TITLE:
Normal-internal resonances in quasi-periodically forced
oscillators: a conservative approach

AUTHORS:
Henk Broer^(1), Heinz Hanssmann^(2), Angel Jorba^(3),
Jordi Villanueva^(4), Florian Wagener^(5)

(1) broer@math.rug.nl
Instituut voor Wiskunde en Informatica (IWI), 
Rijksuniversiteit Groningen, 
Postbus 800, 9700 AV Groningen, The Netherlands.

(2) Heinz@iram.rwth-aachen.de
Institut fur Reine und Angewandte Mathematik der RWTH Aachen,
52056 Aachen, Germany.

(3) angel@maia.ub.es
Departament de Matematica Aplicada i Analisi,
Universitat de Barcelona,
Gran Via 585, 08007 Barcelona, Spain.

(4) jordi@vilma.upc.es
Departament de Matematica Aplicada I,
Universitat Politecnica de Catalunya,
Diagonal 647, 08028 Barcelona, Spain.

(5) f.o.o.wagener@uva.nl
Center for Nonlinear Dynamics in Economics and Finance (CeNDEF),
Department of Quantitative Economics, 
Universiteit van Amsterdam,
Roetersstraat 11, 
1018 WB Amsterdam, The Netherlands.

ABSTRACT:
We perform a bifurcation analysis of normal-internal resonances in
parametrised families of quasi-periodically forced Hamiltonian
oscillators, for small forcing.  The unforced system is a one degree
of freedom oscillator, called the `backbone' system; forced, the
system is a skew-product flow with a quasi-periodic driving with~$n$
basic frequencies.  The dynamics of the forced system are simplified
by averaging over the orbits of a linearisation of the unforced
system.  The averaged system turns out to have the same structure as
in the well-known case of periodic forcing ($n=1$); for a real
analytic system, the non-integrable part can even be made
exponentially small in the forcing strength.  We investigate the
persistence and the bifurcations of quasi-periodic $n$-dimensional
tori in the averaged system, filling normal-internal resonance `gaps'
that had been excluded in previous analyses.  However, these gaps
cannot completely be filled up: secondary resonance gaps appear, to
which the averaging analysis can be applied again. This phenomenon of
`gaps within gaps' makes the quasi-periodic case more complicated than
the periodic case.