Consider the two one-dimensional linear odes $$\dot x=\lambda_1x\qquad\dot x=\lambda_2x$$ Here $\lambda_1\not=\lambda_2$ and they have the same sign.
Now the solutions to those equations are $x_ie^{\lambda_i t}$, where $x_i$ are some initial conditions. $(i=1,2)$
The flows are $\phi_i^t(x)=xe^{\lambda_i t}$. By Hartman-Grobman's theorem, a homeomorphism that conjugates these flows exists. I want to find one in explicit form. Additionally, can I find a diffeomorphism that conjugates the flows?
Any help would be appreciated.
Thank you, Harald. A desired map is $$x\to x^{\frac{\lambda_1}{\lambda_2}}$$ Another is $$x\to x^{\frac{\lambda_2}{\lambda_1}}$$ Since $\lambda_1\not=\lambda_2$, and they have the same sign, the above maps are diffeomorphisms on the domain of positive reals.