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

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

Volume 3, Number 3, 2000
CONTENTS
Valery Romanovski and Marko Robnik
On WKB Series for the Radial Kepler Problem. pp. 214219
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 socalled "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.
JeanFranзois Gouyet and Mathis Plapp
Vacancy Mediated Spinodal Decomposition of a Twocomponent Droplet: Pattern Formation at Surfaces. pp. 220225
Summary: We use meanfield 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:
meanfield kinetic equations, surface patterns.
E.I. Bogdanov
Schrodinger Nonlinear Equations on a Phase Space with a Nonflat Metrics. pp. 226230
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. 231241
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 Jacobianmatrix 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 "sandbox" method. time series, asymptotic estimations, fractal.
V .M. Dubovik, A. G. Galperin, and V. S. Richvitsky
Around LotkaVolterra Kind Equations and Nearby Problems. pp. 242246
Summary: Some object problems concerning the redistribution of common resourse, upon being formalized, lead to
skewsymmetric threedimensional (3D) LotkaVolterra 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 righthand 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, skewsymmetry, 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. 247252
Summary: We study the effects of connectance on the stability of linear and nonlinear 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 nonlinear Hamiltonian systems we show that stability seems to depend mainly on the connectance. Key
words: nonlinear dynamical systems, nonlinear Hamiltonian systems, connectedness, stability.
A.D. Linkevich
Neural Systems, Language, Chaotic Processors for Thinking, and MachinesProgrammers. pp. 253259
Summary: It is discussed the problem how to design a machineprogrammer, 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 socalled "regularized Englishlike 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 machineprogrammer 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. 260267
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. 268283
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 RRIntervals in Human ECG. pp. 284288
Summary: The temporal and frequency peculiarities of the time dependent information Shannon entropy and its power
frequency spectra, describing the dynamics of RRintervals in human ECG has been investigated in details.
Key words:
longrange fluctuations, information, shorttime 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. 289292
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 quasiperiodical 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 nonstationary transfer
of the oxygen concentration which results in wavelike 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 ColorBall, the Complex System of Gluons or Chroms. pp. 293298
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 hadronhadron, photon (gaugeboson)hadron, and (hadronic) photon
(gaugeboson)photon (gaugeboson) scatterings at high energies. In particular, it is proposed that the Pomeron is a
"colorball", the colorsinglet complex object consisting of an arbitrary number of gluons, which can be approximately
described in QCD either by the extended NambuGoto 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, NambuGoto action.
V.I. Kuvshinov and A.V. Kuzmin
Influence of Quantum Higgs Vacuum Fluctuations on the ChaosOrder Transition in the System of YangMills Fields. pp. 299303
Summary: We investigate the classical stochastic dynamics of the spatially homogeneous model system of YangMills 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, YangMills fields.
Andreas Ruffing
Unexpected Supermodels. pp. 304312
Summary: In this article, interrelations between a special type of superpotentials with dilatative scaling behavior and between
qdiscretized harmonic oscillators are investigated. It turns out that there is a certain equivalence principle between qdiscrete
oscillators on the one hand and between selfsimilar supermodels on the other hand. qdiscrete 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, qdiscretized harmonic oscillators.
George Krylov and Andrew Filimonov
On One Model of Random Walk with a Memory of Orientation. pp. 313320
Summary: A model of a random walk with a memory of orientation have been proposed. Fractal properties of the set of
visited sites for 2Dcase 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 rodlike molecules in a viscous media and in a media with random obstacles has been shsortly
discussed. random walk, fractal, memory of orientation, perspistence.