I have found all equilibria, studied their nature, and have been able to make a parametric plot of the following non-linear system along a time axis:
$$r'(t)=i-l.r(t)-\text{ux}. r(t). x(t)-\text{uy}. r(t). y(t) \\ x'(t)=\text{ex}. \text{ux}. r(t). x(t)-\text{mx}. x(t)\\y'(t)=\text{ey}.\text{uy}. r(t). y(t)-\text{my}. y(t)$$
$i,l,ux,uy,mx,my,ex,ey$ are parameters.I am currently reading about dynamical systems and the authour mention uncoupling the systems, for example this one:
$$\dot x =x,\;\; \dot y=y,\;\;\dot z=-z$$
I do not understand what this uncoupling is. Is it about setting one of the variable as constant ?
Is it possible to uncouple all three dimensional autonomous systems (mine for example) ?
How to draw the xy phase plane portrait, from this xyz system, for example?
If your system is written as $\dot{x}=Ax$ (i.e. it is a linear system), $x\in\mathbb R^n$, then you can under some conditions transform your system to $\dot{y}=By$, where $B$ is diagonal matrix, hence uncoupling the system. Here $B=U^{-1}AU$, $U$ is the eigenvector matrix of A, and $x=Uy$.
In case you have a nonlinear system $\dot{x}=f(x)$, you can do a similar transformation by replacing A by Jacobian of $f$, i.e. $A=Df$. Once you carry out the transformation as above, the linear part of the system will be diagonalized, but ofcourse you would have coupling in the higher order terms.