I am taking a course in dynamical systems and I am a bit lost with the following concept. Given the definition of topological equivalence:
Topological equivalence: We say two flow $\phi$ and $\psi$ are topologically equivalent if there is a homeomorphism $f : X \to Y$, mapping orbits of $\psi$ to orbits of $\phi$ homeomorphically, and preserving the orientation of the orbits. In addition, one must line up the flow of time: for each $y \in Y$, there exists a $\delta > 0$ such that, if $0 < |s| < t < \delta$, and if $s$ is such that $\phi(h(y),s) = h \circ \psi(y,t)$, then $s > 0$.
I know the following linear systems are topologically equivalent, but I do not know how to find the homeomorphism $f$ or how to prove their equivalence:
$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$
and
$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & 4 \\ -4 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$
Is there a method to find this homeomorphism or is it just pure intuition?
Besides, I do know, because this was stated as a counterexample to topologically equivalent $\nLeftrightarrow$ topologically conjugate, that the systems are not topologically conjugate, which is the same as topologically equivalent but without the lining up the flow of time. Why is this?