I have a system, which has the form
$\frac{da_i}{dt}=\nu(\eta_i)(\sigma(\eta_i)-a_i)$,
where $\eta=WA+V$ and $A,V,\eta\in\mathbb{R}^N$ are vectors and $W\in\mathbb{R}^{N\times N}$ is some square matrix. Also, $\nu(\eta_i) > 0$.
I can prove that the related system
$\frac{da_i}{dt}=\sigma(\eta_i)-a_i$
has a Lyapunov function, which is of the form
$V(A)=\sum_{i=1}^N F_1(\eta_i) - 1/2\sum_{i=1}^N\sum_{j=1}^Nw_{i,j}\sigma(\eta_i)\sigma(\eta_j)$.
In this case $\nabla V\cdot F$ can be simplified to yield something of the form
$\nabla V\cdot F=\sum_{i=0}^Ng_i(\eta_i)\left(\sigma\left(\eta_i\right)-\eta_i\right)^2$ for some $g_i(\eta_i)>0$.
It is clear that the two systems must have the same fixed points. Is it possible to somehow use the Lyapunov function from the simpler system to derive a Lyapunov function for the more complex system?