Henk Broer

Near-resonances between frequencies notoriously lead to small denominators when trying to prove persistence of invariant tori carrying quasi-periodic 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 quasi-periodic tori. The parameter values for which the Diophantine conditions are not fulfilled form so-called resonance gaps. Normal hyperbolicity can guarantee invariance of the perturbed tori, if not their quasi-periodicity, for larger parameter ranges. For a 1-dimensional parameter space this allows to close almost all resonance gaps.

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.

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 contact-equivalence 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.

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.

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 computer-assisted research on time-periodic 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 multi-stability 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 time-periodic system, our main interest is the chaotic dynamics. We plot several numerical examples of strange attractors.

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.