Research Group for Mathematical and Numerical Analysis of Dynamical Systems



Novel Insights into Nonlinear Dynamical Systems & Chaos


          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.

  • Modelling of Optical Effects

         Optics has traditionally been an important research area for our Group. Our field in optics could be shortly described as modelling of optical effects in virtual computational environments. The utilization of three basic components – numerical models of elastic dynamically interacting bodies, a physical model describing interference effects occurring whenever optical whole field fringe based experimental techniques are used to investigate these bodies, and the construction of digital graphical representations helps to mimic realistic optical processes taking place in those systems. And not only – such tools help to interpret complex optical phenomena and help to develop new measurement techniques.

       We managed to propose several new measurement techniques and improvements of existing measurement techniques – time average photoelasticity; time average stochastic moire; time average geometric super moire; the generalization of Abel transform for vibrating tubes; corrections of classical formulas describing time average projection moire and time average geometric moire. The range of investigated systems is quite wide: from MEMS up to fluids, membranes, or bodies with fractal surface geometry. Investigations of optical phenomena originated new numerical cryptographic methods and initiated the development of specialized numerical techniques.

  • Development of Numerical Methods

       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.

        A new numerical technique for time series forecasting based on fuzzy inference systems produced by symbolic multiplicative operator techniques. The employment of this criterion also gives an answer on the structure of the solution – contrary to the exp-function method where the structure of the solution is initially guessed and only then symbolic computations are used to identify the necessary parameters. Our technique provides a deeper insight into the structure of the original differential equation.

  • Visual Cryptography

       Visual Cryptography (VC) is a cryptographic technique which allows visual information (pictures, text, etc.) to be encrypted in such way that the decryption can be performed by the human visual system, without the aid of computers. Classical VC is developed in the nineties and is based on a visual secret sharing scheme, where an image was broken up into n shares so that only someone with all n shares could decrypt the image, while any n−1 share revealed no information about the original image. Numerous modifications and advancements of the method have been proposed so far: halftone VC, colored VC, secret sharing schemes without pixel expansion, polynomial style sharing, multiple secrets schemes, circular VC, progressive image sharing, etc.

       We have proposed a new concept of Dynamic VC. It is a one share method; the secret is encoded into a single image. The secret can be leaked only when this image is oscillated into a predefined direction at predefined amplitude. We exploit the inability of human visual system to follow rapid oscillatory motion and exploit the optical moiré phenomenon to form the secret in the time-averaged image. The security of the encryption can be increased if a non-harmonic moiré grating is used to form the image.

      Another important development in the area of cryptography is a new class of hash functions based on time-averaging moiré operators. Special properties of algebras of moiré grating functions and time-averaging operators enabled to construct efficient one-way collision-free hash functions. The functional principle of these hash functions is based on the inherent ill-posed inverse problem which is in its turn based on optical moiré phenomena.

  • Nonlinear Dynamical Systems and Chaos

        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.

  • Results and Future Trends

       Our Group has originated a number of interesting solutions, methods and applications. We have developed the concept of dynamic visual cryptography which is a new branch in the science of digital image security. Our fuzzy time series forecasting methods based on the optimal attractor embedding outperform state-of-the-art predictors for such benchmark tests as Mackey-Glass series. We have developed an analytical criterion which determines if solution of nonlinear differential equation can be expressed in a form comprising finite number of standard functions; moreover, this criterion generates the structure of the solution automatically and outperforms homotopy perturbation methods. As mentioned previously, all these developments are closely related and originated by the theory of nonlinear dynamical systems and chaos.

      Modelling of optical effects in virtual computational environments, novel numerical techniques and applications, dynamic visual cryptography, control and characterization of chaotic nonlinear systems build the ground for new challenging possibilities in mathematical and numerical analysis of dynamical systems.


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