Master stability function

In mathematics, the master stability function is a tool used to analyze the stability of the synchronous state in a dynamical system consisting of many identical systems which are coupled together, such as the Kuramoto model.

The setting is as follows. Consider a system with ${\displaystyle N}$ identical oscillators. Without the coupling, they evolve according to the same differential equation, say ${\displaystyle {\dot {x}}_{i}=f(x_{i})}$ where ${\displaystyle x_{i}}$ denotes the state of oscillator ${\displaystyle i}$. A synchronous state of the system of oscillators is where all the oscillators are in the same state.

The coupling is defined by a coupling strength ${\displaystyle \sigma }$, a matrix ${\displaystyle A_{ij}}$ which describes how the oscillators are coupled together, and a function ${\displaystyle g}$ of the state of a single oscillator. Including the coupling leads to the following equation:

${\displaystyle {\dot {x}}_{i}=f(x_{i})+\sigma \sum _{j=1}^{N}A_{ij}g(x_{j}).}$

It is assumed that the row sums ${\displaystyle \sum _{j}A_{ij}}$ vanish so that the manifold of synchronous states is neutrally stable.

The master stability function is now defined as the function which maps the complex number ${\displaystyle \gamma }$ to the greatest Lyapunov exponent of the equation

${\displaystyle {\dot {y}}=(Df+\gamma Dg)y.}$

The synchronous state of the system of coupled oscillators is stable if the master stability function is negative at ${\displaystyle \sigma \lambda _{k}}$ where ${\displaystyle \lambda _{k}}$ ranges over the eigenvalues of the coupling matrix ${\displaystyle A}$.