Florian Wagener

Roetersstraat 11, 1018 WB Amsterdam
University of Amsterdam


Broer H. W., Hanßmann H., Wagener F. O.
Persistence Properties of Normally Hyperbolic Tori
2018, vol. 23, no. 2, pp.  212-225
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, vol. 23, no. 2, pp. 212-225
Broer H. W., Takens F., Wagener F. O.
In the skew Hopf bifurcation a quasi-periodic attractor with nontrivial normal linear dynamics loses hyperbolicity. Periodic, quasi-periodic and chaotic dynamics occur, including motion with mixed spectrum. The case of $3$-dimensional skew Hopf bifurcation families of diffeomorphisms near integrability is discussed, surveying some recent results in a broad perspective. One result, using KAM-theory, deals with the persistence of quasi-periodic circles. Other results concern the bifurcations of periodic attractors in the case of resonance.
Citation: Broer H. W., Takens F., Wagener F. O.,  Integrable and non-integrable deformations of the skew Hopf bifurcation, Regular and Chaotic Dynamics, 1999, vol. 4, no. 2, pp. 17-43

Back to the list