# Topological conjugation between autonomous and non-autonomous systems

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 $M$.

• What is your idea of taking the extension of just such a form? For $f$ constantly equal to zero, the flow of the autonomous part is $x(t) \equiv x_0$, $\tau(t) = \tau_0 e^{t}$. – user539887 Mar 22 '18 at 21:48
• I was interested in studying trajectories of a singular nonlinear system $\tau\dot x =h(\tau,x)$ close to $\tau = 0$ on a compact manifold. So I basically expanded right-hand side in series in $\tau$ and studied the zero-order approximation, which is nice. So I was thinking that one can view the original system as a small perturbation of the truncated one. – Ivan Mar 23 '18 at 8:14
• If the real part of the largest eigenvalue at an equilibrium $x$ of $f$ is less than one then $M \times \{0\}$ is a normally repelling manifold close to $(x, 0)$. Have you tried to apply the theory of normally hyperbolic manifolds? – user539887 Mar 23 '18 at 8:41