The Simplest Normal Form Of Non-resonant Double Hopf Bifurcation System With The Complex Normal Form Method

Normal form method is an important tool for simplifying the ordinary differential equations, and plays an important role in the study of bifurcation and stability behaviors of the nonlinear dynamical systems near the critical equilibrium. Presently the study and application of the normal form (NF) is under its way towards the direction of maximum simplicity and high-dimension. More and more NF of the nonlinear dynamical systems can be further simplified by applying the theory of the simplest normal form (SNF) to research their dynamical behavior near the critical equilibrium profoundly.Based on the Conventional Normal Form (CNF) result, the SNF utilizes the Real Form Matrix Representation Method to further simplify the CNF result by introducing nonlinear transformations. That makes the research of the complicated dynamic behavior belong to the nonlinear systems become more easily. However there are also some drawbacks, similarly to the CNF, it requires too much matrix deduction process, which certainly increases the inconvenience and hardness of computation. So, in this article, the application of the complex normal form method in finding the SNF is developed more efficiently to a higher level, the non-resonance double Hopf bifurcation system. That means the hardness to find the final SNF result can be seriously reduced during the course of computation.Here the normal form of the non-resonance double Hopf bifurcation system and the associated nonlinear transformation are constructed in a complex ordinate form. By applying the Complex Normal Form and transformation, a common expression of the key equations of all phases under the non-resonance circumstance can be achieved without any matrix operation. It also shows that the nonlinear transformation used can be reduced to a more simply style. The computer algebra Mathematica is taken to figure out the SNF expression of this non-resonance double Hopf bifurcation system up to the fifth order and summarizes the rules of selecting the SNF coefficients. With the help of all these achievements the simplest normal forms of this system up to any order should be obtained.