Transition from oscillatory to excitable regime in a system forced at three times its natural frequency

The effect of a temporal modulation at three times the critical frequency on a Hopf bifurcation is studied in the framework of amplitude equations. In this case, these equations are complex Ginzburg-Landau equations with an extra quadratic term, resulting from the strong coupling between the external field and unstable modes. On increasing the intensity of the forcing, one passes from an oscillatory regime to an excitable one with three equivalent frequency locked states. In the first regime, topological defects are one-armed phase spiral, while in the second one they correspond to three-armed excitable amplitude spirals. Analytical results show that the transition between these two regimes occurs at a critical value of the forcing intensity. The transition between phase and amplitude spirals is confirmed by numerical analysis.<