Algorithms for computing normally hyperbolic invariant manifolds

An effcient algorithm is developed for the numerical computation of normally hyperbolic invariant manifolds, based on the graph transform and Newton's method. It fits in the perturbation theory of discrete dynamical systems and therefore allows application to the setting of continuation. A convergence proof is included. The scope of application is not restricted to hyperbolic attractors, but extends to normally hyperbolic manifolds of saddle type. It also computes stable and unstable manifolds. The method is robust and needs only little specification of the dynamics, which makes it applicable to e.g. Poincaré maps. Its performance is illustrated on examples in 2D and 3D, where a numerical discussion is included.

A reversible bifurcation analysis of the inverted pendulum

The inverted pendulum with a periodic parametric forcing is considered as a bifurcation problem in the reversible setting. Parameters are given by the size of the forcing and the frequency ratio. Normal form theory provides an integrable approximation of the Poincaré map generated by a planar vector field. Genericity of the model is studied by a perturbation analysis, where the spatial symmetry is optional. Here equivariant singularity theory is used.

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. KEY WORDS: Reversibility; quasi-periodicity; unfolding parameters. AMS (MOS) Classification Numbers: 58F27, 58F30

Equivariant singularity theory with distinguished parameters: Two case studies of resonant Hamiltonian systems

We consider Hamiltonian systems near equilibrium that can be (formally) reduced to one degree of freedom. Spatio-temporal symmetries play a key role. The planar reduction is studied by equivariant singularity theory with distinguished parameters. The method is illustrated on the conservative spring-pendulum system near resonance, where it leads to integrable approximations of the iso-energetic Poincaré map. The novelty of our approach is that we obtain information on the whole dynamics, regarding the (quasi-) periodic solutions, the global configuration of their invariant manifolds, and bifurcations of these.