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Nonlinear Phenomena in Complex Systems, 1999, Vol.2, No.3

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

Volume 2, Number 3, 1999
CONTENTS
T. Bountis, J.P. Van Der Welle
Subharmonic bifurcations and Melnikov's theory in a system of parametrically driven pendulums. pp. 113
Summary: Melnikov's theory of homoclinic bifurcations has proved to be a most successful tool for predicting the onset of (homoclinic) chaos in periodically
perturbed dynamical systems of two or more dimensions. By contrast, Melnikov's predictions for the occurrence of subharmonic bifurcations have not been as widely tested and
applied to manydimensional systems of physical significance. In this paper we first demonstrate explicitly the remarkable
accuracy of Melnikov's subharmonic theory in predicting the appearance of inphase and counterphase oscillations, as well as
inphase rotations, in a system of two damped, parametrically driven pendulums. Secondly, we propose a generalization of
Melnikov's approach by which more complicated motions can be studied in coupled pendulum systems of 2N (N2) phase space dimensions.
D. Bainov, E. Minchev
Oscillation of solutions of nonlinear parabolic equations in noncylindrical domains. pp. 1418
Summary: Forced oscillations of solutions of nonlinear parabolic equations in noncylindrical domains are
considered. The case when the force term is zero is investigated separately.
M. V. Bartuccelli
Ladder theorems and positivity of solutions for dissipative partial differential equations. pp. 1922
Summary: We obtain ladder estimates for semilinear dissipative differential equations. Then we use these
estimates to find conditions which assure that solutions with positive initial data are (at least) eventually (i.e.
asymptotically) positive. These results are of fundamental importance in many fields of applied mathematics as, for example,
biological dynamics, where the solutions of the equations modelling the systems under investigation must often be positive
functions.
L.A.Cherkas, V.G.Romanovsky and N.L.Shcheglova
On computing of the limit cycles manifold. pp. 2327
Summary: An algorithm to compute a nonlocal analog of AndronovHopf manifold for plane polynomial vector
fields with a rotating field parameter is described AMS classification scheme numbers.
Margarita Kondrat'eva
Dynamics of quantum averages and quantization conditions in semiclassical approximation. pp. 2831
Summary: A system of ODE which describes dynamics of quantum averaves of position and momentum operators
and second moments in semiclassical approximation is considered (average values are calculated w.r.t. approximate solutions of
the evolution Schroedinger equation). It is shown that quantization condition for the Schroedinger operator is related
to periodic solutions of the system of ODE. Known semiclassical quantization condition follows from the one obtained here under
additional assumptions.
Alexander Rauh
Remarks on unsolved basic problemsof the NavierStokes equations. pp. 3237
Summary: There is renewed interest in the question of whether the NavierStokes equations (NSE), one of
the fundamental models of classical physics and widely used in engineering applications, are actually selfconsistent. After
recalling the essential physical assumptions inherent in the NSE, the notion of weak solutions, possible implications for the
energy conservation law, as well as existence and uniqueness in the incompressible case are discussed. Emphasis will be placed on
the possibility of finite time singularities and their consequences for length scales which should be consistent with the continuum hypothesis.
Krzysztof Sacha, Robert Gebarowski and Jakub Zakrzewski
Classical thresholds for H atom ionization by circularly polarized microwave fields. pp. 3841
Summary: Ionization of H atoms by circularly polarized microwaves is studied within the classical mechanics
framework. We compare recent predictions coming from the Chrikov overlap criterion with results of numerical simulations for
finite interaction time. The numerical threshold dependence on the interaction time is shown to obey a power law behaviour for high frequencies.
R. Castro T. D. Sauer
Forecasting and dimension calculations from event timing data. pp. 4251
Summary: Recent methods for recovering nonlinear dynamical invariants from a sequence of event timings
are considered. The analysis is focused on two dynamical invariants: nonlinear forecastability and correlation dimension.
The methodology involves geometric reconstruction of the underlying dynamics through the interevent intervals.
R. M. Yulmetyev, G. P. Vishnevskaya, E. N. Frolova
Manifestation of nonMarkovian behavior of molecular system in electron spin relaxation of paramagnetic ions in solutions. pp. 5260
Summary: NonMarkovian processes and spin memory effects in electron spin relaxation of paramagnetic ions
in aqueous, aquaacid and aquaglicerol solutions have been investigated. For these purposes the spinlattice relaxation time
of some paramagnetic (Gd^{3+}, Mn^{2+}, Cr^{3+}) ions have been measured by two independent methods: nonresonant
paramagnetic absorption method in parallel magnetic fields and EPR. The microscopic mechanism of spin relaxation is the magnetic interparticle dipoledipole interactions of paramagnetic ions
modulated by the translational movement of the particles. Our experimental data for the viscosity dependence of the spin relaxation times exhibited the smooth transition from the
liquidlike (diffusional) to the solidlike (rigidlattice) relaxation scenario in aqueousacid and aqueousglicerol
solutions. We elaborated the statistical nonMarkov theory of paramagnetic relaxation of spins using wellknown ZwanzigMori's
kinetic equations for the normalized spin time correlation functions. Our approach involves the statistical memory effects in combined (spin + solution) system by means of correlation
approximation for the second order spin memory function. The theory developed is in good accordance with our experimental
results. From our investigations it is obvious that nonMarkov processes and spin memory effects play a crucial role in the
determination of the character of paramagnetic relaxation of electron spins in solutions.
L. M. Barkovsky, A. N. Furs
Tensor eikonal approximations in asymptotic expansions for stratified media in the absence of commutation. pp. 6171
Summary: The tensor eikonal method for solving wave equations of optics of stratified anisotropic media is
proposed. Geometrooptical solutions of these equations are presented in the covariant threedimensional evolutional form. Such representation allows to operate with electromagnetic fields
in coordinate free representation without any expansion of the fields into partial waves. The eikonal tensors determined by wave
polarization (photon spin) are involved in the evolution Cauchy operators and are expressed in terms of the normal refraction tensors. These tensors are noncommutative in the
general case and necessity of disentangling the evolution operators arises even if a medium is isotropic. With the use of
standard operator procedures such disentangling is carried out for oblique incidence of polarized monochromatic light to
isotropic layered media. For this case the closed expressions describing the spatial evolution of fields are obtained.
A.P. Blokhin, M.F. Gelin, and A.V. Uvarov
Orientational relaxation of macromolecules immobilized on a surface. pp. 7281
Summary: To investigate the influence of inertial effects on the orientational relaxation of
macromolecules subjected to rigid intersubunit constraints and placed in a buffer medium, the diffusion equation with memory
(DEM) is deduced from the FokkerPlanck equation (FPE) in case of arbitrary generalized coordinates (the FPE has been derived by us
elsewhere [1,2]). The DEM allows us to establish a correct short time behavior for correlation functions (CFs) of generalized
coordinates. The DEM is invoked to perform illustrative calculations of the orientational CFs for immobilized on a
surface once broken nonrigid rod in two dimensions (2D). These calculations reveal that the CFs under certain conditions exhibit
an oscillatory behavior, which is irreproducible within the standard diffusion equation (DE). Several methods are considered
for the approximate solution of the DEM, and their application to three dimensional (3D) DEMs is discussed.
L.F. Babichev, D.V. Klenitsky, V.I. Kuvshinov
Fractal structure in the first order QCD phase transition. pp. 8285
Summary: Dynamical fluctuations, induced by the QCD first order phase transition in nuclear collisions, are
studied. Their fractal structure is discussed.
N.I. Stetyukevich, O.P. Kuznechik, V.N. Gorenkov
IRemission characteristics of crystalline hydrates in the processes of melting and crystallisation. pp. 8690
Summary: The influence of selforganization of molecular structure on emission characteristics of crystalline
hydrates 2methyl2propanols and sodium thiosulphate in the range of 4000400 cm^{1} has been investigated. The
surface flux density of thermal emission in the processes of crystallisation and melting has been measured. Possible
mechanisms of generation of nonequilibrium emission has been discussed. The obtained results may be used for remote control of
the phase transitions.
A. S Dmitriev, A. I. Panas, L.V.Kuzmin
Chaotic synchronization and chaotic communications over a bandpass channel. pp. 9199
Summary: We propose a method for combating the transmitted signal filtering in the communication channel, based
on a special design of chaotic oscillator. In the method, an element that has the same filtering properties as the channel is
added to the feedback loop of the chaotic source. It gives possibility to obtain perfect chaotic synchronous response in the
receiver. The method is illustrated by numerical simulation and physical experiments.
V.I. Kuvshinov, A.V.Kuzmin
Chaosorder transitions in SU(2) YangMillsHiggs systems. pp. 100105
Summary: In this work we briefly discuss the problem of chaos in classical gauge field theories. It is not a
full consideration of such questions. Rather it is an introduction. We give a qualitative description of the methods widely used to study the chaotic behaviour of the classical field
systems and discuss some investigations of these questions.