My book defines topological conjugacy between the flows of two differential equations as follows:
Suppose $X' = AX \text{ and } X' = BX$ have flows $\phi ^A$ and $\phi ^B$. These two systems are topologically conjugate if there exists a homeomorphism $h: \mathbb{R}^2 \to \mathbb{R}^2$ that satisfies $\phi ^B (t, h(X_0))= h(\phi ^A (t,X_0))$.
I understand the idea behind homemorphism, but why is conjugacy defined the way it is? Why is $h$ of the initial condition taken on one side while h of the flow is taken on the other? Why isn't it defined like $\phi ^B (t, X_0)= h(\phi ^A (t,X_0))$ instead?
Best Answer
The idea behind it is that: the image of the orbit is the orbit of the image, with corresponding times. And for that you need to have precisely $$\phi^B (t, h(X_0))= h(\phi ^A (t,X_0)).$$ More precisely, an orbit is $$\mathcal O^A(X_0)=\{\phi ^A (t,X_0):t\in\mathbb R\}.$$ Its image is thus $$h(\mathcal O^A(X_0))=\{h(\phi^A (t,X_0)):t\in\mathbb R\}=\{\phi^B (t,h(X_0)):t\in\mathbb R\}=\mathcal O^B(h(X_0)).$$ Summing up, the image of the orbit is the orbit of the image: $h(\mathcal O^A(X_0))=\mathcal O^B(h(X_0))$.