gdangelmayr.us

Research Areas:  

Dynamical Systems

• Geometrical theory of dynamical systems:
  • Chaotic Dynamics
  • Normal Forms and Unfoldings of Vector Fields and Maps
  • Singularity Theory and Imperfect Bifurcations
• Dynamical Systems with Symmetries
• Algorithms for Center Manifold Reductions and Normal Form Transformations
• Perturbation Techniques:
  • Averaging and Melnikov-Methods
  • Multiple Time Scales
  • Singular Perturbations
• Systems of Nonlinear Oscillators

Instabilities and Pattern Formation:

• Formation of Spatio-Temporal Patterns in Systems of PDE's:
  • Analysis of Instabilities via Center Manifold and Normal Form Theory
  • Spontaneous and Forced Symmetry Breaking
  • Reduction of PDE's to Systems of ODE's
  • Envelope- and Phase Diffusion-Equations
• Application to:
  • Fluid Mechanics
  • Reaction-Diffusion Systems
  • Semiconductors and Superconductors
  • Nonlinear Optics (optical bistability and laser)

Pattern Analysis and Neural Networks:

• Remodeling and Prediction of Dynamical Systems from Data via
  • Topology Preserving Neural Network Algorithms
  • Extraction of Invariant Manifolds
  • Markov Analysis
• Dynamics and Modeling of Continuous Neural Networks
• Systems of Neural Oscillators
• Neural Learning Rules for Storing Patterns and Pattern Cycles

Methods of Mathematical Physics:

• Linear and Nonlinear Boundary- and Eigenvalue-Problems
• Variational Calculus and Optimization
• Asymptotic Expansions for Linear and Nonlinear Waves
• Geometrical Theory of Diffraction and "Singularity Optics"
• Semiclassical Methods of Quantum Mechanics
• Asymptotic Approach to Inverse Scattering Problems