Evolution of the ``last'' invariant curve in a family of area preserving maps

Carles Sim\'o, Departament de Matem\`atica Aplicada i An\`alisi, Universitat
   de Barcelona, Gran Via 585, 08007 Barcelona, Spain

Dmitry V. Treschev, Department of Mathematics, Moscow State University,
   Moscow, Russia

Abstract

Area preserving maps appear in a natural way in Poincar\'e sections of 
Hamiltonian systems with two degrees of freedom. Around an elliptic fixed
point there are, generically, invariant curves bounding a stable region.
This note proposes a numerical method, strongly based on the dynamics, to
locate the ``last'' invariant curve, that is, the boundary of the stable
connected component containing the elliptic fixed point. The variation of
this curve and the corresponding rotation number when a parameter is 
changed are displayed. Several examples illustrate the procedure and some 
properties are discussed.