# Master stability function

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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 $\dot{x}_i = f(x_i)$ 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:

$\dot{x}_i = f(x_i) + \sigma \sum_{j=1}^N A_{ij} g(x_j).$

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 $z$ to the greatest Lyapunov exponent of the equation

$\dot{y} = (Df + \gamma Dg) y.$

The synchronous state of the system of coupled oscillators is stable if the master stability function is negative at $\sigma \lambda_k$ where $\lambda_k$ ranges over the eigenvalues of the coupling matrix $A$.

## References

• Arenas, Alex; Díaz-Guilera, Albert; Kurths, Jurgen; Moreno, Yamir; Zhou, Changsong (2008), "Synchronization in complex networks", Physics Reports 469: 93–153, doi:10.1016/j.physrep.2008.09.002.
• Pecora, Luis M.; Carroll, Thomas L. (1998), "Master stability functions for synchronized coupled systems", Physical Review Leters 80: 2109–2112, doi:10.1103/PhysRevLett.80.2109.