# A question about conjugacy between two systems

We've been doing some examples of showing when two specific systems are conjugate in class. I came across this more general question and I'm having trouble finding such a conjugacy.

Suppose there are two systems $(\mathbb{R}^n, F)$ and $(\mathbb{R}^n, G)$ where $F(x) = Mx + b$ and $G(x) = Mx$ for some matrix $M$ without eigenvalue 1. Then, there is a conjugacy between these two systems, that is, we can find a homeomorphism $h: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $h \circ F = G \circ h$.

Any help would be greatly appreciated. Thank you in advance!

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 without eigenvalue 1 - what's the significance of that, I wonder... – anon Sep 26 '12 at 3:09 Well I am not sure what is exactly that you are asking but on dynamical systems equivalence you could check Arnold's book "Geometrical Methods in the theory of ordinary differential equations", chapter 3 on structural stability, especially 3.10 where he explains the notion of stability. I hope this helps. – tst Nov 6 '12 at 0:28