I am considering the following question:
Suppose I have a coupled ODEs, described by the following scheme:
$$\dot{x}_1 = f(x_1)+H(x_2); \ \ \ \ \dot{x}_2 = f(x_2)+H(x_3)+H(x_1); \ \ \ \ \dot{x}_3 = f(x_3)+H(x_2); $$
where $f$ and $H$ could be nonlinear function.
Now based on that scheme, obviously, node $1$ and $3$ are symmetric and both of them make no difference for the node $2$. So reduce the above three ODEs to the following by renaming a different variable $y$: $$\dot{y}_1 = f(y_1)+H(y_2); \ \ \ \ \dot{y}_2 = f(y_2)+2H(y_1); $$
i.e., I combine nodes $1$ and $3$ to a new node $1$. More precisely, $x_1$ and $x_3$ are replaced by $y_1$, $x_2$ is replaced by $y_2$
My question is:
- Can I say both systems ($x$ and $y$) are equivalent?
- Solution behavior around equilibrium points are the same, such as region of attraction?
Can anyone please give me some directions or related article about questions? Thanks!