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. 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 {displaystyle N} identical oscillators. Without the coupling, they evolve according to the same differential equation, say x ˙ i = f ( x i ) {displaystyle {dot {x}}_{i}=f(x_{i})} where x i {displaystyle x_{i}} denotes the state of oscillator i {displaystyle 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 {displaystyle sigma } , a matrix A i j {displaystyle A_{ij}} which describes how the oscillators are coupled together, and a function g {displaystyle g} of the state of a single oscillator. Including the coupling leads to the following equation: It is assumed that the row sums ∑ j A i j {displaystyle 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 γ {displaystyle gamma } 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 σ λ k {displaystyle sigma lambda _{k}} where λ k {displaystyle lambda _{k}} ranges over the eigenvalues of the coupling matrix A {displaystyle A} .