Let $M$ be a compact manifold and $T=\mathbb{R}$ or $T=\mathbb{Z}$. We assume we are given two continuous flows $$\Phi_1: T \times M \rightarrow M$$ and $$\Phi_2: T \times M \rightarrow M$$ Let's further assume we can choose two continuous functions $h,h'$.
My question is the following: Can we always choose $h,h'$ in a way such that $$\Phi_1(t,x)=h' \circ \Phi_2(t,h(x)) \quad \forall x \in M, t \in T$$
To me it is quite clear that this is always possible locally, but I am not sure if it is also possible globally.