Weak and non-resonant double Hopf bifurcations in m coupled van der Pol oscillators with delay coupling

A system of m   coupled van der Pol oscillators is considered. The dynamical behavior around 1:γ(γ∈R+,γ≠1,2,3) double Hopf bifurcation point is studied analytically. By using the method of multiple scales, the amplitude equations are obtained. Two parameters including time delay and the coupling strength are chosen as the bifurcation parameters, and then the dynamical behavior arising from the bifurcation is classified qualitatively in two-parameter plane. Some interesting dynamical phenomena including amplitude death, periodic solution and coexisting periodic solutions are obtained. By fixing m or γ, the effects of the changes of γ or m on the dynamics are investigated, respectively.