Analysis of interacting local oscillation mechanisms in three-timescale systems

Oscillations in slow/fast systems can be generated locally in phase space from either slow passage of trajectories through fast subsystem Hopf bifurcations, or from solutions tunnelling through twisted slow manifolds due to folded node singularities. In the two-timescale context, the fast subsystem Hopf and folded singularity remain as separate entities. In the three-timescale context, however, these singularities can interact, which allows for an interesting mixing of timescales and complex oscillatory dynamics. To further compound the problem, the folded singularity in these three-timescale problems is always close to the folded saddle-node regime, where canard theory breaks down. In this work, we examine a natural scenario in three-timescale systems where the fast subsystem Hopf and a folded node coexist and interact, as seen for instance in models for the electrical bursting activity in pituitary lactotrophs and for the secretion of gonadotropin releasing hormone by hypothalamic neurons. We analyze a novel type of fold-related singularity of the three-timescale system, which requires three timescales and which we call the canard-delayed-Hopf (CDH) singularity, from which the Hopf and folded node unfold, in order to determine properties of the local oscillatory behavior. We establish three types of results: (i) we prove the existence of canard solutions near the CDH singularity, which we show occurs naturally in the three-timescale context, (ii) we obtain an upper bound on the number of canard solutions and hence on the amount of rotation, and (iii) we estimate the bifurcation delay induced by passage of trajectories through the canard/fast-subsystem-Hopf region.