Analysis And Application Of The Complex Normal Form Method

Normal form method is an important tool for simplifying the ordinary differential equations; it plays an important role in the study of bifurcation and stability behaviors of the nonlinear dynamical systems near the critical equilibrium. In recent years, the methodology of normal form and its application in the dynamical system have made quite great progress with the domestic and foreign scholars'unceasingly thorough research in this filed. More and more nonlinear dynamical systems can be simplified by normal form method as the concept of the simplest normal form (SNF) is presented, and the nonlinear dynamical behavior of these systems near critical equilibrium can be obtained more easily. However, there are still a lot of unsolved problems on the computation and application of normal form method. Such as the application of normal form in the strongly nonlinear oscillation system with multi degrees of freedom,the efficient and simple method to calculate the SNF,and the application of the SNF in the nonlinear oscillation system are eagerly to be solved. In order to deal with those problems on normal form, the following major research were down in this paper:(1) The SNF of double Hopf bifurcation system was studied by using the advanced conventional normal form (CNF) method, and the results were applied to compute the asymptotic solutions of a 2-dimensional strongly nonlinear oscillator. Generally conventional normal form theory has the limitation of dealing with weakly nonlinear vibration systems. So the method of undetermined instantaneous fundamental frequency was developed, which made the CNF method be suitable for the strongly nonlinear oscillation system with two degrees of freedom, as an example a Duffing-Van der Pol oscillator was studied.(2) The SNF of Hopf, Hopf-zero and a resonance double Hopf bifurcation systems have been derived by the complex normal form method. It is well known that matrix representation method is the mostly used approach to obtaining the simplest normal form. However a series of matrix deduction are necessary to get the expression of SNF during the process, which hinders the actual application of that method. The complex normal form method, on the other hand, requires little matrix deduction. As the result, the former...