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Regul. Chaotic Dyn., 2018, Volume 23, Issue 2, Pages 212–225 (Mi rcd319)  

Persistence Properties of Normally Hyperbolic Tori

Henk Broera, Heinz Hanßmannb, Florian Wagenerc

a Johann Bernoulli Institute for Mathematics and Computer Science, Rijksuniversiteit Groningen, 9747 AG Groningen, The Netherlands
b Mathematisch Instituut, Universiteit Utrecht, Postbus 80010, 3508 TA Utrecht, The Netherlands
c Center for Nonlinear Dynamics in Economics and Finance (CeNDEF), Amsterdam School of Economics, Universiteit van Amsterdam, Postbus 15867, 1001 NJ Amsterdam, The Netherlands

Abstract: 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

DOI: https://doi.org/10.1134/S1560354718020065

References: PDF file   HTML file

Bibliographic databases:

Document Type: Article
MSC: 37J40 37D10 37G35 37J20
Received: 18.11.2017
Accepted:09.01.2018
Language: English

Citation: Henk Broer, Heinz Hanßmann, Florian Wagener, “Persistence Properties of Normally Hyperbolic Tori”, Regul. Chaotic Dyn., 23:2 (2018), 212–225

Citation in format AMSBIB
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\by Henk Broer, Heinz Han{\ss}mann, Florian Wagener
\paper Persistence Properties of Normally Hyperbolic Tori
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 2
\pages 212--225
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\crossref{https://doi.org/10.1134/S1560354718020065}
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