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 $\dot_i = f(x_i)$ 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 $A_$ 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:
:$\dot_i = f(x_i) + \sigma \sum_^N A_ g(x_j).$
It is assumed that the row sums $\sum_j A_$ 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
:$\dot = (Df + \gamma Dg) y.$
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

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* {{citation , last1 = Pecora , first1 = Louis M. , last2 = Carroll , first2 = Thomas L. , title = Master stability functions for synchronized coupled systems , journal = Physical Review Letters , year = 1998 , volume = 80 , issue = 10 , pages = 2109–2112 , doi = 10.1103/PhysRevLett.80.2109 , bibcode = 1998PhRvL..80.2109P .

Dynamical systems