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 $\backslash dot\{x\}\_i\; =\; f(x\_i)$ 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 $\backslash sigma$, 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:

:$\backslash dot\{x\}\_i\; =\; f(x\_i)\; +\; \backslash sigma\; \backslash sum\_\{j=1\}^N\; A\_\{ij\}\; g(x\_j).$

It is assumed that the row sums $\backslash sum\_j\; A\_\{ij\}$ 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 $\backslash gamma$ to the greatest Lyapunov exponent of the equation

:$\backslash dot\{y\}\; =\; (Df\; +\; \backslash gamma\; Dg)\; y.$

The synchronous state of the system of coupled oscillators is stable if the master stability function is negative at $\backslash sigma\; \backslash lambda\_k$ where $\backslash lambda\_k$ ranges over the eigenvalues of the coupling matrix $A$.