Henk Broer
P.O. Box 407, 9700 AK Groningen, The Netherlands
Johann Bernoulli Institute for Mathematics and Computer Science University of Groningen
Publications:
Broer H. W., Hanßmann H., Wagener F. O.
Persistence Properties of Normally Hyperbolic Tori
2018, vol. 23, no. 2, pp. 212225
Abstract
Nearresonances between frequencies notoriously lead to small denominators when trying to prove persistence of invariant tori carrying quasiperiodic motion. In dissipative systems external parameters detuning the frequencies are needed so that Diophantine conditions can be formulated, which allow to solve the homological equation that yields a conjugacy between perturbed and unperturbed quasiperiodic tori. The parameter values for which the Diophantine conditions are not fulfilled form socalled resonance gaps. Normal hyperbolicity can guarantee invariance of the perturbed tori, if not their quasiperiodicity, for larger parameter ranges. For a 1dimensional parameter space this allows to close almost all resonance gaps.

Vitolo R., Broer H. W., Simo C.
Quasiperiodic bifurcations of invariant circles in lowdimensional dissipative dynamical systems
2011, vol. 16, no. 12, pp. 154184
Abstract
This paper first summarizes the theory of quasiperiodic 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, saddlenode and perioddoubling type.

Broer H. W., Holtman S. J., Vegter G., Vitolo R.
Dynamics and geometry near resonant bifurcations
2011, vol. 16, no. 12, pp. 3950
Abstract
This paper provides an overview of the universal study of families of dynamical systems undergoing a Hopf–Neimarck–Sacker bifurcation as developed in [1–4]. The focus is on the local resonance set, i.e., regions in parameter space for which periodic dynamics occurs. A classification of the corresponding geometry is obtained by applying Poincaré–Takens reduction, Lyapunov–Schmidt reduction and contactequivalence singularity theory, equivariant under an appropriate cyclic group. It is a classical result that the local geometry of these sets in the nondegenerate case is given by an Arnol’d resonance tongue. In a mildly degenerate situation a more complicated geometry given by a singular perturbation of a Whitney umbrella is encountered. Our approach also provides a skeleton for the local resonant Hopf–Neimarck–Sacker dynamics in the form of planar Poincaré–Takens vector fields. To illustrate our methods a leading example is used: A periodically forced generalized Duffing–Van der Pol oscillator.

Lukina O. V., Takens F., Broer H. W.
Global properties of integrable Hamiltonian systems
2008, vol. 13, no. 6, pp. 602644
Abstract
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.

Broer H. W., Naudot V., Roussarie R., Saleh K.
A predatorprey model with nonmonotonic response function
2006, vol. 11, no. 2, pp. 155165
Abstract
We study the dynamics of a family of planar vector fields that models certain populations of predators and their prey. This model is adapted from the standard Volterra–Lotka system by taking into account group defense, competition between prey and competition between predators. Also we initiate computerassisted research on timeperiodic perturbations, which model seasonal dependence.
We are interested in persistent features. For the planar autonomous model this amounts to structurally stable phase portraits. We focus on the attractors, where it turns out that multistability occurs. Further, we study the bifurcations between the various domains of structural stability. It is possible to fix the values of two of the parameters and study the bifurcations in terms of the remaining three. We find several codimension 3 bifurcations that form organizing centers for the global bifurcation set. Studying the timeperiodic system, our main interest is the chaotic dynamics. We plot several numerical examples of strange attractors. 
Broer H. W., Takens F., Wagener F. O.
Integrable and nonintegrable deformations of the skew Hopf bifurcation
1999, vol. 4, no. 2, pp. 1743
Abstract
In the skew Hopf bifurcation a quasiperiodic attractor with nontrivial normal linear dynamics loses hyperbolicity. Periodic, quasiperiodic 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 KAMtheory, deals with the persistence of quasiperiodic circles. Other results concern the bifurcations of periodic attractors in the case of resonance.
