Henk Broer et al 2003 Nonlinearity 16 1751 doi:10.1088/0951-7715/16/5/312
Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach
Henk Broer1, Heinz Hanßmann2, Àngel Jorba3, Jordi Villanueva4 and Florian Wagener5
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We perform a bifurcation analysis of normal-internal resonances in parametrized 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 linearization 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.
- PACS
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05.45.Xt Synchronization; coupled oscillators
- MSC
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34C15 Nonlinear oscillations, coupled oscillators
37J40 Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
- Subjects
- Dates
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Issue 5 (September 2003)
Received 23 December 2002, in final form 14 May 2003
Published 18 July 2003
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Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach
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