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This article is cited in 3 scientific papers (total in 3 papers)
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.
Received: 18.11.2017 Accepted: 09.01.2018
Citation:
Henk Broer, Heinz Hanßmann, Florian Wagener, “Persistence Properties of Normally Hyperbolic Tori”, Regul. Chaotic Dyn., 23:2 (2018), 212–225
Linking options:
https://www.mathnet.ru/eng/rcd319 https://www.mathnet.ru/eng/rcd/v23/i2/p212
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