Hopf bifurcations in fast/slow systems with rate-dependent tipping

We analyze rate-dependent tipping in a fast/slow system with an equilibrium near the fold of a critical manifiold. We find a Hopf bifurcation as the rate parameter increases in the reduced co-moving system. This implies the growth of a limit cycle as the system changes from tracking a quasi-static equilibrium to tipping. Rather than trajectories diverging at a critical rate, they continue to track the quasi-static equilibrium in a spiral corresponding to an emerging limit cycle at the Hopf bifurcation. We apply the same analysis to a forced van der Pol oscillator to show this phenomenon in a familiar system where the growth of this limit cycle is well understood.