Phase control of dendrytic neural networks, evolutionary algorithms for time series forecasting, dynamic visual cryptography, adaptive quadratures for realtime applications, non-uniform embeddings in multi-dimensional phase spaces, chaotic time-averaged moire fringes – is there anything in common among these diverse fields? The answer is a definite Yes. It all perfectly fits into the interests of the Research Group for Mathematical and Numerical Analysis of Dynamical Systems Nonlinear dynamical systems and chaos play an important role in many areas of science and engineering. Thus it is quite natural that our research interests are quite wide. Moreover, a healthy balance between rigorous theoretical analysis and practical applications produces fruitful results. In general, research directions in our Group can be conditionally classified into four subject areas: optics, numerical methods, cryptography and nonlinear dynamics. Though some aspects of different subject areas are highly intertwined, separate classification can help to crystallize the essential qualifiers of the each subject area.

We have been always heavily dependent on the development of novel numerical methods for the visualization of optical effects. One of the important results in this area worth noting is a new method based on conjugate smoothing used to represent a discontinuous field on a finite element mesh. Since our main interest lies in dynamical systems, time-averaging operators play an important role in our research (these operators also lead to applications in cryptography). We have proposed new quadrature rules which are especially effective in real time computational experiments.

Since the title of our research group is Mathematical and Numerical Analysis of Dynamical Systems, dynamical systems traditionally occupy an important part of our research. Attractor control strategies based on small external impulses have been developed for adaptive control of particles in a field of propagating waves; for particles and films conveyed by an undulating membrane; for the parametric identification of complex systems. The bouncing ball model was generalized for Rayleigh surface waves; for electrophoresis and control of biomedical particles. The principles of the developed attractor control techniques are used to control chaotic networks of neurons with dendrytic dynamics.