Abstract
We consider a continuum of phase oscillators on the circle interacting through an impulsive instantaneous coupling. In contrast with previous studies on related pulse-coupledmodels, the stability results obtained in the continuum limit are global. For the nonlinear transport equation governing the evolution of the oscillators, we propose (under technical assumptions) a global Lyapunov function which is induced by a total variation distance between quantile densities. The monotone time evolution of the Lyapunov function completely characterizes the dichotomic behavior of the oscillators: either the oscillators converge in finite time to a synchronous state or they asymptotically converge to an asynchronous state uniformly spread on the circle. The results of the present paper apply to popular phase oscillators models (e.g., the well-known leaky integrate-and-fire model) and show a strong parallel between the analysis of finite and infinite populations. In addition, they provide a novel approach for the (global) analysis of pulse-coupled oscillators.
| Original language | English |
|---|---|
| Pages (from-to) | 1154-1166 |
| Number of pages | 13 |
| Journal | IEEE Transactions on Automatic Control |
| Volume | 58 |
| Issue number | 5 |
| DOIs | |
| Publication status | Published - 2013 |
| Externally published | Yes |
Keywords
- Global stability
- Impulsive coupling
- Lyapunov function
- Partial differential equations
- Phase oscillators
- Synchronization
- Total variation distance
- Transport equation