Bifurcations in Hamiltonian systems

The type of bifurcation can often be deduced from the normal symplectic behaviour. Where the latter has one degree of freedom the bifurcation may be explained by the universal unfolding of the corresponding singularity of the Hamiltonian function. KAM theory can then be used to show that the bifurcation scenario is a valid description of the quasi-periodic dynamics.

My publications on this subject

Heinz Hanßmann

The reversible umbilic bifurcation

p. 81-94 in :

Time-reversal symmetry in dynamical systems, Warwick 1996 (ed. J.S.W. Lamb)

Physica D 112(1-2) (1998)

The abstract is available from the publisher and there is a review in Mathematical Reviews 99b:58171. A preprint version can be downloaded as PostScript file (3.3 M) or in a gzipped version (941 K).

Heinz Hanßmann

The Quasi-Periodic Centre-Saddle Bifurcation

Journal of Differential Equations 142(2), p. 305-370 (1998)

The abstract is available from the publisher and is reprinted in Mathematical Reviews 98m:58099. Furthermore there is a review in the Zentralblatt der Mathematik 936.37028. A preprint version can be downloaded as PostScript file (2.0 M) or in a gzipped version (328 K).

Henk W. Broer, Heinz Hanßmann, Àngel Jorba, Jordi Villanueva and Florian Wagener

Normal-internal resonances in quasi-periodically forced oscillators

a conservative approach

Nonlinearity 16(5), p. 1751-1791 (2003)

The abstract is available from the publisher and there are reviews in the Zentralblatt der Mathematik 1049.34044 and in Mathematical Reviews 2004i:37111. A preprint version can be downloaded as PostScript file (12 M) or in a gzipped version (3.5 M).

Heinz Hanßmann

A Survey on Bifurcations of Invariant Tori

p. 109-121 in :

New Advances in Celestial Mechanics and Hamiltonian systems, Guanajuato 2001 (eds. J. Delgado et al.)

Kluwer/Plenum (2004)

The abstract is reprinted in Mathematical Reviews 2005c:37101. A preprint version can be downloaded as PostScript file (485 K) or in a gzipped version (98 K).

Heinz Hanßmann

Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems

Results and Examples

Habilitationsschrift, RWTH Aachen (2004)

Heinz Hanßmann

Hamiltonian Torus Bifurcations Related to Simple Singularities

p. 679-685 in :

Proceedings of Dynamic Systems and Applications 4, Atlanta 2003 (eds. G.S. Ladde et al.)

Dynamic Publishers (2004)

The abstract is reprinted both in Mathematical Reviews 2005j:37089 and the Zentralblatt der Mathematik 1071.37035. A preprint version can be downloaded as PostScript file (314 K) or in a gzipped version (123 K).

Henk W. Broer, Heinz Hanßmann, Àngel Jorba, Jordi Villanueva and Florian Wagener

Quasi-periodic response solutions at normal-internal resonances

p. 702-707 in :

Equadiff 2003, Hasselt 2003 (eds. F. Dumortier, H.W. Broer, J. Mawhin, A. Vanderbauwhede and S. Verdyun Lunel)

World Scientific (2005)

The abstract is available from the publisher and there is a review in the Zentralblatt der Mathematik 1106:37040. A preprint version can be downloaded as PostScript file (11 M) or in a gzipped version (3.1 M).

Heinz Hanßmann

Hamiltonian bifurcations of invariant tori with a vanishing Floquet exponent

p. 732-737 in :

Equadiff 2003, Hasselt 2003 (eds. F. Dumortier, H.W. Broer, J. Mawhin, A. Vanderbauwhede and S. Verdyun Lunel)

World Scientific (2005)

The abstract is available from the publisher and there is a review in the Zentralblatt der Mathematik 1107:37043. A preprint version can be downloaded as PostScript file (154 K) or in a gzipped version (65 K).

Henk W. Broer, Heinz Hanßmann and Jiangong You

Bifurcations of normally parabolic tori in Hamiltonian systems

Nonlinearity 18(4), p. 1735-1769 (2005)

The abstract is available from the publisher and is reprinted in the Zentralblatt der Mathematik 1181.37072. Furthermore there is a review in Mathematical Reviews 2006a:37050. A preprint version can be downloaded as PostScript file (799 K) or in a gzipped version (337 K).

Henk W. Broer, Heinz Hanßmann and Jiangong You

Umbilical Torus Bifurcations in Hamiltonian Systems

Journal of Differential Equations 222(1), p. 233-262 (2006)

The abstract is available from the publisher and is reprinted in the Zentralblatt der Mathematik 1085.37048. Furthermore there is a review in Mathematical Reviews 2006i:37122. A preprint version can be downloaded as PostScript file (2.4 M) or in a gzipped version (556 K).

Heinz Hanßmann

On Hamiltonian bifurcations of invariant tori with a Floquet multiplier -1

Dynamical systems 21(2), p. 115-145 (2006)

The abstract is available from the publisher and appeared slightly changed in the Zentralblatt der Mathematik 1136.37346. Furthermore there is a review in Mathematical Reviews 2007f:37078. A preprint version can be downloaded as PostScript file (2.2 M) or in a gzipped version (710 K).

Heinz Hanßmann

Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems

Results and Examples

LNM 1893, Springer (2007)

The table of contents is available from the publisher and there are reviews in the Zentralblatt der Mathematik 1120.37036 and in Mathematical Reviews 2008a:37061.

Henk W. Broer, Heinz Hanßmann and Jun Hoo

The quasi-periodic Hamiltonian Hopf bifurcation

Nonlinearity 20(2), p. 417-460 (2007)

The abstract is available from the publisher and is reprinted in the Zentralblatt der Mathematik 1121.37042. Furthermore there is a review in Mathematical Reviews 2007m:37126. A preprint version can be downloaded as PostScript file (6.2 M) or in a gzipped version (1.2 M).

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

The quasi-periodic reversible Hopf bifurcation

p. 2605-2623 in :

Recent Advances in Nonlinear Dynamics: Theme section dedicated to André Vanderbauwhede (eds. E. Doedel, B. Krauskopf and J. Sanders)

International Journal of Bifurcation and Chaos 17(8) (2007)

The abstract is available from the publisher and reprinted both in Mathematical Reviews 2008i:37102 and the Zentralblatt der Mathematik 1141.37342. A preprint version can be downloaded as PostScript file (2.3 M) or in pdf (924 K).

Henk W. Broer and Heinz Hanßmann

Perturbation theory (dynamical systems)

Scholarpedia 3(9), p. 2399-240? (2008)

A preprint version can be downloaded as gzipped PostScript file (260 K) or in pdf (186 K).

Heinz Hanßmann

Dynamics of Hamiltonian Systems

p. 2261-2284 in :

Encyclopedia of Complexity and System Science (ed. R.A. Meyers)

Springer (2009)

A preprint version can be downloaded as PostScript file (22.6 M) or in pdf (1.5 M).

Henk W. Broer and Heinz Hanßmann

Hamiltonian Perturbation Theory (and Transition to Chaos)

p. 4515-4540 in :

Encyclopedia of Complexity and System Science (ed. R.A. Meyers)

Springer (2009)

A preprint version can be downloaded as gzipped PostScript file (628 K) or in pdf (713 K).

Heinz Hanßmann

Quasi-periodic Bifurcations in Reversible Systems

p. 51-60 in :

New Directions in Dynamical Systems, Leiden 2009 (eds. H. Hanßmann, A.J. Homburg and S. van Strien)

Regular and Chaotic Dynamics 16(1-2) (2011)

The abstract is available from the publisher and there are reviews in the Zentralblatt der Mathematik 1225.37059 and in Mathematical Reviews 2012d:70026. A preprint version can be downloaded as PostScript file (348 K) or in pdf (166 K).

Heinz Hanßmann

Non-degeneracy conditions in KAM theory

Indagationes Mathematicae 22(3-4), p. 241-256 (2011)

The abstract is available from the publisher and is reprinted in the Zentralblatt der Mathematik 1253.37058. Furthermore there is a review in Mathematical Reviews MR2853608. A preprint version can be downloaded as PostScript file (1.8 M) or in pdf (302 K).

Maikel Bosschaert and Heinz Hanßmann

Bifurcations in Hamiltonian Systems with a Reflecting Symmetry

p. 67-87 in :

Hamiltonian Systems and Celestial Mechanics, Castro Urdiales 2011 (eds. J.F. Palacián, P. Yanguas and H.S. Dumas)

Qualitative Theory of Dynamical Systems 12(1) (2013)

The abstract is available from the publisher and is reprinted in the Zentralblatt der Mathematik Zbl 06196270. Furthermore there is a review in Mathematical Reviews MR3038566. A preprint version can be downloaded as PostScript file (34 M) or in pdf (15 M).

Henk W. Broer, Heinz Hanßmann and Jiangong You

On the Destruction of Resonant Lagrangean Tori in Hamiltonian Systems

p. 317-333 in :

Recent Trends in Dynamical Systems, München 2012 (eds. A. Johann et al.)

Proceedings in Mathematics & Statistics 35, Springer (2013)

The abstract is reprinted both in Mathematical Reviews MR3110135 and the Zentralblatt der Mathematik 1314.37044. A preprint version can be downloaded as PostScript file (476 K) or in pdf (217 K).

Henk W. Broer, Heinz Hanßmann and Florian Wagener

Persistence Properties of Normally Hyperbolic Tori

Preprint, Mathematisch Instituut, Universiteit Utrecht (2017)

A 20 pages preprint can be downloaded as PostScript file (735 K) or in pdf (225 K).