Normal Form Theory Of Hopf Bifurcation And It's Application In The Nonlinear Vibration Systems

In this thesis the autor makes a thorough study to normal form theory of hopf bifurcation. The normal form of two dimensional generatic and degeneratic hopf bifurcation has been studied by matrix represenation method in real range . The algebra expression between the coefficients of original equations and normal form equations has been given . The simple and convenience new method of solving nonlinear vibration problems has been obtained by useing this expression, and it is found that the precision of solution is equal by using this expression or K-B averaging method when the original equation doesn't exist polinomials of order even.The normal form of arbitrary n dimentional generatic and degeneratic hopf bifurcation has been studied. The process of calculating normal form coefficients has been greatly simplified because we express the nonlinear parts of original equation in the form of "sum series' . The algebratic expression between the coefficients of original equations and normal form equations has been induced . These results are also be used to solve multi-degree-freedom nonlinear self-excited vibration system, a new method has been set up for solving nonlinear vibration problems. Because this method is simple, easy and is easy formularization the programs of FORTRAN and REDUCE have been made.The fourth focal value of hopf bifurcation of ordinary differential equations has been calculated by using normal form theory , this is very usefull for studying the stablity of degeneratic ordinary differential equations and the Hubert's 16th problem.As an application of the theoretical results of this thesis the shimmy problem of automobil has been studied . The sensitive parameters of causing shimmy and the main parameters of reducing the amplitude of shimmy have been found . The theoretical basts have been provided for eliminating and preventing shimmy.