Appendix 2. Poincare return mappings
L«j,t p be a point lying on a closed orbit of a Cr(rSs l) flow ip over a manifold M and suppose the period of the orbit through p is r so <A(r, p) = p. Now let X be an open disc embedded in M and intersecting p so that (v(x)) @tpX - tpM (where © is the direct sum, tp the tangent map at p and v(x) is the velocity of </> at x). The submanifold X is said to be transverse to the orbit through p and is a cross-section to the flow at p.
For some small open neighbourhood U about p in X there exists a unique, continuous function it: U-+R so that ir(p) = r and 4>(tt(s), X where s EX. The function ir is the first return function for X. Roughly speaking, rr(x) gives the time taken by a point beginning at x to traverse its orbit forwards in time along the flow to the next point of intersection with X, see fig. 7. If this new point of intersection is P(s) then P: defined by 
is the Poincare map for X. For U sufficiently small the first return function ir is well-defined and C and the Poincare map P is a well-defined C diffeomorphism of U onto an open subset of X.