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

${\displaystyle {\dot {x}}_{i}=f(x_{i})+\sigma \sum _{j=1}^{N}A_{ij}g(x_{j}).}${\displaystyle {\dot {x}}_{i}=f(x_{i})+\sigma \sum _{j=1}^{N}A_{ij}g(x_{j}).}

It is assumed that the row sums ${\displaystyle \sum _{j}A_{ij}}${\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 ${\displaystyle \gamma }${\displaystyle \gamma } to the greatest Lyapunov exponent of the equation

${\displaystyle {\dot {y}}=(Df+\gamma Dg)y.}${\displaystyle {\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 ${\displaystyle \sigma \lambda _{k}}${\displaystyle \sigma \lambda _{k}} where ${\displaystyle \lambda _{k}}${\displaystyle \lambda _{k}} ranges over the eigenvalues of the coupling matrix ${\displaystyle A}${\displaystyle A}.

• Arenas, Alex; Díaz-Guilera, Albert; Kurths, Jurgen; Moreno, Yamir; Zhou, Changsong (2008), "Synchronization in complex networks", Physics Reports, 469 (3): 93–153, arXiv:0805.2976 , Bibcode:2008PhR...469...93A, doi:10.1016/j.physrep.200809002, S2CID 14355929 .
• Pecora, Louis M.; Carroll, Thomas L. (1998), "Master stability functions for synchronized coupled systems", Physical Review Letters, 80 (10): 2109–2112, Bibcode:1998PhRvL..80.2109P, doi:10.1103/PhysRevLett.80.2109 .

• Dynamical systems