The Normal Form And Versal Unfolding Of Degenerate Planar Hamiltonian System

The normal form,an important tool to simplify nonlinear dynamical system,is widely used in the study of bifurcation problems of dynamical system.Its key idea is by constructing properly invertible nonlinear coordinate transformations to reduce the nonlinear dynamical system to be studied as another one as simple as possible,which is easy to analyze.The obtained system is viewed as a normal form of original system,which provides great convenience for the study of local bifurcation properties of the original system.On the other hand,because of the complexity of calculating normal form,we usually can only obtain finite order normal form for a dynamical system.Due to the influence of the truncated part for the normal form cannot be overlooked,we need consider the codimension and versal unfolding of truncated system.A large number of privious research on dynamical system were based on the dynam-ical system with non-zero linear part at an equilibrium.Classical Hamiltonian system is an important dynamical system with symplectic structure,for which the normal form and codimension 2 have been investigated by lots of researchers,but there are not so many investigation relatively about degenerate classical Hamiltonian system with zero linear part and some system with codimension 3.This master thesis mainly studies normal form for degenerate planar classical Hamil-tonian system with,zero linear part.For such,dynamical system,classical Poincare normal form theory fails to calculate its normal form.Firstly,we utilize the linear symplectic transformation to reduce second order terms for such degenerate system.Secondly,we employ the near-identity nonlinear transformation to simplify its three order terms.After these simplification,we can obtain different type of normal form up to three order under different condition.Then,we prove that the codimension is three for corresponding de-generate system when its coefficients of second order terms in the original system satisfy?g? 0.If the coefficients of the second order terms are not all zero and ?g= 0,then for corresponding degenerate system its codimension is four.Finally,we give the cor-responding versal unfolding for above various normal form and then analyze bifurcation behavior of their dynamics and related phase portraits.