Bifurcation in coupled Hopf oscillators

Two identical dynamical systems, representing the normal form corresponding to the Hopf bifurcation, were coupled using two parameters. The 4D dynamical system obtained possesses additional equilibria. Our study concerns the bifurcations of this system around the origin. We found that Hopf bifurcation takes place in two cases and it is of the same type as the Hopf bifurcation of the single model. In the first case the center manifold is a 2-plane and the limit cycle does not depend on the coupling parameters. In the second case, if the coupling parameters are equal, limit cycles with four regimes of behavior emerge, while if the coupling parameters are different, limit cycles with eight regimes of behavior are emphasized and different amplitudes of the oscillations occur in addition. For some values of the parameters, other bifurcations are present: degenerated fold bifurcation, degenerated double-zero bifurcation and symmetric Hopf bifurcation.