Resonant double Hopf bifurcation in a diffusive Ginzburg–Landau model with delayed feedback

We investigate the resonant double Hopf bifurcation in a diffusive complex Ginzburg–Landau model with delayed feedback and phase shift. The conditions for the existence of resonant double Hopf bifurcation are obtained by analyzing the roots’ distribution of the characteristic equation, and a general formula to determine the bifurcation point is given. For the cases of 1:2 and 1:3 resonance, we choose time delay, feedback strength and phase shift as bifurcation parameters and derive the normal forms which are proved to be the same as those in non-resonant cases. The impact of cubic terms on the unfolding types is discussed after obtaining the normal form till 3rd order. By fixing phase shift, we find that varying time delay and feedback strength simultaneously can induce the coexistence of two different periodic solutions, the existence of quasi-periodic solutions and strange attractors. Also, the effects on the existence of transient quasi-periodic solution exerted by the phase shift are illustrated.