The stroboscopic map of a periodic model and the Poincare return map of a transversal section: domains, iterates, fixed points and the local C1 diffeomorphism property

(i) The stroboscopic map. Let a $P$-periodic stability model be fixed ($P>0$; The model vocabulary of stability and chaos: the associated flow of a declared force model, equilibria, periodic and homoclinic orbits, and invariant sets clause (i)). Its restriction domain $D\subseteq \mathbb{R}\times Q\times {\mathbb{R}}^{3N}$ is invariant under time shifts by $P$. Define its time-zero state slice $S_P:=\{z_0\in Q\times {\mathbb{R}}^{3N}:(0,z_0)\in D\};$ periodicity gives the same admissible slice at every integer multiple of $P$. For $z_0\in S_P$, let $z_D(\cdot ;z_0):I_{z_0}^D\to Q\times {\mathbb{R}}^{3N}$ be the unique $D$-maximal state curve through $z_0$ at time $0$, with interval $I_{z_0}^D$ as in The model vocabulary of stability and chaos: the associated flow of a declared force model, equilibria, periodic and homoclinic orbits, and invariant sets clause (i). The stroboscopic map at period $P$ is the partial map $\mathrm{\Psi }:\mathrm{\Sigma }⟶S_P,\mathrm{\Sigma }:=\{z_0\in S_P:[0,P]\subseteq I_{z_0}^D\},\mathrm{\Psi }(z_0):=z_D(P;z_0),$ with domain $\mathrm{\Sigma }$ the states whose $D$-maximal motion remains in the declared restriction domain throughout $[0,P]$. Its endpoint belongs to $S_P$ because the restriction domain is preserved by the shift from $P$ to $0$. Its iterates are defined recursively by ${\mathrm{\Psi }}^1:=\mathrm{\Psi }$ and, where ${\mathrm{\Psi }}^n(z_0)\in \mathrm{\Sigma }$, ${\mathrm{\Psi }}^{n+1}(z_0):=\mathrm{\Psi }({\mathrm{\Psi }}^n(z_0))$; the domain of ${\mathrm{\Psi }}^n$ is therefore the set of states whose $D$-maximal motion remains in $D$ over $[0,nP]$, and it may be strictly smaller than $\mathrm{\Sigma }$. This domain identity follows by induction from the $P$-periodicity of the field, shift invariance of $D$ and uniqueness of the $D$-maximal solution. A state $z_{\ast }\in \mathrm{\Sigma }$ with $\mathrm{\Psi }(z_{\ast })=z_{\ast }$ is a fixed point of $\mathrm{\Psi }$. It is a $P$-periodic state in the full-time sense: the motion through $z_{\ast }$ extends to a motion on $\mathbb{R}$ satisfying $z(t+P)=z(t)$ for every $t\in \mathbb{R}$. Indeed, write the state curve on $[0,P]$ as $z_{\ast }(t)$; for $k\in \mathbb{Z}$ and $s\in [0,P]$ set $\tilde{z}(kP+s):=z_{\ast }(s)$. The endpoint states agree because $z_{\ast }(P)=z_{\ast }(0)$, and the vector field agrees at the seams by its $P$-periodicity. The restriction domain is preserved by those time shifts, so the translated segments solve the same equation; uniqueness glues them to a global motion. No completeness of $\mathrm{\Psi }$ is asserted: $\mathrm{\Sigma }$ need not be all of $S_P$, and iterates can leave the domain.

(ii) Autonomous special case. If the model is autonomous, then its stroboscopic map at any period $P>0$ is the time-$P$ map ${\mathrm{\Phi }}_P$ of its local flow restricted to ${\mathrm{\Omega }}_P=\{z\in S_0:(P,z)\in \mathrm{\Omega }\}$ (The model vocabulary of stability and chaos: the associated flow of a declared force model, equilibria, periodic and homoclinic orbits, and invariant sets clause (ii)); this is the reading used for the isotropic-oscillator check on the page.

(iii) The Poincare return map. Let an autonomous model with force map of class $C^1$ in the state be fixed, with associated $C^1$ field $G$ on $S_0$ and local flow $\mathrm{\Phi }:\mathrm{\Omega }\to S_0$ (The flow domain of a C¹ autonomous field is open and the flow is C¹ on it). Let $z_0\in S_0$, $t_0>0$ with $(t_0,z_0)\in \mathrm{\Omega }$, put $z_1:=\mathrm{\Phi }(t_0,z_0)$, let $W\subseteq S_0$ be open with $z_0,z_1\in W$, and let $h:W\to \mathbb{R}$ be of class $C^1$ with $h(z_0)=h(z_1)=0$ and $Dh(z_0)G(z_0)\mathrm{\ne }0$, $Dh(z_1)G(z_1)\mathrm{\ne }0$. The section is ${\mathrm{\Sigma }}_h:=\{z\in W:h(z)=0\}$, and the reference orbit meets it transversally at $z_0$ and again at $z_1$. Then A transversal return of a C¹ flow yields a local C¹ Poincaré map, with inverse given by the reversed crossing supplies open sets $A\ni z_0$, $B\ni z_1$ and $C^1$ functions $\tau :A\to \mathbb{R}$, $\sigma :B\to \mathbb{R}$ with $\tau (z_0)=\sigma (z_1)=t_0$, and an open time interval $J\ni t_0$ as in clause 1 of that proposition. Restrict $A$ to the open positive-time branch $A_{+}:=A\cap {\tau }^{-1}((0,\mathrm{\infty }))$, which contains $z_0$ by continuity and $t_0>0$, and restrict the return map to $A_{+}\cap {\mathrm{\Sigma }}_h$. Then $P:A_{+}\cap {\mathrm{\Sigma }}_h⟶{\mathrm{\Sigma }}_h,P(z):=\mathrm{\Phi }(\tau (z),z)$ is a $C^1$ diffeomorphism onto $P(A_{+}\cap {\mathrm{\Sigma }}_h)$, which is open in the subspace topology of ${\mathrm{\Sigma }}_h$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace), with inverse $Q{\mathrm{\mid }}_{P(A_{+}\cap {\mathrm{\Sigma }}_h)}$, where $Q(z)=\mathrm{\Phi }(-\sigma (z),z)$; moreover $h(\mathrm{\Phi }(t,z))=0$ for $(t,z)\in J\times A_{+}$ holds exactly for $t=\tau (z)$, so $P$ is the local return map at the reference crossing. The section near the two crossings has the embedded $C^1$ charts established by the proposition. If the open reference segment satisfies $\mathrm{\Phi }(s,z_0)\mathrm{\notin }{\overline{{\mathrm{\Sigma }}_h}}^{S_0}$ for every $0<s<t_0$ (closure in the state domain), clause 4 of the proposition supplies persistence of the absence of earlier hits: for $z\in A_{+}\cap {\mathrm{\Sigma }}_h$ the positive return time $\tau (z)$ is the least positive hitting time. Under this explicit additional hypothesis $P$ is the local first-return map. Transversality alone gives the return branch near $t_0$, and permits earlier hits. A state $z_{\ast }\in A_{+}\cap {\mathrm{\Sigma }}_h$ with $P(z_{\ast })=z_{\ast }$ is a fixed point of the return map, and the motion through $z_{\ast }$ is periodic with period $\tau (z_{\ast })$.

(iv) Conventions and scope. (a) Both maps are attached to the declared model and its flow; the stroboscopic map depends on the chosen period $P$ and the return map on the chosen section and reference crossing, and different choices give different (conjugate only under the stated conditions) maps. (b) The return time $\tau$ is a $C^1$ function on the domain, not a constant; the reference crossing time $t_0$ is only one value of it. (c) Neither map is asserted to be defined on all of phase space, and no statement here asserts that a fixed point, a periodic orbit or a return exists for any particular model. (d) Periods and return times are measured in the frame time $[t]=\tau$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework); the positivity required for the periodicity reading of clause (iii) is built into the chosen local return branch.