Hamiltonian structure and Darboux theorem for families of generalized Lotka-Volterra systems

Abstract

This work is devoted to the establishment of a Poisson structure for a format of equations known as Generalized Lotka-Volterra systems. These equations, which include the classical Lotka-Volterra systems as a particular case, have been deeply studied in the literature. They have been shown to constitute a whole hierarchy of systems, the characterization of which is made in the context of simple algebra. Our main result is to show that this algebraic structure is completely translatable into the Poisson domain. Important Poisson structures features, such as the symplectic foliation and the Darboux' canonical representation, arise as result of rather simple matrix manipulations.

References in corpus (2)

• Nonpolynomial vector fields under the Lotka-Volterra normal form
• A constant of motion in 3D implies a local generalized Hamiltonian structure

Cited by in corpus (17)

• New solutions of the Jacobi equations for three-dimensional Poisson structures
• Separation of variables in the Jacobi identities
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• Characterization and global analysis of a family of Poisson structures
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• Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems
• New solution family of the Jacobi equations: Characterization, invariants, and global Darboux analysis
• Symplectic Realizations and Symmetries of a Lotka Volterra Type System
• Computing Casimir invariants from Pfaffian systems
• Characterization, global analysis and integrability of a family of Poisson structures
• New four-dimensional solutions of the Jacobi equations for Poisson structures
• Quasipolynomial generalization of Lotka-Volterra mappings
• Generalized results on the role of new-time transformations in finite-dimensional Poisson systems
• Algebraic decoupling of variables for systems of ODEs of quasipolynomial form
• Generalization of the separation of variables in the Jacobi identities for finite-dimensional Poisson systems
• New global solutions of the Jacobi partial differential equations
• Linear Darboux polynomials for Lotka-Volterra systems, trees and superintegrable families