View source for Poincaré return map

''successor mapping, of a smooth or at least continuous [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p0730901.png" /> and a hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p0730902.png" /> transversal to it''

A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p0730903.png" /> that assigns to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p0730904.png" /> that point of the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p0730905.png" /> and the positive semi-trajectory of the flow starting at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p0730906.png" /> that comes first in time (it is defined for those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p0730907.png" /> for which such an intersection exists). (The hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p0730908.png" /> is called a section, an intersecting surface or a transversal.) If the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p0730909.png" /> (so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309010.png" /> is a flow in the plane or on a two-dimensional surface, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309011.png" /> is called a contactless arc) and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309012.png" /> is parametrized by a numerical parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309013.png" />, then the shift of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309014.png" /> under the Poincaré return map is described by some numerical function in one variable (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309015.png" /> corresponds to the parameter value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309017.png" /> corresponds to the parameter value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309018.png" />), which is called the successor function. This mapping was used for the first time by H. Poincaré (see ), that is why it is called the Poincaré return map.

If all semi-trajectories intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309019.png" />, then the Poincaré return map (in this case defined on the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309020.png" />) determines the behaviour of all trajectories of the flow to a considerable extent. However, such  "global"  sections are far from being common (in particular, a Hamiltonian system on a manifold of constant energy which does not pass through critical points of the Hamiltonian, i.e. through equilibrium positions (cf. [[Equilibrium position|Equilibrium position]]), does not have closed — as manifolds — global sections, see [[#References|[3]]], Chapt. 8, Sect. 4.7).

For a non-autonomous system with a periodic right-hand side,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

there exists an analogue of the Poincaré return map: to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309022.png" /> corresponds the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309024.png" /> is the solution of (*) with initial value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309025.png" />. This  "map of the shift by a period"  can be considered even formally as a Poincaré return map if (*) is considered as an autonomous system in  "cylindrical"  phase space. The map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309026.png" /> is defined everywhere if the solutions of (*) are defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309027.png" />.

More often one has to deal with a  "local"  section — it is cut only by a part of the trajectories and often only part of the trajectories intersecting it return again to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309028.png" />. As an example one can consider a small smooth  "surface element"  of codimension 1 intersecting transversally some periodic trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309029.png" />. In this case the Poincaré return map is defined near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309030.png" /> and characterizes the behaviour of the trajectories near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309031.png" />.

In the theory of foliations one can also introduce a Poincaré return map (see [[#References|[2]]]), which is a generalization of the above example (and includes the Poincaré return map for ordinary differential equations in a complex domain).

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  H. Poincaré,   "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''7'''  (1881)  pp. 375–422</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  H. Poincaré,   "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''8'''  (1882)  pp. 251–296</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  H. Poincaré,   "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''1'''  (1885)  pp. 167–244</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top">  H. Poincaré,   "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''2'''  (1886)  pp. 151–217</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I. Tamura,   "Topology of foliations" , Iwanami Shoten  (1976)  (In Japanese)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Godbillion,   "Géométrie différentielle et mécanique analytique" , Hermann  (1969)</TD></TR></table>



====Comments====
The return map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309032.png" /> faithfully reflects many properties of the flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309033.png" />. For example, a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309034.png" /> which is periodic under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309035.png" /> is necessarily periodic under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309036.png" />, with a possibly different period. Moreover, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309037.png" />-orbit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309038.png" /> is asymptotically stable if and only if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309039.png" />-orbit has the same property. In many ways the discrete-time dynamics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309040.png" /> are easier to analyze than the continuous-time dynamics of the original flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309041.png" />. Poincaré exploited these ideas in his study of homoclinic orbits in the three-body problem [[#References|[a7]]].

Every diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309042.png" /> of a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309043.png" /> can be identified with a return map: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309044.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309045.png" /> by the identifications <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309046.png" />; the flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309048.png" /> is induced by the flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309049.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309050.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309051.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309052.png" /> is a global section with return map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309053.png" />. By this construction many results proved for flows can be applied to diffeomorphisms.

For the case when all semi-trajectories intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309054.png" /> see [[#References|[a8]]].

The  "cylindrical"  phase space mentioned above is defined as follows. Consider the autonomous system associated with (*), i.e.,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

Identify the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309056.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309057.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309058.png" /> in the domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309059.png" />; note that the latter is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309061.png" /> is a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309062.png" /> (when (*) is defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309063.png" />). Then (a1) defines a dynamical system on the  "cylinder"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309065.png" /> is the closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309066.png" /> with the end-points identified, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309067.png" /> is a circle. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309068.png" /> considered above now coincides with the Poincaré map of the system (a1) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309069.png" /> in the hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309070.png" />.

For the existence of global sections, see e.g. [[#References|[a2]]], Sect. IV.2, and [[#References|[a3]]]. In the context of more general transformation groups one speaks of  "global sliceglobal slices" ; see e.g. [[#References|[a1]]]. As to the existence of local sections in non-differentiable dynamical systems, see [[#References|[a4]]], Sect. VI.2. In the theory of foliations one can recover the generalization of the Poincaré return map in the generators of the (leaf) holonomy groups. See e.g. [[#References|[a6]]].

For applications of the Poincaré return map in the theory of differential equations (behaviour near a periodic orbit), see e.g. [[#References|[a5]]] (so-called  "Floquet theoryFloquet theory" ).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Abels,   "Parallelizability of proper actions, global <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073090/p07309071.png" />-slices and maximal compact subgroups"  ''Math. Ann.'' , '''212'''  (1974)  pp. 1–19</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.P. Bhatia,   G.P. Szegö,   "Stability theory of dynamical systems" , Springer  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O. Hajek,   "Parallelizability revisited"  ''Proc. Amer. Math. Soc.'' , '''27'''  (1971)  pp. 77–84</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  O. Hajek,   "Dynamical systems in the plane" , Acad. Press  (1968)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Hector,   U. Hirsch,   "Introduction to the geometry of foliations" , Vieweg  (1981)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Poincaré,   "Les méthodes nouvelles de la mécanique céleste" , '''I''' , Gauthier-Villars  (1899)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  D. Fried,   "The geometry of cross-sections to flows"  ''Topology'' , '''21'''  (1982)  pp. 353–371</TD></TR></table>