**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

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