I was reading Synchronization of coupled oscillators and I came by a nice technique of having an estimation of the parameters in the coupled oscillators for which there is Synchronization.The techniques is Master Stability Function.
But while reading the paper of "Master Stability Functions for Synchronized Coupled Systems by Louis M. Pecora and Thomas L. Carroll".I was thinking about how equation (3) is derived from equation(1).
That is $$\dot{x^{i}} = F(x^{i}) - K \sum_{j} g_{ij} H(x^{j}) \rightarrow (1)$$
is the reference model of the MSF.
$i=1,...,N$ , $x^{i}$ is the $m$ dimensional vector of dynamical variables of the $ith$ node.
$\dot{x^{i}} = F(x^{i})$ is the isolated dyanmics of each node (uncoupled dyanmics)
$K$ is the coupling strength
$H :\Bbb{R}^{m } \rightarrow \Bbb{R}^{m }$ is the coupling function
$G =[g_{ij}]$ is the zero sum $N\times N $ matrix modelling network connection
$A$ is the Adjacency matrix
$G = [G_{ij}]$ = 0 if $i$ and $j$ are not connected $= -1$ if $i $and $j $ are linked $ = k_{i} $if $i=j$ ,$k_{i}$ is the degree of node $i$ = $k_{i} = \sum_{j}A_{ij}$
Now since $G$ is diagonalizable so how it obtains a block diagonalized variational equation of the form
$\dot{\eta_{h}} = [DF -k\gamma_{h} DH \eta_{h}] \rightarrow (3)$
where $\gamma_{h}$ is the eigenvalue of matrix $G$ ,$h=1,2,3,...,N$ How the above variational equation (3) was derived from equation (1)?.