| A generalized Hamiltonian system defined on Poisson manifold is a generalization of a classical Hamiltonian system defined on symplectic manifold with even-dimension, while maintaining the most properties of classical Hamiltonian systems. They can be used to describe broader dynamical model, and have been used in a wide range of applications, such as mathematical physics, biology, aerospace and other fields.Normal form theory is one of powerful tools for researching local bifurcation of non-linear dynamic system, and has a long history. For the classical Hamiltonian system, through a variety of symplectic transformation method, the reduction problem of Hamil-tonian systems can be converted to the reduction of Hamiltonian functions. This is a very important field of dynamical systems, and has a long history and abundant research results.In this paper, based on the fact that the flow defined by generalized Hamiltonian system is a one-parametric structure preserving nonlinear transformation and some pre-vious related works, we further study the normal form of generalized Hamiltonian system with Lie-Poisson structure, and obtain the general theoretical calculation method. In addition, the theoretical methods are used to study a kind of generalized Hamiltonian systems defined on the Lie algebra dual space (so(3))*, obtain the normal form of these systems up to third order, the corresponding third-order truncated normal form system is a generalized Hamiltonian system with three parameters. Finally, based on the foliation structure of phase space for generalized Hamilton system and the nature of the Hamilton function, we carefully analyse the phase portrait of the third order truncation system by using bifurcation theory and obtain all possible distributions of their phase orbits. |
