Master stability function explained

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 • 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:• x _{i} = f(x _{i)} + \sigma \sum ^{N} _{j =1} 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• 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 λ _{k} where λ _{k} ranges over the eigenvalues of the coupling matrix A .

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