Hendrik Broer

Dynamical systems and chaos

Examples and definitions of dynamical phenomena.- Qualitative properties and predictability of evolutions.- Persistence of dynamical properties.- Global structure of dynamical systems.- On KAM Theory.- Reconstruction and time series analysis.

Bifurcational Aspects of Parametric Resonance

Generic nonlinear oscillators with parametric forcing are considered near resonance. This can be seen as a case-study in the bifurcation theory of Hamiltonian systems with or without certain discrete symmetries. In the analysis, among other things, structure preserving normal form or averaging techniques are used, as well as equivariant singularity theory and theory of flat perturbations.

A normally elliptic Hamiltonian bifurcation

A universal local bifurcation analysis is presented of an autonomous Hamiltonian system around a certain equilibrium point. This central equilibrium has a double zero eigenvalue, the other eigenvalues being in general position. Main emphasis is given to the 2 degrees of freedom case where these other eigenvalues are purely imaginary. By normal form techniques and Singularity Theory unfoldings are obtained, having ‘integrable’ approximations related to the Elliptic and Hyperbolic Umbilic Catastrophes

Unfoldings of Quasi-periodic Tori in Reversible Systems

A general KAM-theory for reversible systems is given. The cases of both maximal and lower-dimensional tori are covered. In some cases parameters are needed for persistence, therefore an unfolding theory is developed.

Formally Symmetric Normal Forms and Genericity

We consider several classes of dynamical systems on manifolds, like Hamiltonian systems, volume preserving systems, etc. On each of these classes there is a more or less natural topology. The word generic is used for properties of dynamical systems which hold for almost all elements, in the topological sense, of a given class. Formal definitions will be given below. Generic properties are known which imply a certain local simplicity of the system, e.g. that fixed points or equilibria are isolated. Still for such a system the global dynamics can be very complicated.

Quasi-periodic Bifurcations of Invariant Circles in Low-dimensional Dissipative Dynamical Systems

This paper first summarizes the theory of quasi-periodic bifurcations for dissipative dynamical systems. Then it presents algorithms for the computation and continuation of invariant circles and of their bifurcations. Finally several applications are given for quasiperiodic bifurcations of Hopf, saddle-node and period-doubling type.

KAM theory: The legacy of Kolmogorov's 1954 paper

Kolmogorov-Arnold-Moser (or KAM) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and KAM theory establishes persistence results for such tori, which carry quasi-periodic motions. We sketch this theory, which begins with Kolmogorovs pioneering work.

A proof of the isoenergetic KAM-theorem from the “ordinary” one

A proof is given of the isoenergetic KAM-theorem for Hamiltonian systems, using the “ordinary” KAM-theorem and a transversality argument.

On a quasi-periodic Hopf bifurcation

Abstract In this paper we study quasi-periodic Hopf bifurcations for the model problem of a quasi-periodically forced oscillator, where the frequencies remain fixed. For this purpose we first consider Stoker’s problem for small damping.

Global analysis of dynamical systems: Festschrift dedicated to Floris Takens for his 60th birthday

Preface. 1 Forced oscillations and bifurcations Floris Takens. 2 Historical behaviour in smooth dynamical systems David Ruelle. References. 3 Implicit formalism for affine-like maps and parabolic composition Jacob Palis and Jean-Christophe Yoccoz. On Floris as a friend. 3-1 On homoclinic bifurcations. 3-2 Implicit formalism for affine-like maps. 3-3 Parabolic composition. References. 4 Strong resonances and Takenss Utrecht preprint Bernd Krauskopf. 4-1 Setting and notation. 4-2 Zq-equivariant normal forms. 4-3 Weak resonance. 4-4 Strong resonances. 4-5 Strong resonance for q = 4. 4-6 From the normal form to the full dynamics. References. 5 Semi-local analysis of the k : 1 and k : 2 resonances in quasi-periodically forced systems Florian Wagener. 5-1 Preliminaries. 5-2 Normal form analysis. 5-3 Semi-local bifurcation analysis of the k: 1 resonance. 5-4 Semi-local bifurcation analysis of the k: 2 resonance. 5-5 Conclusions.vi . Acknowledgements. References. 6 Generic unfolding of the nilpotent saddle of codimension four Freddy Dumortier, Peter Fiddelaers and Chengzhi Li. 7 Exponential confinement of chaos in the bifurcation sets of real analytic difeomorphisms. 8 Takens-Bogdanov bifurcations without parameters and oscillatory shock profiles. 9 Global birfurcations of periodic orbits in the forced Van der Pol equation. 10 An unfolding theory approach to bursting in fast-slow systems. 11 The intermittency route to chaotic dynamics. 12 Homoclinic points in complex dynamical systems. Excitation of elliptic normal modes of invariant tori in volume perserving flows. 14 On the global dynamics of Kirchhoffs equations: rigid body models for underwater vehicles. 15 Global dynamics and fast indicators. 16 A general nonparametric bootstrap test for Granger causality. 17 Birkhoff averages and bifurcations. 18 The multifractal analysis of Birkhoff averages and large deviations. 19 Existence of absolutely continuous invariant probability measures for multimodal maps. 20 On the dynamics of the renormalization operator. Index