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.
Without the coupling, they evolve according to the same differential equation, say
denotes the state of oscillator
which describes how the oscillators are coupled together, and a function
of the state of a single oscillator.
Including the coupling leads to the following equation: It is assumed that the row sums
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
to the greatest Lyapunov exponent of the equation The synchronous state of the system of coupled oscillators is stable if the master stability function is negative at
ranges over the eigenvalues of the coupling matrix