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.

  • $\begingroup$ 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 $x(t) \equiv x_0$, $\tau(t) = \tau_0 e^{t}$. $\endgroup$ – user539887 Mar 22 '18 at 21:48
  • $\begingroup$ 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. $\endgroup$ – Ivan Mar 23 '18 at 8:14
  • $\begingroup$ 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? $\endgroup$ – user539887 Mar 23 '18 at 8:41
  • $\begingroup$ No, I have never seen this theory =) Could you suggest some references? $\endgroup$ – Ivan Mar 23 '18 at 9:02
  • $\begingroup$ There is a book by Stephen Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Springer, 1994. Perhaps you could ask also on MO (to me, this a research problem). $\endgroup$ – user539887 Mar 23 '18 at 14:23

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