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

/ Publications / Scientific Journals
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

## CONTENTS

A.P. Blokhin, M.F. Gelin
Rotational Dynamics of Chiral Molecules. pp. 1--6

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. 7--10

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. 11--27

Summary: Two-dimensional polynomial dynamical systems are mainly considered. We develop Erugin's two-isocline method for the global analysis of such systems, construct canonical systems with field-rotation 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 higher-dimensional dynamical systems and how to apply it for systems with more complicated dynamics.
Key words: Hilbert's 16th Problem, Erugin's two-isocline method, Wintner's principle of natural termination, Perko's planar termination principle, field-rotation parameter, bifurcation, limit cycle, separatrix cycle

V.I. Kuvshinov, V.A. Shaparau
Squeezed States of Colour Gluons in QCD Isolated Jet. pp. 28--36

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 sub-Poissonian and super-Poissonian 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. 37--40

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. 41--48

Summary: The tuning curves, angular and spectral phase-matching 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 phase-matching conditions of the type II along x and y crystallographic axes. The effective nonlinear coefficients as a function of phase-matching angles are calculated in the KTP principal planes.
Key words: nonlinear crystal, laser, phase-matching.

R.M. Yulmetyev, F.M. Gafarov
Dynamical Behaviour and Frequency Spectra of the Short-time Human Memory. pp. 49--54

Summary: The kinetic behaviour of dynamic information Shannon entropy is discussed for the fractal dynamics of long-range in short-time 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 short-time human memory.
Key words: long-range information, short-time human memory, chaos, Shannon entropy.

S.V. Zhestkov, V.I. Kuvshinov
Analysis and Generalization of Modern Methods of Constructing Soliton Solutions. pp. 55--60

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. 61--70

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. 71--80

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 ever-increasing 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 fine-scale 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 fine-scale 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. 81--86

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
Off-lattice Simulation of the Fractal Growth with Attractive Radial Drift and Mobility. pp. 87--92

Summary: In the perfect off-lattice DLA model of the fractal growth the centre-attractive 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. 93--98

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. 99--103

Summary: After reviewing the WKB series for the 1-dim 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. 104--108

Summary: The equation of motion for the time-dependent hedgehog for a generalized Skyrme model including sixth-order 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 self-excitation of soliton because of the fast development of fluctuation.
Key words: Skyrme model, soliton, self-excitation.

Designed and maintained by Dr. Nikolai N. Kostyukovich. Last updated: October 27, 2000
Created with assistance of Dr. Leonid F. Babichev