Persistence Properties of Normally Hyperbolic Tori

    2018, Volume 23, Number 2, pp.  212-225

    Author(s): Broer H. W., Hanßmann H., Wagener F. O.

    Near-resonances between frequencies notoriously lead to small denominators when trying to prove persistence of invariant tori carrying quasi-periodic motion. In dissipative systems external parameters detuning the frequencies are needed so that Diophantine conditions can be formulated, which allow to solve the homological equation that yields a conjugacy between perturbed and unperturbed quasi-periodic tori. The parameter values for which the Diophantine conditions are not fulfilled form so-called resonance gaps. Normal hyperbolicity can guarantee invariance of the perturbed tori, if not their quasi-periodicity, for larger parameter ranges. For a 1-dimensional parameter space this allows to close almost all resonance gaps.
    Keywords: KAM theory, normally hyperbolic invariant manifold, van der Pol oscillator, Hopf bifurcation, center-saddle bifurcation
    Citation: Broer H. W., Hanßmann H., Wagener F. O., Persistence Properties of Normally Hyperbolic Tori, Regular and Chaotic Dynamics, 2018, Volume 23, Number 2, pp. 212-225

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