Computation of the simplest normal form of a resonant double Hopf bifurcation system with the complex normal form method

Abstract

The complex normal form method is presented to find the simplest normal form (SNF) of an odd ratio double Hopf bifurcation system. New nonlinear transformations and rules of getting the form of the SNF coefficients are presented to find an efficient approach to get the final explicit expressions of the SNF. During the course of obtaining the key equations, based on the algorithm of complex operations, all the matrix deduction is substituted by complex operation and no former matrix operation is involved compared with the matrix representation method. Explicit iterative formulas have been derived to determine the coefficients of the SNF and associated nonlinear transformations up to an arbitrary order. Meanwhile, a rule of analyzing the general expression of the SNF is also summarized, which enables to predict the components of the final explicit equations before any substantive analyses. As the result of these innovations is deduced, the computation process is basically simplified. An ordinary differential equation (ODE) system shows the feasibility and convenience of the new method.