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 $$ N $$ 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 $$ A $$.