out to be Hamiltonian with respect to a Poisson-Lie structure on the dual Lie group D associated to the given Lie bialgebra. Moreover, by making use of the group law on the dual Lie group D, completely integrable Hamiltonian systems on N copies of D can be obtained, thus giving rise to a clustering method for the integrable coupling of dynamical systems, with N arbitrary. If the initial dynamical system is bihamiltonian, a compatibility condition between the two Lie algebras has to be imposed in order to preserve the bihamiltonian structure under deformation. This approach is exemplified by constructing integrable deformations and cluster generalizations of the Lorenz system, the Euler top system or the SIR epidemiological model. These results have been obtained in collaboration with A. Blasco, I. Gutierrez-Sagredo, J.C. Marrero, F. Musso and Z. Ravanpak.