P.O. Box 800, 9700 AV Groningen, Netherlands
University of Groningen Department of Mathematics
Lukina O. V., Takens F., Broer H. W.
Global properties of integrable Hamiltonian systems
2008, vol. 13, no. 6, pp. 602-644
This paper deals with Lagrangian bundles which are symplectic torus bundles that occur in integrable Hamiltonian systems. We review the theory of obstructions to triviality, in particular monodromy, as well as the ensuing classification problems which involve the Chern and Lagrange class. Our approach, which uses simple ideas from differential geometry and algebraic topology, reveals the fundamental role of the integer affine structure on the base space of these bundles. We provide a geometric proof of the classification of Lagrangian bundles with fixed integer affine structure by their Lagrange class.
Takens F., Verbitskiy E.
Multifractal Analysis of Dimensions and Entropies
2000, vol. 5, no. 4, pp. 361-382
The theory of dynamical systems has undergone a dramatical revolution in the 20th century. The beauty and power of the theory of dynamical systems is that it links together different areas of mathematics and physics.
In the last 30 years a great deal of attention was dedicated to a statistical description of strange attractors. This led to the development of notions of various dimensions and entropies, which can be associated to the attractor, dynamical system or invariant measure.
In this paper we review these notions and discuss relations between those, among which the most prominent is the so-called multifractal formalism.
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
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.