In nonlinear dynamical systems theory, if an equilibrium point is classified as a sink or a source using one system of coordinates, that is, the differential equations are written using one set of variables/coordinates, do the points retain their sink/source/saddle behaviour under a transformed set of variables/coordinates?
Thanks.
Yes, assuming the transformation is a $C^{k}, \quad k\geq 1$, diffeomorphism.
Suppose we have two systems $\dot{x} = f(x)$ and $\dot{y} = g(y)$ on $\mathbb{R}^{n}$, related by a change of coordinates $y=h(x)$. Then, assuming $h(x)$ is a smooth diffeomorphism, we have $f(x) = (Dh(x))^{-1}g(h(x))$. Now let $x_{0}$ and $y_{0}=h(x_{0})$ be the corresponding equilibria. Then
$$Df(x_{0}) = Dh(x_{0})^{-1}Dg(y_{0})Dh(x_{0})$$
and the eigenvalues of $Df(x_{0})$ and $Dg(y_{0})$ are the same.