Weakly resonant double Hopf bifurcation in coupled nonlinear systems with delayed freedback and application of homotopy analysis method

In this paper, we study the dynamical behaviors of coupled nonlinear systems with delay coupling by the multiple scales method and the homotopy analysis method. Firstly, we analyze the distribution of the eigenvalues of its linearized characteristic equations, and obtain the critical value for the occurrence of double Hopf bifurcation, which is caused by time delay and strength of coupling. Second, we obtain the normal form equations by the multiple scales method, and study the dynamical behaviors around the 3:5 weakly resonant double Hopf bifurcation point by analyzing the normal form equations. Finally, using the homotopy analysis method, we obtain analytical approximate solutions of the system with parameter values located in different regions. The periodic solution obtained by the homotopy analysis method is compared with the periodic solution obtained by the Runge-Kutta method, we found that the Runge-Kutta method does not get unstable periodic solutions, but the homotopy analysis method can be. So the homotopy analysis method is a powerful tool for studying coupled nonlinear systems with delay coupling.