Heinz Hanßmann
Mathematisch Instituut
Universiteit Utrecht
Postbus 80.010
3508 TA Utrecht
The Netherlands
email :
Heinz.Hanssmann@math.uu.nl
Bifurcations in Hamiltonian systems
Quasi-periodic motions of Hamiltonian systems densily fill invariant
tori, and these invariant tori form families parametrised by the
actions conjugate to the toral angles.
Therefore, bifurcations can already be observed in a single
Hamiltonian system.
This is in sharp contrast with dissipative dynamical systems where one
observes bifurcations only under variation of external parameters.
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)
A preprint version can be downloaded as PostScript file
(476 K)
or in pdf
(217 K).