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 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 σ , a matrix A i j 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:
x ˙ i = f ( x i ) + σ ∑ j = 1 N A i j g ( x j ) . It is assumed that the row sums ∑ j A i j 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 γ to the greatest Lyapunov exponent of the equation
y ˙ = ( D f + γ D g ) y . The synchronous state of the system of coupled oscillators is stable if the master stability function is negative at σ λ k where λ k ranges over the eigenvalues of the coupling matrix A .
