• Source: 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


{\displaystyle N}

identical oscillators. Without the coupling, they evolve according to the same differential equation, say







x
˙




i


=
f
(

x

i


)


{\displaystyle {\dot {x}}_{i}=f(x_{i})}

where




x

i




{\displaystyle x_{i}}

denotes the state of oscillator



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



σ


{\displaystyle \sigma }

, a matrix




A

i
j




{\displaystyle A_{ij}}

which describes how the oscillators are coupled together, and a function



g


{\displaystyle g}

of the state of a single oscillator. Including the coupling leads to the following equation:








x
˙




i


=
f
(

x

i


)
+
σ

∑

j
=
1


N



A

i
j


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




∑

j



A

i
j




{\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



γ


{\displaystyle \gamma }

to the greatest Lyapunov exponent of the equation







y
˙



=
(
D
f
+
γ
D
g
)
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



σ

λ

k




{\displaystyle \sigma \lambda _{k}}

where




λ

k




{\displaystyle \lambda _{k}}

ranges over the eigenvalues of the coupling matrix



A


{\displaystyle 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.

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