Dynamics, integrability and topology for Lotka-Volterra Hamiltonian systems in R^4

Abstract

Agraïments: The second author is partially supported by the National Natural Science Foundation of China (No. 11371248 & No. 11431008) and the RFDP of Higher Education of China grant (No. 20130073110074).

In this paper first we give the sufficient and necessary conditions in order that two classes of polynomial Kolmogorov systems in \R_ ^4 are Hamiltonian systems. After 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 a x y b z w c x^2 y d x y^2 e z^2 w f z w^2= constant.