Regular and Chaotic Dynamics

, Volume 23, Issue 2, pp 212–225 | Cite as

Persistence Properties of Normally Hyperbolic Tori

  • Henk BroerEmail author
  • Heinz Hanßmann
  • Florian Wagener


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.


KAM theory normally hyperbolic invariant manifold van der Pol oscillator Hopf bifurcation center-saddle bifurcation 


37J40 37D10 37G35 37J20 


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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Johann Bernoulli Institute for Mathematics and Computer Science Rijksuniversiteit GroningenGroningenThe Netherlands
  2. 2.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherlands
  3. 3.Center for Nonlinear Dynamics in Economics and Finance (CeNDEF) Amsterdam School of EconomicsUniversiteit van AmsterdamAmsterdamThe Netherlands

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