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My question is the following: Assume that the flow of an autonomous ODE $\dot{x} = f(x)$ ($f$ is $C^1$) has a periodic hyperbolic orbit $\varphi^t(x_0)$, $\varphi^{t+T}(x_0) = \varphi^t(x_0)$. Then consider a ($C^1$) small perturbation of the ODE, e.g., $\dot{x} = f(x) + \epsilon(t)$. Without any further assumption on $\epsilon$ (except for regularity type assumptions), can we say what happens to the periodic orbit? Will there be, e.g., a non-periodic hyperbolic orbit in the vicinity of $\{\varphi^t(x_0)\}$, that corresponds to the given periodic orbit (via shadowing)?

More general, is there a theorem that describes what happens to a uniformly hyperbolic set of a flow under time-varying perturbations?

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  • $\begingroup$ It seems probable, however that "non-periodic hyperbolic orbit" should be (the projection on the fibers of) an orbit of some skew-product flow, with the base space equal to the (compact) closure of the time-translates of $\epsilon$ in an appropriate topology. $\endgroup$ – user539887 Dec 12 '18 at 20:03
  • $\begingroup$ Thank you. Actually, I was thinking of systems that are already embedded in a skew-product, where the base space consists of the possible perturbations $\epsilon(\cdot)$ with the shift flow. Do you have a specific shadowing result in mind that is applicable to continuous-time skew-products? $\endgroup$ – C. Kawan Dec 13 '18 at 9:03
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    $\begingroup$ No, I do not have any specific results in mind. Perhaps D. Dragičević Admissibility, a general type of Lipschitz shadowing and structural stability? Or (some) papers by L. Barreira and C. Valls? There was a special issue of DCDS-B on Nonautonomous hyperbolicity and related aspects. Unfortunately, the majority of the papers are behind a paywall. $\endgroup$ – user539887 Dec 14 '18 at 9:23

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