Math 6410 • ODE

Let \(\phi_t\) and \(\psi_t\) be two flows on \(\mathbb{R}^2\setminus\{0\}\) induced by the autonomous linear ODEs
\[
\frac{d}{dt}(x,y)=(x,y),\qquad \frac{d}{dt}(u,v)=(\alpha u,\beta v),
\]

where \(\alpha, \beta > 0\) are constants.
\begin{enumerate}

\item
Show that for any \(\alpha,\beta>0\), the flows \(\phi_t\) and \(\psi_t\) are topologically conjugate on
\(\mathbb{R}^2\setminus\{0\}\); i.e., there exists a homeomorphism
\(h:\mathbb{R}^2\setminus\{0\}\to\mathbb{R}^2\setminus\{0\}\) such that
\[
h(\phi_t(x,y))=\psi_t(h(x,y))
\]
for all \((x,y)\ne(0,0)\) and all \(t\in\mathbb{R}\).


\item
Find an explicit homeomorphism \(h\) realizing the conjugacy and prove it is a bijective
continuous map with continuous inverse.


\item
Discuss whether the conjugacy can be chosen differentiable (i.e., a \(C^1\) diffeomorphism).
Under what conditions on \(\alpha,\beta\) does a differentiable conjugacy exist?

\end{enumerate}