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2013
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# Florian Wagener

Roetersstraat 11, 1018 WB Amsterdam
University of Amsterdam

## Publications:

 Broer H. W., Hanßmann H., Wagener F. O. Persistence Properties of Normally Hyperbolic Tori 2018, vol. 23, no. 2, pp.  212-225 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 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 DOI:10.1134/S1560354718020065
 Broer H. W., Takens F., Wagener F. O. Integrable and non-integrable deformations of the skew Hopf bifurcation 1999, vol. 4, no. 2, pp.  17-43 Abstract 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 DOI:10.1070/RD1999v004n02ABEH000103