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?

specificresults 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