The Normal Form Of One Class Of Generalized Three-dimensional Hamiltonian System

The normal form is a useful tool to study the local behaviors of the nonlinear dy-namical system. Its basic idea is to simplify dynamical systems by employing proper reversible nonlinear coordinate transformations.In existed references, analysis for the normal form of nonlinear dynamical systems focus on general dynamical systems or some systems with specific structure such as the classical Hamiltonian systems. Much more advances in this field have been made. On the other hand, the generalized Hamiltonian systems widely exist in mathematical models originated from celestial mechanics, biological science, and plasma physics. This kind of systems is featured with more generalized Poisson structure and generalizes the classical Hamiltonian systems. The study on the normal form for the generalized Hamiltonian systems is less reported, both in theoretical and applied research. Thus, detailed analysis of the normal form for generalized Hamiltonian systems should be received more attention.In this master thesis, we detailedly study the normal form of generalized Hamilton normal form and its computation. First, based on the fact that the transformation de-fined by the flow of generalized Hamilton systems is structure-preserving, the problem to compute normal form of generalized Hamilton system is transformed into the one to compute normal form of the Hamiltonian by constructing structure-preserving transfor-mation. Second, by deducting the structure-preserving transformation defined by the generalized Hamilton system, the Lie series linked the original and transformed gener-alized Hamilton system is provided. Third, by defining Lie operator and deducting the direct sum decomposition of homogeneous polynomial spaces, the general method and procedure to compute the normal form of generalized Hamiltonian systems are obtained. Finally, in order to illustrate our achieved results, up to three-order terms, we calculated the normal forms for three-dimensional generalized Hamilton systems with one type of Lie-Poisson structure, analyzed its three-order truncated normal form system and plotted all phase portraits with bounded orbits.