Stability of Splay States in Coupled Oscillator Networks

Nesky, Amy Lynn. “Stability of Splay States in Coupled Oscillator Networks”, Boston College, 2013. http://hdl.handle.net/2345/3022.

Abstract

There are countless occurrences of oscillating systems in nature. Climate cycles and planetary orbits are a few that humans experience daily. Man has also incorporated, to his benefit, oscillation into his craft; the grandfather clock, for example, can keep track of time with astounding accuracy using the period of a long pendulum. Such systems can range in complexity in a number of ways. The governing equation for a given oscillator could be as simple as a sine curve, or its motion could appear so erratic that oscillatory motion is undetectable to viewers. The number of oscillators in a system can also vary, and oscillators can be coupled; that is, oscillators can be affected by the motion of neighboring oscillators. It is this last case we wish to study. We will briefly look at the case of finitely many oscillators and then move to analyzing a model consisting of infinitely many identical oscillators. Synchrony is the simplest collective behavior. We will study a more complicated pattern called splay states in which oscillators are equally staggered in phase, i.e. phase locked such that the system will return to this pattern if it is disturbed by an arbitrarily small amount. Mathematically, this requires us to find attracting fixed points in the system. We will approximate the local behavior of our model by linearizing the system near its fixed points. We will then apply our findings to a few specific cases of such models including: uniform density, linear distribution, alpha-function pulses, and integrate-and-fire.