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Heinz Hanßmann

Postbus 80010, 3508 TA Utrecht, The Netherlands
Mathematisch Instituut, Universiteit Utrecht


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
Hanßmann H.
Quasi-periodic bifurcations in reversible systems
2011, vol. 16, no. 1-2, pp.  51-60
Invariant tori of integrable dynamical systems occur both in the dissipative and in the conservative context, but only in the latter the tori are parameterized by phase space variables. This allows for quasi-periodic bifurcations within a single given system, induced by changes of the normal behavior of the tori. It turns out that in a non-degenerate reversible system all semi-local bifurcations of co-dimension 1 persist, under small non-integrable perturbations, on large Cantor sets.
Keywords: invariant tori, KAM theory, versal unfolding, persistence
Citation: Hanßmann H.,  Quasi-periodic bifurcations in reversible systems, Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, pp. 51-60
Hanßmann H., Homburg A. J., van Strien S.
2011, vol. 16, no. 1-2, pp.  1
Citation: Hanßmann H., Homburg A. J., van Strien S.,  Foreword, Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, pp. 1
Cushman R., Dullin H. R., Hanßmann H., Schmidt S.
The $1:\pm2$ Resonance
2007, vol. 12, no. 6, pp.  642-663
On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Non-linear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio $\pm 1/2$ and its unfolding. In particular we show that for the indefinite case ($1:-2$) the frequency ratio map in a neighborhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is non-degenerate (i.e. the Kolmogorov non-degeneracy condition holds) for every torus in a neighborhood of the equilibrium point. As a by product of our analysis of the frequency map we obtain another proof of fractional monodromy in the $1:-2$ resonance.
Keywords: resonant oscillators, normal form, singular reduction, bifurcation, energy-momentum mapping, monodromy
Citation: Cushman R., Dullin H. R., Hanßmann H., Schmidt S.,  The $1:\pm2$ Resonance, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 642-663
Duistermaat J. J., Hanßmann H.
SPECIAL ISSUE. Dedicated to Richard Cushman on the occasion of his 65th birthday
2007, vol. 12, no. 6, pp.  577-577
Citation: Duistermaat J. J., Hanßmann H.,  SPECIAL ISSUE. Dedicated to Richard Cushman on the occasion of his 65th birthday, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 577-577
Hanßmann H.
Quasi-periodic Motions of a Rigid Body. I. Quadratic Hamiltonians on the Sphere with a Distinguished Parameter
1997, vol. 2, no. 2, pp.  41-57
The motion of a dynamically symmetric rigid body, fixed at one point and subject to an affine (constant+linear) force field is studied. The force being weak, the system is treated as a perturbation of the Euler top, a superintegrable system. Averaging along the invariant $2$-tori of the Euler top yields a normal form which can be reduced to one degree of freedom, parametrized by the corresponding actions. The behaviour of this family is used to identify quasi-periodic motions of the rigid body with two or three independent frequencies.
Citation: Hanßmann H.,  Quasi-periodic Motions of a Rigid Body. I. Quadratic Hamiltonians on the Sphere with a Distinguished Parameter, Regular and Chaotic Dynamics, 1997, vol. 2, no. 2, pp. 41-57

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