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

/ Publications / Scientific Journals
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

## CONTENTS

Alexander M. Krot
Matrix Decompositions of Vector Functions and Shift Operators on the Trajectories of a Nonlinear Dynamical System. pp. 106-115

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. 116-122

Summary: We obtain the necessary and sufficient conditions of isochronicity (linearisability) of centers of two eight-parametric 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
Spatial-Temporal Dynamics of the Concentration Response of Polarizable Particles Acted upon by a Gradient Force in a Laser Radiation Field. pp. 123-128

Summary: On the basis of Smoluchowski equation the theory of the spatial-temporal 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 spatially-modulated 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 particle-concentration spatial structures induced by the two interfering coherent Gaussian beams and a Bessel beam of zeroth order has been investigated.

George G. Krylov
Transverse Field Effects and Electron Transport in Quasi-One-Dimensional Systems. pp. 129-135

Summary: The existence of the regime of metal-insulator transition in quantum ballistic transport in low-dimensional 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, metal-insulator transition, polymers

O.P. Kuznechik, V.N. Gorenkov, and O.O. Kuznechik
Model of the Scattered Moonligh. pp. 136-139

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. 140-149

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. 150-156

Summary: One from actual problems of the percolation theory which has many applications is considered. The Monte-Carlo 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 two-dimensional 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, Monte-Carlo method, site problem, spanning cluster

Mitja Perus
Multi-level Synergetic Computation in Brain. pp. 157-193

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 most-strongly correlated with some environmental pattern. Neuronal patterns-qua-attractors have both a material and a virtual aspect. As neuronal patterns, on the one hand, patterns-qua-attractors are explicit carriers of informational contents. As attractors, on the other hand, patterns-qua-attractors 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, re-construction and Hebbian short-term storage is realized in a net. Perceptual processes are here modeled using Kohonen's topology-preserving 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 multi-level 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 spatio-temporal 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 pattern-correlations represent memory. (Re)construction of a neuronal pattern-qua-attractor is analogous to the "wave-function collapse". Transformations of memory (or sub-conscious) representations to a conscious representation is modeled in the same way. Found mathematical analogies allow translation of the neural-net "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 sub-cellular 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. 194-205

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. 206-211

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

Designed and maintained by Dr. Nikolai N. Kostyukovich. Last updated: September 12, 2001
Created with assistance of Dr. Leonid F. Babichev