| The basic idea of normal form is to simplify the vector field near the equilibrium point and the periodic solution to the simplest equivalent form through a series of smoothing transformations.Therefore,it is a necessary prerequisite for qualitative research and numerical analysis to simplify the relatively complex dynamical system by using the normal form theory.The traditional normal form theory cannot give the simplest standard shape,and the coefficients in the traditional canonical form are generally not determined only by the original equation.Therefore,it is necessary to study the further simplification of normal form.In this paper,we use the method of combining parametric transformation and multiple Lie brackets to study the hypernormal form and coefficient calculation of three dimensional nilpotent vector fields.The main contents of this paper are as follows:(1)The research background and research status of normal form theory are reviewed.At the same time,the basic theories and basic concepts of traditional normal form and hypernormal form are also outlined.(2)The hypernormal form and the associated coefficients for a class of three dimensional nilpotent vector field with degree 2 are investigated.By using the methods of parameter transformation and multiple Lie brackets,the specific form of hypernormal form under different conditions is given by means of Maple software.The computation results are compared with the existing ones,and numerical simulations are carried out to verify the effectiveness of the results.(3)The hypernormal form and the associated coefficients for a class of three dimensional nilpotent vector field with degree 3 are investigated.By using the methods of parameter transformation and multiple Lie brackets,the specific form of hypernormal form under different conditions is given.Maple software is used for numerical simulation to obtain the trajectory diagram,which verifies the correctness of the result. |
