form of a stable hyperbolic periodic orbit of a flow. The result is proved using a combination of Proposition \[prop:sternberg\] and stable manifold theory [@fenichel1974asymptotic; @fenichel1977asymptotic; @hirsch1977; @de1995irwin] specialized to the theory of isochrons [@isochrons].

\[prop:floq-norm-form\] Fix $k\in {\mathbb{N}}_{\geq 1}\cup \{\infty\}$ and $0 \leq \alpha \leq 1$. Let $\Phi\colon Q \times {\mathbb{R}}\to Q$ be a ${C_{\textnormal{loc}}^{k,\alpha}}$ flow having a globally attracting hyperbolic $\tau$-periodic orbit with image $\Gamma \subset Q$, where $Q$ is a smooth manifold. Fix $x_0 \in \Gamma$ and let $E^s_{x_0}\subset {\mathsf{T}}_{x_0}Q$ denote the unique ${\mathsf{D}}_{x_0}\Phi^\tau$-invariant subspace complementary to ${\mathsf{T}}_{x_0}\Gamma$. Assume that $\nu({\mathsf{D}}_{x_0} \Phi^\tau|_{E^s_{x_0}},{\mathsf{D}}_{x_0} \Phi^\tau|_{E^s_{x_0}}) < k + \alpha$, and assume that $({\mathsf{D}}_{x_0} \Phi^\tau|_{E^s_{x_0}},{\mathsf{D}}_{x_0} \Phi^\tau|_{E^s_{x_0}})$ is $k$-nonresonant.

Then if we write ${\mathsf{D}}_{x_0}\Phi^\tau|_{E^s_{x_0}}=e^{\tau A}$ for some complex linear $A\colon (E^s_{x_0})_{\mathbb{C}}\to (E^s_{x_0})_{\mathbb{C}}$, there exists a unique proper ${C_{\textnormal{loc}}^{k,\alpha}}$ embedding $\psi = (\psi_\theta, \psi_z) \colon Q\to S^1 \times (E^s_{x_0})_{\mathbb{C}}$ such that $\psi_\theta(x_0) = 1$, $({\mathsf{D}}_{x_0}\psi_z)|_{E^s_{x_0}} = (E^s_{x_0} \hookrightarrow (E^s_{x_0})_{\mathbb{C}})$, and $$\label{eq:floquet}
        \forall t \in {\mathbb{T}}\colon \psi_\theta \circ \Phi^t(x) = e^{2\pi i \frac{t}{\tau}}\psi_\theta(x), \qquad \psi_z\circ \Phi^t(x) = e^{tA} \psi_z(x),$$ where $S^1\subset {\mathbb{C}}$ is the unit circle. If $A\colon E^s_{x_0}\to E^s_{x_0}$ is real, then $\psi_z\in {C_{\textnormal{loc}}^{k,\alpha}}(Q, E^s_{x_0})$ is real, and the codomain-restricted map $\psi \colon Q \to S^1 \times E^s_{x_0}\subset S^1\times (E^s_{x_0})_{\mathbb{C}}$ is a 