Controlling and anticipating transitions in oscillatory systems

Rhythmic activity is found throughout nature, and oscillatory systems can undergo abrupt qualitative changes. In this talk, I will discuss two such transitions: the suppression of oscillation in a single oscillator and the emergence of synchrony in a population of coupled oscillators. First, I ask whether periodic forcing can suppress the oscillatory state of a bistable oscillator, in which a stable limit cycle coexists with a stable steady state. Using the averaging method, I show that a periodic force combined with quadratic feedback can quench the oscillation through a homoclinic bifurcation. Second, I examine whether the onset of synchronization in oscillator networks can be anticipated when only a few nodes are observed. Using a stochastic Kuramoto model, I show that, with an appropriate early warning signal, monitoring only ⌊ln N⌋ selected nodes performs nearly as well as observing all N nodes.