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Adiabatic invariants of slowly varying three-dimensional systems and existence of invariant tori of Lotka-Volterra equation

Adiabatic invariants of slowly varying three-dimensional systems and existence of invariant tori... In this paper, we use the theory of generalized Poisson bracket (GPB) to build the Poisson structure of three-dimensional “frozen” systems of Hamiltonian systems with slow time variable, and show that under proper conditions, there exists an adiabatic invariant on every closed simply connected symplectic leaf for the time-dependent Hamiltonian systems. If the HamiltonianH(p,q,τ) on these symplectic leaves are periodic with respect to τ and the frozen systems are in some sense strictly nonisochronous, then there are perpetual adiabatic invariants. To illustrate these results, we discuss the classical Lotka-Volterra equation with slowly periodic time-dependent coefficients modeling the interactions of three species. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Adiabatic invariants of slowly varying three-dimensional systems and existence of invariant tori of Lotka-Volterra equation

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References (22)

Publisher
Springer Journals
Copyright
Copyright © 1996 by Science Press
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF02007739
Publisher site
See Article on Publisher Site

Abstract

In this paper, we use the theory of generalized Poisson bracket (GPB) to build the Poisson structure of three-dimensional “frozen” systems of Hamiltonian systems with slow time variable, and show that under proper conditions, there exists an adiabatic invariant on every closed simply connected symplectic leaf for the time-dependent Hamiltonian systems. If the HamiltonianH(p,q,τ) on these symplectic leaves are periodic with respect to τ and the frozen systems are in some sense strictly nonisochronous, then there are perpetual adiabatic invariants. To illustrate these results, we discuss the classical Lotka-Volterra equation with slowly periodic time-dependent coefficients modeling the interactions of three species.

Journal

Acta Mathematicae Applicatae Sinica – Springer Journals

Published: Jul 13, 2005

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