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 $N$ 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 $\gamma$ 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 (3): 93–153, arXiv:0805.2976, Bibcode:2008PhR...469...93A, doi:10.1016/j.physrep.2008.09.002, 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.