DFG Research Center MATHEON on Mathematics for key technologies: Modelling, simulation, and optimization of real-world processes Research Project Project D3: Global singular perturbations Research director Dr. Jörg Härterich

Former Member

Dr. Astrid Huber

Summary

Optical fibre communication is a promising new technology to transfer large packages of data in a very short time. Modelling the dynamics of semiconductor lasers and optoelectronical devices involves very different space and time scales. This leads to singularly perturbed systems with one or several parameters tending to zero.

The main mathematical tools to treat singularly perturbed systems are geometric singular perturbation theory and asymptotic expansions. Mostly, these tools apply only to trajectories over bounded slow temporal or spatial scales, and require some hyperbolic behavior of the corresponding degenerate system.

Moreover, known techniques are mostly limited to ordinary differential equations. Driven by applications in laser technology, the project aims at overcoming both barriers.

Major breakthroughs in our understanding of global heteroclinic bifurcations are strongly based on work by the Shilnikov school. Recently, new technical developments relating spatial slow-fast transitions with PDE stability of pulses have been obtained in Berlin. A combinatorial description of global attractors for scalar parabolic PDEs in one space dimension has been achieved by Fiedler, Rocha, and Wolfrum.

On the field of singular perturbed systems new global phenomena related to a loss of normal hyperbolicity have been examined previously.

The main goal of the project is a systematic combination of global bifurcation theory with the singular perturbation analysis. We will combine the two approaches and extend the tools available in bifurcation theory to the specific slow-fast systems.

Specific topics to be examined are

1. global analysis of singular homoclinic and heteroclinic orbits;
2. stability of singular fronts and pulses in partial differential equations;
3. singular limits of global attractors.