COMPLETE POINCARE SECTIONS AND TANGENT SETS

Trying to extend a local definition of a surface of a section, and the corresponding Poincare map to a global one, one can encounter severe difficulties. We show that global transverse sections often do not exi st for Hamiltonian systems with two degrees of freedom. As a consequen ce we present a method to generate the so-called W-section, which by c onstruction will be intersected by (almost) all orbits. Depending on t he type of tangent set in the surface of the section, we distinguish f ive types of W-sections. The method is illustrated by a number of exam ples, most notably the quartic potential and the double pendulum. W-se ctions can also be applied to higher dimensional Hamiltonian systems a nd to dissipative systems.