Emergence of Supercritical Hopf Bifurcations in Coupled Oscillator Ensembles and How Desynchronization Can Facilitate Energy-Optimal Phase Resetting

Abstract. Phase resetting, involving shifting the timing of oscillations by a prespecified amount, is a fundamental control problem arising in the investigation of oscillatory dynamical systems. This work investigates the relationships between phase resetting of the collective rhythm and desynchronization in populations of coupled oscillators in the context of an optimal control problem. Conditions on the emergence of a supercritical Hopf bifurcation are considered for a general model of coupled oscillators, and the Hopf normal form is used to simplify the analysis. Two solution archetypes for phase resetting emerge: weak resetting, which does not influence interoscillator synchrony, and strong resetting, whereby the network is first desynchronized before resynchronizing with the correct phase. Analytical expressions for the energy expenditure associated with these solutions yield general conditions for which desynchronization is expected during the course of phase resetting. In particular, strong phase resetting is more efficient than weak resetting when the following conditions are met: 1) the relaxation rate of the collective rhythm to its limit cycle is sufficiently slow relative to its natural frequency, and 2) the required phase shifts are sufficiently large.