Double Hopf bifurcation and quasi-periodic attractors in delay-coupled limit cycle oscillators

We investigate the double Hopf bifurcation at zero equilibrium point. Firstly, we give the critical values of Hopf and double Hopf bifurcations. Secondly, we implement the normal form method and the center manifold theory for delay-coupled limit cycle oscillators, and derive the universal unfolding and a complete bifurcation diagram of the system. Thirdly, many interesting phenomena, such as attractive periodic motion and three-dimensional invariant torus, are observed using numerical simulation. Finally, the normal forms of several strong resonant cases are listed.