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Nonlinear Phenomena in Complex Systems, 2000, Vol.3, No.3

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Nonlinear Phenomena in Complex Systems, 2000, Vol.3, No.3

An Interdisciplinary Journal

Published by The "Education and Upbringing" Publishing Company, Minsk, Republic of Belarus

Volume 3, Number 3, 2000

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Valery Romanovski and Marko Robnik
On WKB Series for the Radial Kepler Problem. pp. 214--219

Summary: We obtain the rigorous WKB expansion to all orders for the radial Kepler problem, using the residue calculus in evaluating the WKB quantization condition in terms of a complex contour integral in the complexified coordinate plane. The procedure yields Nonlinear Phenomena in Complex Systems 2000.240 Kazunari Shima: Nonlinear representation of supersymmetry ... the exact energy spectrum of this Schrцdinger eigenvalue problem and thus resolves the controversies around the so-called "Langer correction". The problem is nontrivial also because there are only a few systems for which all orders of the WKB series can be calculated, yielding a convergent series whose sum is equal to the exact result, and thus sheds new light to similar and more difficult problems.
Key words: radial Kepler problem, WKB expansion, Schrцdinger eigenvalue problem.

Jean-Franзois Gouyet and Mathis Plapp
Vacancy Mediated Spinodal Decomposition of a Two-component Droplet: Pattern Formation at Surfaces. pp. 220--225

Summary: We use mean-field kinetic equations to study the dynamics of spinodal decomposition of a ’droplet’ consisting of two components A and B and a small percentage of vacancies. We focus our attention on the vacancy mechanism, which is a more faithful picture of diffusion in solids than the more widely studied exchange mechanism. We study interfaces between an unstable mixture and a stable vapour phase which exhibit surface modes and lead to specific surface patterns.
Key words: mean-field kinetic equations, surface patterns.

E.I. Bogdanov
Schrodinger Nonlinear Equations on a Phase Space with a Nonflat Metrics. pp. 226--230

Summary: The Schrцdinger nonlinear equation has been proven to be obtained via variation of such functionals that are directly constructed from a geometrical structure of the physical system’s phase space.
Key words: Schrцdinger nonlinear equation, geometrical structure, phase space.

Victor F. Dailyudenko
Characterization of the Topological Structure and Stability for a Vector Map Derived from a Delay Differential Equation. pp. 231--241

Summary: Statistical characteristics of the attractor for investigated complex system have been defined by dispersion analysis of obtained topological curves in a form of asymptotic estimations of convergence to exact stabilization. The delay functional and matrix operator and corresponding Jacobian-matrix for successive calculation of segments of nonlinear time series have been obtained, the vector field divergence is modeled by the obtained analytical model. As the structure of phase trajectories is shown to be fractal, some components of the multifractal dimensions spectrum have been defined applying modified "sand-box" method. time series, asymptotic estimations, fractal.

V .M. Dubovik, A. G. Galperin, and V. S. Richvitsky
Around Lotka--Volterra Kind Equations and Nearby Problems. pp. 242--246

Summary: Some object problems concerning the redistribution of common resourse, upon being formalized, lead to skew-symmetric three-dimensional (3D) Lotka--Volterra systems (LVS) with an additive planar first integral relevant to the conservation of that resource. If the domain of variables is extended to the negative range, one can construct an atlas of solutions based on the classes of equivalence connected with solution diffeomorphism. That division entails the division of the system coefficient space into zones. Each point belonging to a zone defines the system and the family of its solutions in the solution space. The physical meaning of the ratios of that system coefficients can be understood as a measure of "coherence" of the swapping between its constituents. The maximum of coherence corresponds to equality to unity of absolute values of those ratios. For 3D LVS with an antisymmetric right-hand side, the projective properties of functional space and the space of its parameters with respect to families of the integral curves in functional space are investigated.
Key words: nonlinearity, conservation law, skew-symmetry, projective properties, oscillations, election.

Claude Froeschle , Elena Lega, and Elke Lohinger
Connectance and Stability. Application to Linear and Nonlinear Dynamical Systems with Increasing Number of Degrees of Freedom. pp. 247--252

Summary: We study the effects of connectance on the stability of linear and non-linear dynamical systems. We recover and extend previous results showing that the stability of a system decreases when increasing the connectance and the size of the system. In the case of non-linear Hamiltonian systems we show that stability seems to depend mainly on the connectance. Key words: non-linear dynamical systems, non-linear Hamiltonian systems, connectedness, stability.

A.D. Linkevich
Neural Systems, Language, Chaotic Processors for Thinking, and Machines-Programmers. pp. 253--259

Summary: It is discussed the problem how to design a machine-programmer, i.e. an artificial system capable to produce computer programs. Making a computer program is treated as a translation of a text that describes an algorithm in a natural language into a text of a program in some algorithmic language, whereas the essense of the developed approach is an intermediate stage of this transition, i.e. comprehension of the meaning of the original text. It is shown that generalized frames can be used for analysis of meanings, which removes the intractable problem of finding a set of independent meanings. This enables to suggest an outline of a dynamical mechanism of thinking based on associative generation of information in a semantic space which employs hyperbolity of chaotic attractors. It is a so-called "regularized English-like language" (RELL) that is stated to describe customer requirements and to formulate algorithms. The abovementioned points are combined and embodied in an architecture of a machine-programmer composed of lexical, semantic and operator neural systems which are implemented as a neural network, a neural field and a neural network respectively.
Key words: neural systems, language, chaotic processors.

Tomaz Prosen
Quantum Poincare Mapping: Systems with Smooth Potentials. pp. 260--267

Summary: Exact quantization of the Poincare mapping over the surface of section of an autonomous Hamiltonian system is shortly reviewed. The method reduces the number of freedoms in the system by one. Further we propose an efficient and stable numerical scheme for the computation of the unitary quantum Poincare mapping in the general case of smooth potentials. We illustrate the proposed method by working out the two examples: The diamagnetic Kepler problem (hydrogen atom in a uniform magnetic field) and the Nelson potential. It is demonstrated explicitly that the results of the method (say the system’s energy spectrum) do not depend on the choice of surface of section. The latter may even lie in the classically forbidden part of phase space.
Key words: quantum Poincare mapping, quantization, diamagnetic Kepler problem representation of supersymmetry.

J.P. van der Weele and E.J. Banning
Mode Interaction in a Cup of Coffee and Other Nonlinear Oscillators. pp. 268--283

Summary: This paper is about mode interaction in systems of coupled nonlinear oscillators. That is to say, about the interaction of two "pure" modes via a mixed motion with a lesser degree of symmetry, in many cases leading eventually to chaos. This nonlinear interaction is obviously a much more intricate affair than a simple superposition of the contributing modes, and we will use group theory to gain some general insight in it. It will be demonstrated that not just any two modes can interact with each other, but only those which are linked in the system’s symmetry hierarchy by a common daughter mode; further we shall see that the interaction strongly depends on the nonlinearities Nonlinear Phenomena in Complex Systems 2000.Kazunari Shima: Nonlinear representation of supersymmetry ... 235 of the system. Our model system consists of two coupled, parametrically driven pendulums but we also pay attention to mode interaction in the Faraday experiment (as observed by Ciliberto and Gollub) and in animal locomotion.
Key words: nonlinear oscillators, coupled parametrically driven pendulums, chaos.

R.M. Yulmetyev, D.G. Yulmetyeva, and F.M. Gafarov
Dynamical and Frequency Peculiarities of the Shannon Entropy for the Chaotic Dynamics of RR-Intervals in Human ECG. pp. 284--288

Summary: The temporal and frequency peculiarities of the time dependent information Shannon entropy and its power frequency spectra, describing the dynamics of RR-intervals in human ECG has been investigated in details.
Key words: long-range fluctuations, information, short-time human memory, chaos, Shannon entropy.

V.V. Litvinov, A.N. Petuch, and Ju.M. Pokotilo
On the Origin of Bigscale Longitudinal Oscillations of the Oxygen and Doped Impurity Concentration in Silicon Crystals Grown by the Czochralski Method. pp. 289--292

Summary: The frequency spectrum of the longitudinal concentration oscillations of the oxygen in silicon mono crystals doped by phosphorus was investigated. Two modes with independent frequencies f1 = 1.6 ·10-3 and f2 = 6.0 · 10-4 Hertz were found under quasi-periodical regime of melt convection. Comparison of oxygen and phosphorus concentration oscillations allowed to make connection between f1 mode and convection roll oscillations along temperature gradient which results in periodical change of both impurity segregation coefficients. Lower frequency mode f2 is associated with non-stationary transfer of the oxygen concentration which results in wave-like displacement of melt convection rolls across temperature gradient. Oxygen distribution coefficient in silicon was estimated about 0.25 using comparison analysis of the oxygen and phosphorus profiles.
Key words: melt convection, longitudinal concentration oscillations, silicon crystals doped by phosphorus.

Hidezumi Terazawa
Pomeron as a Color-Ball, the Complex System of Gluons or Chroms. pp. 293--298

Summary: Various pictures of glueballs in a generic sense, the hadrons consisting of gluons only, are reviewed and discussed in some detail in the light of their relevance to hadron-hadron, photon (gauge-boson)-hadron, and (hadronic) photon (gauge-boson)-photon (gauge-boson) scatterings at high energies. In particular, it is proposed that the Pomeron is a "color-ball", the color-singlet complex object consisting of an arbitrary number of gluons, which can be approximately described in QCD either by the extended Nambu--Goto action or by the extended Polyakov action for the hadronic membrane or for the hadronic bundle.
Key words: QCD, glueballs, hadronic membrane, hadronic bundle, Polyakov action, Nambu--Goto action.

V.I. Kuvshinov and A.V. Kuzmin
Influence of Quantum Higgs Vacuum Fluctuations on the Chaos-Order Transition in the System of Yang--Mills Fields. pp. 299--303

Summary: We investigate the classical stochastic dynamics of the spatially homogeneous model system of Yang-Mills fields in the Higgs vacuum. We are interested in the study of the in uence of vacuum quantum uctuations of Higgs fields on the dynamics of such model system.
Key words: quantum Higgs vacuum, uctuations, stochastic dynamics, Yang--Mills fields.

Andreas Ruffing
Unexpected Supermodels. pp. 304--312

Summary: In this article, interrelations between a special type of superpotentials with dilatative scaling behavior and between q-discretized harmonic oscillators are investigated. It turns out that there is a certain equivalence principle between q-discrete oscillators on the one hand and between selfsimilar supermodels on the other hand. q-discrete oscillators thus contribute to a better understanding of quantum mechanical superpotentials, indeed they can be regarded as unexpected supermodels. The main approach in our context will be the combination of spectral methods on Jackson square integrable functions with a formalization of supersymmetric Hamilton operators. A further new observation is this one: By the outlined methods, one can introduce completely new types of lattices to discretize Schrцdinger equations.
Key words: lattice quantum mechanics, Schrцdinger operator, superpotentials, supermodels, q-discretized harmonic oscillators.

George Krylov and Andrew Filimonov
On One Model of Random Walk with a Memory of Orientation. pp. 313--320

Summary: A model of a random walk with a memory of orientation have been proposed. Fractal properties of the set of visited sites for 2D-case have been obtained based on computer simulations. It has been shown that the model gives a method for the construction of a fractal set with fractal dimensions smoothly dependent upon a parameter. The applicability of the models to the description of rod-like molecules in a viscous media and in a media with random obstacles has been shsortly discussed. random walk, fractal, memory of orientation, perspistence.

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