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

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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.

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Authors and Affiliations

  1. Institute of Applied Mathematics of Yunnan Province, 650091, Kunming, China

    Li Jibin, Zhao Xiaohua & Liu Zhengrong

  2. Department of Mathematics of Yunnan University, 650091, Kunming, China

    Li Jibin, Zhao Xiaohua & Liu Zhengrong

Authors
  1. Li Jibin
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  2. Zhao Xiaohua
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  3. Liu Zhengrong
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Additional information

This project was supported partially by the National Natural Sciences Foundation of China and the STCF of Yunnan Province.

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Li, J., Zhao, X. & Liu, Z. Adiabatic invariants of slowly varying three-dimensional systems and existence of invariant tori of Lotka-Volterra equation. Acta Mathematicae Applicatae Sinica 12, 201–208 (1996). https://doi.org/10.1007/BF02007739

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  • DOI: https://doi.org/10.1007/BF02007739

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