Persistence Properties of Normally Hyperbolic Tori

Henk Broer*, Heinz Hanßmann, Florian Wagener

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


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.

Original languageEnglish
Pages (from-to)212-225
Number of pages14
JournalRegular and Chaotic Dynamics
Issue number2
Publication statusPublished - 1 Mar 2018


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


Dive into the research topics of 'Persistence Properties of Normally Hyperbolic Tori'. Together they form a unique fingerprint.

Cite this