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

Abstract

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 quasiperiodic 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 nonintegrable 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 quasiperiodic case more complicated than the periodic case.

Resonance and Fractal Geometry

“…Generally on the covering space equivariant Singularity Theory can be practised, see [26,56,57] and references therein, as well as equivariant KAM Theory [23,27,33,37,44]. In the present case this construction is only needed for p/q = k/2.…”

“…with q quasi-periodic is dealt with in [27]. In comparison with the case of periodic f the averaged, approximating situation, is identical.…”

On the Computation of Reducible Invariant Tori on a Parallel Computer

“…be a norm over R n , and, if u : T d → R n is a smooth function, we use the standard notation u ∞ = sup θ u(θ) . We assume that we have a parametrization x 0 : T d → R n such that, if we define (4) y 0 (θ) = x 0 (θ + ω) − f (x 0 (θ)), then y 0 ∞ is small. We will denote this as y 0 ∞ ≈ ε.…”

“…Condition (9) has a more technical role: it is required to obtain reducible tori, but its failure is not related to the destruction of the tori. A study of the role of (9) along a family of invariant tori can be found in [4], where the dynamical effects of the failure of this condition are studied. Here, we need condition (9) to compute the Floquet transformation that reduces the linearized flow along the torus to a constant coefficients matrix.…”

“…When ε is small but outside of the set E, the torus can still exist, but with different stability properties and/or it does not need to be unique. For a discussion of this, see [4]. Here we select a small value for ε, and we will try to compute an invariant torus close to the origin, with frequencies ω 0 , .…”