Master stability function

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 [math]\displaystyle{ N }[/math] identical oscillators. Without the coupling, they evolve according to the same differential equation, say [math]\displaystyle{ \dot{x}_i = f(x_i) }[/math] where [math]\displaystyle{ x_i }[/math] denotes the state of oscillator [math]\displaystyle{ i }[/math]. 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 [math]\displaystyle{ \sigma }[/math], a matrix [math]\displaystyle{ A_{ij} }[/math] which describes how the oscillators are coupled together, and a function [math]\displaystyle{ g }[/math] of the state of a single oscillator. Including the coupling leads to the following equation:

[math]\displaystyle{ \dot{x}_i = f(x_i) + \sigma \sum_{j=1}^N A_{ij} g(x_j). }[/math]

It is assumed that the row sums [math]\displaystyle{ \sum_j A_{ij} }[/math] 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 [math]\displaystyle{ \gamma }[/math] to the greatest Lyapunov exponent of the equation

[math]\displaystyle{ \dot{y} = (Df + \gamma Dg) y. }[/math]

The synchronous state of the system of coupled oscillators is stable if the master stability function is negative at [math]\displaystyle{ \sigma \lambda_k }[/math] where [math]\displaystyle{ \lambda_k }[/math] ranges over the eigenvalues of the coupling matrix [math]\displaystyle{ A }[/math].

References

• Arenas, Alex; Díaz-Guilera, Albert; Kurths, Jurgen; Moreno, Yamir; Zhou, Changsong (2008), "Synchronization in complex networks", Physics Reports 469 (3): 93–153, doi:10.1016/j.physrep.2008.09.002, Bibcode: 2008PhR...469...93A .
• Pecora, Louis M.; Carroll, Thomas L. (1998), "Master stability functions for synchronized coupled systems", Physical Review Letters 80 (10): 2109–2112, doi:10.1103/PhysRevLett.80.2109, Bibcode: 1998PhRvL..80.2109P .