# Master stability function

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

The setting is as follows. Consider a system with $N$ identical oscillators. Without the coupling, they evolve according to the same differential equation, say ${\dot {x}}_{i}=f(x_{i})$ where $x_{i}$ denotes the state of oscillator $i$ . 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 $\sigma$ , a matrix $A_{ij}$ which describes how the oscillators are coupled together, and a function $g$ of the state of a single oscillator. Including the coupling leads to the following equation:

${\dot {x}}_{i}=f(x_{i})+\sigma \sum _{j=1}^{N}A_{ij}g(x_{j}).$ It is assumed that the row sums $\sum _{j}A_{ij}$ 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 $\gamma$ to the greatest Lyapunov exponent of the equation

${\dot {y}}=(Df+\gamma Dg)y.$ The synchronous state of the system of coupled oscillators is stable if the master stability function is negative at $\sigma \lambda _{k}$ where $\lambda _{k}$ ranges over the eigenvalues of the coupling matrix $A$ .