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.