Topological conjugacy between linear ODE systems

by Joaquin C.   Last Updated May 16, 2019 04:20 AM

I'm trying to learn how to find the homeomorphism that defines the topological conjugacy between two ODE systems. By definition, we say that two ODE systems with matrices $A,B \in \mathbb{R^{n \times n}}$ are topologically conjugate iff there exists a homeomorphism $h: \mathbb{R^n} \longrightarrow \mathbb{R^n}$ such that $h( \varphi_{A} (t,x) ) = \varphi_{B} (t,h(x))$, i.e. $h(e^{At}x) = e^{Bt} h(x)$.

In practice I'm having trouble finding these explicit homeomorphisms. A simple example in the $A,B \in \mathbb{R^{2 \times 2}}$ case would be $$ A = \begin{bmatrix} 1 && 0 \\ 1 && -2 \end{bmatrix}$$ $$ B = \begin{bmatrix} -1 && 1 \\ 0 && 2 \end{bmatrix}$$ I figured it would be easier to work with their diagonalized forms at first. By diagonalizing both matrices (basically just taking their main diagonals since they are triangular) and finding their flows, I obtain, if $x = (x_{1} , x_{2}) \in \mathbb{R^{n}}$: $$\varphi_{D_{A}}(t,x) = (x_{1}e^{t},x_{2}e^{-2t}) , \varphi_{D_{B}}(t,x) = (x_{1}e^{-t},x_{2}e^{2t})$$ If I'm not mistaken, the phase portait should look like two hyperbolas, the first one approaches to the x-axis and grows to infinity horizontally, while the second one approaches to the y-axis and grows vertically towards infinity. From this graphic intuition, I tried finding a homeomorphism that rotates the plane and scales the exponentials appropriately, and came up with: $$h(x,y) = (\text{sgn}(y) \sqrt{|y|},\text{sgn}(x) |x|^{2})$$ Because the exponential function is always positive, it seems to hold the equality $h(\varphi_{D_{A}}(t,x)) = \varphi_{D_{B}}(t,h(x))$. In that case, all that's left is returning to the original matrices $A$ and $B$. I'm having a bit of trouble with this step, though. My questions are:

  1. Is my process right so far? And how can I adjust $h$ so it fits the non-diagonal matrices I have to work with?
  2. Is there a more direct way to find such homeomorphisms? I feel like I wouldn't be able to find them without drawing the phase portaits

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