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 formula_1 identical oscillators. Without the coupling, they evolve according to the same differential equation, say formula_2 where formula_3 denotes the state of oscillator formula_4. 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 formula_5, a matrix formula_6 which describes how the oscillators are coupled together, and a function formula_7 of the state of a single oscillator. Including the coupling leads to the following equation:
It is assumed that the row sums formula_9 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 formula_10 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 formula_12 where formula_13 ranges over the eigenvalues of the coupling matrix formula_14.