Dynamics, integrability and topology for some classes of Kolmogorov Hamiltonian systems in R<inf>+</inf>4

Jaume Llibre; Dongmei Xiao

doi:https://doi.org/10.1016/j.jde.2016.10.042

Dynamics, integrability and topology for some classes of Kolmogorov Hamiltonian systems in R<inf>+</inf>4

Jaume Llibre, Dongmei Xiao

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

2 Citations (Scopus)

Abstract

© 2016 Elsevier Inc. In this paper we first give the sufficient and necessary conditions in order that two classes of polynomial Kolmogorov systems in R+4 are Hamiltonian systems. Then we study the integrability of these Hamiltonian systems in the Liouville sense. Finally, we investigate the global dynamics of the completely integrable Lotka–Volterra Hamiltonian systems in R+4. As an application of the invariant subsets of these systems, we obtain topological classifications of the 3-submanifolds in R+4 defined by the hypersurfaces axy+bzw+cx2y+dxy2+ez2w+fzw2=h, where a,b,c,d,e,f,w and h are real constants.

Original language	English
Pages (from-to)	2231-2253
Journal	Journal of Differential Equations
Volume	262
Issue number	3
DOIs	https://doi.org/10.1016/j.jde.2016.10.042
Publication status	Published - 5 Feb 2017

Keywords

Global dynamics
Hamiltonian system
Hypersurfaces
Liouvillian integrability
Topological classification

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title = "Dynamics, integrability and topology for some classes of Kolmogorov Hamiltonian systems in R+4",

abstract = "{\textcopyright} 2016 Elsevier Inc. In this paper we first give the sufficient and necessary conditions in order that two classes of polynomial Kolmogorov systems in R+4 are Hamiltonian systems. Then we study the integrability of these Hamiltonian systems in the Liouville sense. Finally, we investigate the global dynamics of the completely integrable Lotka–Volterra Hamiltonian systems in R+4. As an application of the invariant subsets of these systems, we obtain topological classifications of the 3-submanifolds in R+4 defined by the hypersurfaces axy+bzw+cx2y+dxy2+ez2w+fzw2=h, where a,b,c,d,e,f,w and h are real constants.",

keywords = "Global dynamics, Hamiltonian system, Hypersurfaces, Liouvillian integrability, Topological classification",

author = "Jaume Llibre and Dongmei Xiao",

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T1 - Dynamics, integrability and topology for some classes of Kolmogorov Hamiltonian systems in R+4

AU - Llibre, Jaume

AU - Xiao, Dongmei

PY - 2017/2/5

Y1 - 2017/2/5

N2 - © 2016 Elsevier Inc. In this paper we first give the sufficient and necessary conditions in order that two classes of polynomial Kolmogorov systems in R+4 are Hamiltonian systems. Then we study the integrability of these Hamiltonian systems in the Liouville sense. Finally, we investigate the global dynamics of the completely integrable Lotka–Volterra Hamiltonian systems in R+4. As an application of the invariant subsets of these systems, we obtain topological classifications of the 3-submanifolds in R+4 defined by the hypersurfaces axy+bzw+cx2y+dxy2+ez2w+fzw2=h, where a,b,c,d,e,f,w and h are real constants.

AB - © 2016 Elsevier Inc. In this paper we first give the sufficient and necessary conditions in order that two classes of polynomial Kolmogorov systems in R+4 are Hamiltonian systems. Then we study the integrability of these Hamiltonian systems in the Liouville sense. Finally, we investigate the global dynamics of the completely integrable Lotka–Volterra Hamiltonian systems in R+4. As an application of the invariant subsets of these systems, we obtain topological classifications of the 3-submanifolds in R+4 defined by the hypersurfaces axy+bzw+cx2y+dxy2+ez2w+fzw2=h, where a,b,c,d,e,f,w and h are real constants.

KW - Global dynamics

KW - Hamiltonian system

KW - Hypersurfaces

KW - Liouvillian integrability

KW - Topological classification

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DO - https://doi.org/10.1016/j.jde.2016.10.042

M3 - Article

VL - 262

SP - 2231

EP - 2253

IS - 3

ER -

Dynamics, integrability and topology for some classes of Kolmogorov Hamiltonian systems in R<inf>+</inf><sup>4</sup>

Abstract

Keywords

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