| The theory of normal form for ordinary differential equation is the important partof the theories for bifurcation about modern vector fields, normal form theory cansimplify the ordinary differential equation near the balance point or periodic solutionto a lager extent and keep topological equivalence of the original equation, therefore,it is also one of the important strategy to study the bifurcation phenomena of vectorfields. In recent years, the theory of normal form has been obvious effect in actualapplication. Therefore, the studies of normal form of simplification and hypernormalform (simplest normal form, unique normal form) have been advanced subjects for therecent years.The research of normal form with application is developing towardshigh-dimensional. However, simplifying this kind of equation is a very complex andcumbersome process. At present, there are more research results for two-dimensionaland three-dimensional vector fields, but it is a puzzle to research hypernormal formand its application for the higher-dimensional vector fields. To address this issue, thefollowing work has been done:(1) the study of hypernormal form to a class of four-dimensional vector fieldswith nilpotent linear part and symmetrical property is presented via the use of multipleLie bracket and new grading function, through the mathematical software Maple withsymbolic manipulation, the uniqueness of the first order normal form is proved incertain condition. We also obtain the interceptive form of the first order hypernormalform at cubic and quintic;(2)Based on wings flutter on flying aircraft, the mechanical model of therectangular symmetric cross-ply composite laminated plates is studied. Frist, themethod of multiple scales is employed to obtain the four-dimensional averagedequations of the model. Then, the method of new grading function and multiple Liebrackets is utilized to obtain the hypernormal form at quintic of above averagedequations. |
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