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Nonlinear Phenomena in Complex Systems, 2001, Vol.4, No.2

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal
Published by The "Education and Upbringing" Publishing Company, Minsk, Republic of Belarus

Volume 4, Number 2, 2001
CONTENTS
Alexander M. Krot
Matrix Decompositions of Vector Functions and Shift Operators on the Trajectories of a Nonlinear Dynamical System. pp. 106115
Summary: Decomposition relationships for nonlinear dynamical system operators into
state space based on matrix series are proposed. Analysis of attractors of
complex dynamical systems based on proposed decomposition was carried out.
The representation for a shift operator on trajectories of nonlinear
dynamical system through shift operators on trajectories of multidimensional
linear dynamical systems has been obtained.
Key words: nonlinear dynamical systems, state space, matrix series,
attractors of complex dynamical systems, shift operator on the trajectories.
Valery G. Romanovski, Marko Robnik, and Olga I. Romanovskaya
The Isochronicity Conditions for Some Cubic Systems. pp. 116122
Summary: We obtain the necessary and sufficient conditions of isochronicity
(linearisability) of centers of two eightparametric cubic systems of differential equations.
Key words: isochronicity, centers, linear normal forms
A.N. Rubinov, A.A. Afanas'ev, Yu.A. Kurochkin, and S.Yu. Mikhnevich
SpatialTemporal Dynamics of the Concentration Response of Polarizable Particles Acted upon by a Gradient Force in a Laser Radiation Field. pp. 123128
Summary: On the basis of Smoluchowski equation the theory of the spatialtemporal
dynamics of the concentration response of the polarizable particles in a liquid under the action of the gradient force in the field of a nonresonance
spatially modulated laser radiation has been developed. It is shown that the action of a nonresonance radiation with a spatiallymodulated intensity on the ansemble of particles causes their motion, which leads to the formation
of concentrational spatial structures corresponding to the intensity
modulation. The kinetic of formation and subsequent relaxation of particleconcentration spatial structures induced by the two interfering coherent Gaussian beams and a Bessel beam of zeroth order has been investigated.
Key words: spatialtemporal dynamics, gradient force, laser radiation,
polarizable particles, diffusion
George G. Krylov
Transverse Field Effects and Electron Transport in QuasiOneDimensional Systems. pp. 129135
Summary: The existence of the regime of metalinsulator transition in quantum
ballistic transport in lowdimensional system in an external electric field has been shown. The discussion of possible realization of the effect as so
as limitations on the system has been performed.
Key words: quantum waveguide, ballistic transport, metalinsulator transition, polymers
O.P. Kuznechik, V.N. Gorenkov, and O.O. Kuznechik
Model of the Scattered Moonligh. pp. 136139
Summary: Overall performance of astronomical complexes is mainly influenced by the
night sky background radiation. The scattered light from the Moon makes an
important contribution to this one. For taking into account such background radiation in the case of the night cloudless sky a model describing the
characteristics of the scattered light the Moon has been offered. In
addition, a rather valuable empirical expression represented the spatial distribution of the background radiation parameters in arbitrary units with
spectral transparency of the atmosphere $p_\lambda \geq 0.7$ has been proposed. Despite its simplicity the model is more evident, logical and
precise than those proposed earlier.
Key words: night sky, moonlight, brightness, atmospheric extinction, model
H.V. Grushevskaya and I.I. Kopts
Reconstruction of Neural Network from MEG Data of Cortical Activity. pp. 140149
Summary: It was established an averaged dynamics of neuron behavior based on
magnetoencephalography (MEG) experimental data. It was shown as
the characters of individual neuron influence on a collective behavior of the neural network which was simulated by the network consisting of phase
oscillator. It was estimated the optimal organization of the structures of brain cortex.
Key words: neural network, phase oscillator, coupled oscillatory network,
cortical activity, magnetoencephalography, recognition
P.S.Grinchuk, N.V.Pavlyukevich, S.Borodetsky, and S.Rozin
Coexistence of the Spanning Clusters with Percolation on a Square Lattice above Percolation Threshold. pp. 150156
Summary: One from actual problems of the percolation theory which has many
applications is considered. The MonteCarlo method is used to investigate
the possibility of simultaneous existence of two clusters above the percolation threshold, when the clusters connect the opposite sides of a
percolation system. Consideration is given to the site problem on a twodimensional square lattice. An algorithm is suggested for such an
investigation. It is found that the probability of coexistence of these
clusters is a nonmonotonic function of the relative density of occupied sites that has a maximum at the point differing from the percolation
threshold. It is shown that the position of this maximum depends on the lattice size. As the lattice size increases, the point of the maximum of
probability tends to the percolation threshold. A qualitative explanation is
given concerning the specific features of the behavior of the investigated
probability, which have been revealed in computer simulation.
Key words: percolation theory, MonteCarlo method, site problem, spanning cluster
Mitja Perus
Multilevel Synergetic Computation in Brain. pp. 157193
Summary: Patterns of activities of neurons serve as attractors, since they are those
neuronal configurations which correspond to minimal 'free energy' of the
whole system. Namely, they realize maximal possible agreement among
constitutive neurons and are moststrongly correlated with some environmental pattern. Neuronal patternsquaattractors have both a material
and a virtual aspect. As neuronal patterns, on the one hand, patternsquaattractors are explicit carriers of informational contents. As
attractors, on the other hand, patternsquaattractors are implicit mental
representations which acquire a meaning in contextual relations to other possible patterns. Recognition of an external pattern is explained as a
(re)construction of the pattern which is the most relevant and similar to a given environmental pattern. The identity of the processes of pattern
construction, reconstruction and Hebbian shortterm storage is realized in
a net. Perceptual processes are here modeled using Kohonen's
topologypreserving feature mapping onto cortex where further associative
processing is continued. To model stratification of associative processing because of influence from higher brain areas, Haken's multilevel synergetic
network is found to be appropriate. The hierarchy of brain processes is of "software"type, i.e. virtual, as well as it is of "hardware"type, i.e.
physiological. It is shown that synergetic and attractor dynamics can
characterize not only neural networks, but also underlying quantum networks.
Neural nets are alone not sufficient for consciousness, but interaction with the quantum level might provide effects necessary for consciousness, like,
for instance, ultimate binding of perceptual features into an unified
experience. It is mathematically demonstrated that associative neural
networks realize information processing analogous to the quantum dynamics.
Parallels in the formalism of neural models and quantum theory are listed.
Basic elements of the quantum versus neural system (modeled by formal
neurons and connections) are very different, but their collective processes
obey similar laws. Specifically, it is shown that neuron's weighted
spatiotemporal integration of signals corresponds to the Feynman's version
of the Schrödinger equation. In the first case weights are synaptic
strengths determined by the Hebb or delta correlation rule; in the second
case weights are Green functions or density matrices. In both cases encodings of patterncorrelations represent memory. (Re)construction of a
neuronal patternquaattractor is analogous to the "wavefunction collapse".
Transformations of memory (or subconscious) representations to a conscious representation is modeled in the same way. Found mathematical analogies
allow translation of the neuralnet "algorithm", which in author's
simulations works very well, into a quantum one. This indicates how such
quantum networks, which might be exploited by the subcellular levels of
brain, could process information efficiently and also make it conscious.
Key words: neural net, quantum, brain, associative, synergetic, perception, consciousness
G. Contopoulos
Distribution of Regular and Irregular Orbits. pp. 194205
Summary: We study the distributions of periodic orbits in two simple dynamical
systems of two degrees of freedom. We distinguish between regular periodic
orbits (bifurcating from the periodic families of the unperturbed problem)
and irregular periodic orbits (independent of the above). Most periodic
orbits are unstable for large perturbations. The average instability is larger in regions of larger density of periodic orbits. This fact can be
explained by an ergodic argument. Many regular periodic orbits bifurcate
from the central family of periodic orbits. Irregular families are generated
in pairs (one stable  one unstable). On a Poincaré surface of section
the points representing the periodic orbits form two types of characteristic
lines, namely lines of regular orbits forming Farey trees, and lines of
irregular orbits close to the asymptotic curves of the main unstable
periodic orbits. All irregular orbits probably appear inside lobes of the
homoclinic tangle. But some regular orbits are also trapped in the
homoclinic tangle, although they do not belong to lobes. Inside the tangle
there are some stable orbits.
Key words: periodic orbits, bifurcation, homoclinic points
A. Dabrowski and T. Kapitaniak
Using Chaos to Reduce Oscillations. pp. 206211
Summary: Chaotic dynamical systems are usually controlled in a way which allows the
replacement of chaotic behavior by the desired periodic motion. We give the
example in which an originally regular (periodic) system is controlled in
such a way as to make it chaotic. This approach based on the idea of
dynamical absorber allows the significant reduction of the amplitude of the
oscillations in the neighborhood of the resonance.
Key words: chaotic behaviour, resonance