TITLE:
The parametrically forced pendulum: a case study in one and one half degrees
of freedom

AUTHORS:
H.W. Broer (broer@math.rug.nl), I. Hoveijn (I.Hoveijn@math.rug.nl), M. van Noort (M.van.Noort@math.rug.nl), 
   Dept. of Mathematics, University of Groningen,
   PO Box 800, 9700 AV Groningen, The Netherlands,
Carles Sim\'o (carles@maia.ub.es) 
   Dept. de Matem\`atica Aplicada i An\`alisi, Universitat
   de Barcelona, Gran Via 585, 08007 Barcelona, Spain
   and
G. Vegter (G.Vegter@cs.rug.nl)
   Dept. of Mathematics, University of Groningen,
   PO Box 800, 9700 AV Groningen, The Netherlands

ABSTRACT:
  
  This paper is concerned with the global coherent (i.e., non-chaotic) dynamics
  of the parametrically forced pendulum.  The system is studied in a one and one
  half degree of freedom Hamiltonian setting with two parameters, where a
  spatio-temporal symmetry is taken into account.  Our explorations are
  restricted to sufficiently large regions of coherent dynamics in phase space
  and parameter plane.  At any given parameter point we restrict to a bounded
  subset of phase space, using KAM theory to exclude an infinitely large region
  with trivial dynamics.
  
  In the absence of forcing the system is integrable.  Analytical and numerical
  methods are used to study the dynamics in a parameter region away from
  integrability, where the results of a perturbation analysis of the nearly
  integrable case are used as a starting point.  We organize the dynamics by
  dividing the parameter plane in fundamental domains, guided by the linearized
  system at the upper and lower equilibria.
  
  Away from integrability some features of the nearly integrable coherent
  dynamics persist, while new bifurcations arise.  On the other hand, the
  chaotic region increases.