I am pretty sure that the answer should be known, but I can't find a reference.
Let $M$ be a manifold. Consider an ODE on the extended phase space $M\times \mathbb{R}$ of the form $$ \dot x = f(x) + \tau g(\tau,x) $$ $$ \dot \tau = \tau $$ where $x\in M,\tau \in \mathbb{R}$ and $f,g$ are smooth maybe even analytic vector fields. Is it true that in the neighbourhood of submanifold $\tau = 0$ this flow is topologically conjugate to the flow of the autonomous part $$ \dot x = f(x)\\ \dot \tau = \tau $$
In my case all fixed points of $f(x)$ are hyperbolic, so one can use the Grobman-Hartman theorem to deduce a local result. I don't understand if it is possible perhaps to glue this into something global on $M$.
Thank you in advance.