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Nonlinear Phenomena in Complex Systems, 2000, Vol.3, No.1

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

Volume 3, Number 1, 2000
CONTENTS
A.P. Blokhin, M.F. Gelin
Rotational Dynamics of Chiral Molecules. pp. 16
Summary: A theory of the rotation (R)  translational (T) motion of chiral molecules is developed, and various RT and orientational correlation functions are calculated. It is demonstrated that the difference between the RT dynamics of enantiomer and racemic mixtures is a direct manifestation of the combined e ect of the RT coupling and violation of the parity
invariance.
Key words: Rotational diffusion, chiral molecules.
K. Murali, A. Tamaševicius, G. Mykolaitis, Namajunas, E. Lindberg
Hyperchaotic System with Unstable Oscillators. pp. 710
Summary: A simple electronic system exhibiting hyperchaotic behaviour is described. The system includes two nonlinearly coupled 2nd order unstable
oscillators, each composed of an LC resonance loop and an amplifier. The system is investigated by means of numerical integration of appropriate
differential equations, PSPICE simulations and hardware experiments. The Lyapunov exponents are presented to confrm hyperchaotic mode of the
oscillations.
Key words: hyperchaotic behaviour, coupled unstable oscillators, Lyapunov exponents.
Valery A. Gaiko
On Global Bifurcations and Hilbert's Sixteenth Problem. pp. 1127
Summary: Twodimensional polynomial dynamical systems are mainly considered.
We develop Erugin's twoisocline method for the global analysis of such systems, construct canonical systems with fieldrotation parameters and
study various limit cycle bifurcations. In particular, we show how to carry out the classification of separatrix cycles and consider the most
complicated limit cycle bifurcation: the bifurcation of multiple limit cycles. Using the canonical systems, cyclicity results and Perko's
termination principle, we outline a global approach to the solution of Hilbert's 16th Problem. We discuss also how to
generalize this approach for the study of higherdimensional dynamical systems and how to apply it for systems with more complicated dynamics.
Key words: Hilbert's 16th Problem, Erugin's twoisocline method, Wintner's principle of natural termination, Perko's planar termination principle,
fieldrotation parameter, bifurcation, limit cycle, separatrix cycle
V.I. Kuvshinov, V.A. Shaparau
Squeezed States of Colour Gluons in QCD Isolated Jet. pp. 2836
Summary: We study evolution of colour gluon states in isolated QCD jet and
prove the possibility of existence of the gluon squeezed states at the nonperturbative stage of the jet evolution. Angular and rapidity
dependencies of gluon squeezed second correlation function are studied. We demonstrate that these new gluon states can have both subPoissonian and superPoissonian statistics corresponding to
antibunching and bunching of gluons.
Key words: squeezed states, nonperturbative stage, QCD jet, correlation function.
V.B. Nemtsov
Thermodynamical Description of the Thermochemical Mechanisms of DNA Deformation. pp. 3740
Summary: The thermodynamic description of thermochemical mechanisms of DNA
molecule deformation is developed. The expressions for bending and twisting moments applied to a DNA molecule due to variation of concentrations of the
composition of the environment are obtained. The estimation of energetical deformation effects is given.
Key words: DNA molecule, thermochemical mechanisms of deformation.
V.N. Belyi, N.S. Kazak, V.A. Orlovich, B.B. Sevruk
Performance of the Optical Parametric Generation in Nonlinear KTP Crystal at Nd:YAG Laser Pumping. pp. 4148
Summary: The tuning curves, angular and spectral phasematching widths are
calculated for all possible kinds of OPG in the principal planes of biaxial KTP crystal with Nd:YAG laser pumping at 1.064 $\mu$ wavelength. The plots
presented enable the maximal nonmonochromaticity and radiation divergence to be found for any value of the signal wave and different interaction types. A
particular emphasis has been placed on the OPG investigation under the noncritical phasematching conditions of the type II along x and y
crystallographic axes. The effective nonlinear coefficients as a function of phasematching angles are calculated in the KTP principal planes.
Key words: nonlinear crystal, laser, phasematching.
R.M. Yulmetyev, F.M. Gafarov
Dynamical Behaviour and Frequency Spectra of the Shorttime Human Memory. pp. 4954
Summary: The kinetic behaviour of dynamic information Shannon entropy is
discussed for the fractal dynamics of longrange in shorttime human memory. In order to get the most out of chaotic behaviour of memory parameters,
dynamic functions and frequency spectra of information entropy were constructed. Using then for data analysis we obtained further light on
chaotic kinetics of shorttime human memory.
Key words: longrange information, shorttime human memory, chaos, Shannon entropy.
S.V. Zhestkov, V.I. Kuvshinov
Analysis and Generalization of Modern Methods of Constructing Soliton Solutions. pp. 5560
Summary: The method of the inverse problem of scattering (MISP) and Hirota's
method (HM) are analyzed. It is shown that these methods use not the most common form of representation of soliton solutions of Schrodinger equation.
This fact allows to generalize them and to construct the new variant of soliton scattering theory which enables to operate by some parameters of
scattering.
Key words: solitons, inverse problem, scattering, Schrodinger equation.
N. Berglund
Classical Billiards in a Magnetic Field and a Potential. pp. 6170
Summary: We study billiards in plane domains, with a perpendicular magnetic
field and a potential. We give some results on periodic orbits, KAM tori and adiabatic invariants. We also prove the existence of bound states in a related scattering problem.
Key words: magnetic field, billiards, scattering.
Steven R. Bishop
The Use of Low Dimensional Models of Engineering Dynamical Systems. pp. 7180
Summary: The discovery of chaos in low dimensional dynamical systems has provided a renewed interest in dynamics. The advance in computer technology
allows us to solve (numerically) nonlinear problems of everincreasing complexity. In some instances the need to explain specific numerical
evidence has, in turn, promoted a resurgence in theoretical issues which provide insight into dynamical and bifurcational complexity. This paper
considers two case studies in which low dimensional models have proved useful in examining the dynamical response of engineering systems. In the
first example a pendulum system is modelled which enables finescale quantitative details to be established, allowing the zones in parameter
space to be identified in which various solutions occur. The second treatment is the `broad brush' modelling of fire growth in a room. Here
finescale details are not available but a qualitative insight into the dynamics can be used to guide more complex investigations. This dual
approach to modelling exemplifies the merits of low dimensional modelling.
Key words: the low dimensional dynamical systems, the parametrically excited pendulum, chaos
F.K. Diakonos, P. Schmelcher
The Turning Point Dynamics and the Organization of Chaos in Iterative Maps. pp. 8186
Summary: We study the dynamics of one dimensional iterative maps in the
regime of fully developed chaos. Introducing the concept of the turning points we extract from the chaotic trajectories the corresponding turning
point trajectories which represent a strongly correlated part of the chaotic dynamics of the system. The density of turning points exhibits step like
structures at the positions of the unstable fix points. The turning point dynamics is discussed and the corresponding turning point map which
possesses an appealing asymptotic scaling property is investigated. As a first application of the turning point concept we demonstrate its usefulness
for the analysis of time series and provide an algorithm which allows to locate the fixed points in one dimensional time series.
Key words: chaotic dynamics, iterative maps, asymptotic scaling property.
V.S. Gurin, V.P. Poroshkov
Offlattice Simulation of the Fractal Growth with Attractive Radial Drift and Mobility. pp. 8792
Summary: In the perfect offlattice DLA model of the fractal growth the
centreattractive drift factor and mobility of particles are introduced. Simulation indicates variation of fractal patterns and their characteristics
depending on these new factors. Noticeable e ects of the drift factor upon fractal dimension, growth effciency and frozen zone are shown. The mobility
behaves in the competitive manner with the drift. The combination of drift and mobility is considered as analogs of potential and kinetic energies of
particles.
Key words: fractal growth, the diffusion limited aggregation.
Rudolph C. Hwa
Fluctuation Index as a Measure of Heartbeat Irregularity. pp. 9398
Summary: A method is proposed to analyze the heartbeat waveform that can
yield a reliable characterization of the structure of the pulses. The measure suggested is index that is related to another measure found
effective in describing chaotic behaviors in a wide variety of physical systems. When applied to the ECG data that include ventricular brillation,
the index is shown to change drastically within a few pulses. Wavelet analysis is used to exhibit different scaling behaviors in different phases.
Sensitivity to the pulse shape makes this method an effective tool to diagnose heartbeat irregularities.
Key words: wavelet analysis, heartbeat waveform.
M. Robnik, L. Salasnich
Semiclassical Expansion for the Angular Momentum. pp. 99103
Summary: After reviewing the WKB series for the 1dim Schrodinger equation
we calculate the semiclassical expansion for the eigenvalues of the angular momentum operator. This is the first systematic semiclassical treatment of
the angular momentum for terms beyond the leading torus approximation.
Key words: semiclassical expansion, Schrodinger equation.
Nguyen Vien Tho, To Ba Ha
Higher Order Term and Dynamical Behavior of Skyrmion. pp. 104108
Summary: The equation of motion for the timedependent hedgehog for a
generalized Skyrme model including sixthorder stabilizing term is derived and integrated numerically. The time evolution of soliton is obtained for
two cases: with and without an initial perturbation. For the considered model, besides thermalization which has been found in the Skyrme model, we
have observed the selfexcitation of soliton because of the fast development of fluctuation.
Key words: Skyrme model, soliton, selfexcitation.