Dynamics of oscillators in resonance

In applications coupled oscillators are only to first order harmonic, and the aim is to understand what happens under perturbation by the nonlinear (higher order) terms. The same reason that makes the unperturbed motion particularly easy renders the perturbation analysis much more complicated when there are resonances among the (unperturbed) frequencies.

My publications on this subject

Richard Cushman, Sebastián Ferrer and Heinz Hanßmann

Singular reduction of axially symmetric perturbations of the isotropic harmonic oscillator

Nonlinearity 12(2), p. 389-410 (1999)

The abstract is available from the publisher and there is a review in the Zentralblatt der Mathematik 952.70017. Another review appeared in Mathematical Reviews 2000h:37093. A preprint version can be downloaded as PostScript file (3.8 M) or in a gzipped version (385 K).

Heinz Hanßmann and Britta Sommer

A degenerate bifurcation in the Hénon-Heiles family

Celestial Mechanics and Dynamical Astronomy 81(3), p. 249-261 (2001)

The abstract is available from the publisher and appeared slightly changed in the Zentralblatt der Mathematik 999.70012. Furthermore there is a review in Mathematical Reviews 2002k:37100. A preprint version can be downloaded as PostScript file (583 K) or in a gzipped version (132 K).

Sebastián Ferrer, Heinz Hanßmann, Jesús Palacián and Patricia Yanguas

On perturbed oscillators in 1-1-1 resonance

the case of axially symmetric cubic potentials

Journal of Geometry and Physics 40(3-4), p. 320-369 (2002)

The abstract is available from the publisher and appeared slightly changed in the Zentralblatt der Mathematik 1037.34031. Furthermore there is a review in Mathematical Reviews 2003f:34076. A preprint version can be downloaded as PostScript file (1.9 M) or in a gzipped version (390 K).

Heinz Hanßmann and Jan-Cees van der Meer

On the Hamiltonian Hopf bifurcations in the 3D Hénon-Heiles family

Journal of Dynamics and Differential Equations 14(3), p. 675-695 (2002)

The abstract is available from the publisher and there are reviews in the Zentralblatt der Mathematik 1035.37038 and in Mathematical Reviews 2003d:37079. A preprint version can be downloaded as PostScript file (359 K) or in a gzipped version (128 K).

Heinz Hanßmann and Jan-Cees van der Meer

Algebraic Methods for Determining Hamiltonian Hopf Bifurcations in Three-Degree-of-Freedom Systems

Journal of Dynamics and Differential Equations 17(3), p. 453-474 (2005)

The abstract is available from the publisher and is reprinted in the Zentralblatt der Mathematik 1078.37041. Furthermore there is a review in Mathematical Reviews 2006e:37103. A preprint version can be downloaded as PostScript file (405 K) or in a gzipped version (181 K).

Richard H. Cushman, Holger R. Dullin, Heinz Hanßmann and Sven Schmidt

The 1:±2 Resonance

p. 642-663 in :

Dynamics and Hamiltonian systems, Utrecht 2007 (eds. J.J. Duistermaat and H. Hanßmann)

Regular and Chaotic Dynamics 12(6) (2007)

The abstract is available from the publisher and is reprinted in Mathematical Reviews 2009i:37138. A preprint version can be downloaded as PostScript file (5.8 M) or in pdf (684 K).

Henk W. Broer, M. Cristina Ciocci, Heinz Hanßmann and André Vanderbauwhede

Quasi-periodic stability of normally resonant tori

Physica D 238(3), p. 309-318 (2009)

The abstract is available from the publisher and is reprinted in the Zentralblatt der Mathematik 1156.3731. Furthermore there is a review in Mathematical Reviews 2011a:37111. A preprint version can be downloaded as PostScript file (452 K) or in pdf (223 K).

Heinz Hanßmann and Jianguo Si

Quasi-periodic solutions and stability of the equilibrium for quasi-periodically forced planar reversible and Hamiltonian systems under the Bruno condition

Nonlinearity 23(3), p. 555-577 (2010)

The abstract is available from the publisher and there are reviews in the Zentralblatt der Mathematik 1196.37101 and in Mathematical Reviews 2011j:37118. A preprint version can be downloaded as PostScript file (489 K) or in pdf (221 K).

Heinz Hanßmann and Carles Simó

Dynamical stability of quasi-periodic response solutions in planar conservative systems

p. 151-166 in :

Coherent Structures in Dynamical Systems, Leiden 2011 (eds. F.J. Beron Vera et al.)

Indagationes Mathematicae 23(3) (2012)

The abstract is available from the publisher and is reprinted in the Zentralblatt der Mathematik Zbl 06070474. Furthermore there is a review in Mathematical Reviews MR2948618. A preprint version can be downloaded as PostScript file (5.4 M) or in pdf (3.6 M).

Holger R. Dullin and Heinz Hanßmann

The degenerate C. Neumann system I : symmetry reduction and convexity

p. 1627-1654 in :

Finite dimensional integrable systems, dynamics, and Lie theoretic methods in Geometry and Mathematical Physics, Jena 2011 (eds. Andreas Čap et al.)

Central European Journal of Mathematics 10(5) (2012)

The abstract is available from the publisher and is reprinted in the Zentralblatt der Mathematik Zbl 06137098. Furthermore there is a review in Mathematical Reviews MR2949643. A preprint version can be downloaded as PostScript file (578 K) or in pdf (357 K).