An analytical method for studying double Hopf bifurcations induced by two delays in nonlinear differential systems

An analytical method is introduced to investigate double Hopf bifurcations induced by two delays qualitatively and quantitatively. As an illustrative example, the clear procedure is demonstrated to study delay-induced weak resonant double Hopf bifurcation in a nonlinear system with multiple delays. When two delays are close to double Hopf bifurcation point, all solutions derived from the bifurcation are classified qualitatively and expressed explicitly. Numerical simulations are a good agreement with our theoretical analysis, and also already work in references. The results show that our work in this paper proposes a simple and valid method for investigating delay-induced double Hopf bifurcations. The important feature of our work is that the explicit expression of periodic solutions is easy to be obtained by solving algebraic equations.