**
Novel Insights into Nonlinear Dynamical
Systems & Chaos**

**n**

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.

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.

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 (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.

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.

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.