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Nonlinear Phenomena in Complex Systems, 2000, Vol.3, No.2

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

Volume 3, Number 2, 2000
CONTENTS
Drumi Bainov, Emil Minchev
On the Stability of Solutions of Impulsive Partial DifferentialFunctional Equations of First Order via Lyapunov Functions. pp. 109116
Summary: Stability and asymptotic stability of solutions of impulsive partial
differentialfunctional equations of first order are studied via Lyapunov functions.
Key words and phrases: stability of solutions,
impulsive partial differentialfunctional equations, Lyapunov functions.
M. Morandi Cecchi, L. Salasnich
NonHyperbolic Dynamics: a Family of Special Functions. pp. 116121
Summary: We study the iterative dynamics of a family of special functions from R^{2} into R^{2}
with a nonhyperbolic fixed point in the origin. The characterization by the eigenvalues is analyzed and discussed.
Key words: iterative dynamics, discrete dynamics, basin of attraction, nonhyperbolic fixed point.
A.T. Vlassov
Bifurcations in Dynamical Systems with Inputs. pp. 122131
Summary: Bifurcations for dynamical systems with inputs are discussed. Some examples of
behavior of trajectories for cyclic varying of parameters are considered.
Key words: bifurcation, connection, fixed point, cycle.
Z. Haba
Quantum Noise and Quantum Maps. pp. 132134
Summary: We represent quantum evolution as a random classical evolution described by a
stochastic differential equation defining a Markov process. We discuss a discrete version of the dynamics corresponding to a
discrete map q(n + 1) = T_q(n)_+ r(n), where r(n) are independent random variables. We indicate an
extension of the formalism to quantum dynamics on a manifold.
Alexander D. Linkevich
Application of Some Concepts and Methods of Functional Analysis in Experimental Studies of Nonlinear Phenomena. pp. 135152
Summary: Experimental data are considered in terms of Hilbert space and measure theory. As
illustrations, we take the three tasks: least squares approximation of data, prediction of time series, and
estimation of probability distributions. To these ends, we use datatuned bases composed of functions which are orthonormal with
respect to a specially defined inner product. This greatly reduces computations tobe performed and enables one to calculate
corrections to improve an available description (fit) of data.Two methods of constructing above mentioned bases are presented: a
generalization of Chebyshev polynomial expansions and a generalization of the GramSchmidt orthogonalization procedure.
It is demonstrated that the Dirac representation formalism of quantum theory can be useful to process data owing to a convenient
technique of transitions between various orthogonal bases in a measurablerigged separable Hilbert space. As a more wide tool, we
suggest a generalization of generalized frames involved in wavelets analyses of signals. It is shown that entropylike
functionals can be introduced as quantities associated with an arbitrary Hermitian operator and any partition of its spectrum.
We define also entropylike functionals connected with expansions of a signal vector over orthonormal basis functions and over
generalized frames.
Key words: Hilbert space, prediction of time series, GramSchmidt orthogonalization, entropylike functional.
Claude Froeschle, Elena Lega
Reestablishing Equilibrium Paths in a NeoAustrian Model: Analysis of Transitory Regimes. pp. 153170
Summary: Using a neoaustrian representation of production processes we focus on
the methods of analysis ofthe dynamical evolution of the system. We study the effects of a technological change starting fromthe
classical approach of static comparison of steady states. We then study the shock directly running numerical simulations and
analyzing the variation with time of the economic observables. Finally, inorder to clarify and simplify the analysis, we
introduce a different method of analysis which consists in considering a goal fixed by a social planner and in looking at
the economical coherence of the timeevolution of some economic key variables or control parameters. We claim that this method
gives results easier to understand than the direct one and therefore can be used for the study of transitory paths as a
guideline for a fruitful use of the direct analysis of numerical simulations.
Andreas Spille , Alexander Rauh, Heiko Buhring
Critical Curves of Plane Poiseuille Flow with Slip Boundary Conditions. pp. 171173
Summary: We investigate the linear stability of plane Poiseuille ow in 2D under slip boundary
conditions. The slip s is defined by the tangential velocity at the wall in units of the maximal ow velocity. As it turns out, the
critical Reynolds number depends smoothly on s but increases quite rapidly.
S. A. ElHakim
An Experimental Determination of the Critical Exponents at the MetalInsulator Transition in Germanium, ”Doped” by Radiation Defects. pp. 174176
Summary: This work deals with a determination of the magnitude of the critical exponents and
their ratio from an analysis of the temperature dependence of the conductivity (variable range hopping, VRH) in the temperature
range 1.5300 K in germanium, disordered by large uences of the fast reactorneutrons, with the radiation defect concentration
making it an insulator near the metalinsulator transition. The critical behavior of the activation conductivity and also the
parameters which define them, i.e. the dielectric constant, localization radius and coefficient To are found and compared
withthe predictions of the scaling theory.
P. Schmelcher
The SelfIonization Process for Atomic Ions in a Magnetic Field. pp.177180
Summary: In the presence of a magnetic field a permanent exchange of energy between the center
of mass and electronic degrees of freedom of atomic ions is shown to take place. For large values of theinitial center of mass
velocity the energy transfer from the center of mass to the electronic degrees of freedom is strong enough to allow the atom
to ionize. This dynamical selfionization effect is studied in some detail. Although the study is based on classical mechanics
it is argued that the effect should be observable in laboratory experiments.
V.R. Sobol, O.N. Mazurenko, M. Zoli
Nonlinear dynamics of carriers in aluminum cryoconductors under the action of crossed electric and magnetic field. pp. 181187
Summary: Magnetic field as a reason of nonlinear electric phenomena is studied experimentally
for normal metalconductors. For material under investigation being pure aluminum Corbino geometry of current own and sample
arrangement is applied. This approach gives a possibility to exclude an electric Hall field and to enlarge a sensitivity of
conductor resistance to self magnetic field stimulated by transverse Hall drift under the action of Lorentz force. In
accordance with a direction of radial current through the disk sample the own magnetic field coincides on direction with an
external magnetic field or not. As a result the electric kinetic properties are modified by this additional action of own
carrier movement. The self magnetic field distribution on sample surface is measured at different conditions for the current ow
value and external magnetic field intensity. Nonlinear additions to the electric field potential distribution due to the own
magnetic field are determined on the base of data for potential curves for collinear and anticollinear geometry. The physical
reasons of nonlinearity and their connection with carrier dispersion law are discussed by the means of analytical
investigation of high density charge transport under Corbino geometry. Phenomenological theory of electric field potential
distribution under similar charge transport is constructed with indication of metal specific difficulties for experimental
realization of this phenomenon.
Key words: electric nonlinearity, electric field, magnetic field, aluminum.
A. Namajunas, A. Tamaševicius, G. Mykolaitis, A. Cenys
Smoothing Chaotic Spectrum of Nonautonomous Oscillator. pp. 188191
Summary: A technique for smoothing power spectra of chaotic oscillations is described. The
nonautonomous Duffing oscillator is considered. The signals are processed by means of a linear feedback shift register. Even very
short register of length m = 4 makes the spectrum smooth and at.
Key words: hyperchaotic behaviour, coupled unstable oscillators, Lyapunov exponents.
Andreas Ruffing
Schroedinger Difference Operators with Quadratic Potentials on Geometric Progressions. pp. 192207
Summary: We consider the spectral problem for a Schroedinger difference operator, being
defined on stair functions on a geometric progression of type {+q ^{n},  q ^{n}  n 2 in Z } , q > 1. The potentials under considerationare quadratic and can be considered as discrete
generalizations of the oscillator potential. In contrast to the corresponding situation in continuum quantum mechanics, the
methods to determine the spectral properties turn out to be totally different. Existence results for eigensystems of the discrete oscillator are derived by combining results from the theory of bilateral Jacobi operators
with efficient methods from the theory of holomorphic perturbations. The point spectrum of the Schroedinger operator under consideration turns out to grow exponentially. It can be used
for regularizations in lattice quantum mechanics.
Eric Goles, Oliver Schulz, Mario Markus
A Biological Generator of Prime Numbers. pp. 208213
Summary: In the present work, we shall allow for the merging of two seemingly unrelated
subjects, one beingperiodical insects and the other the theory of prime numbers. The fact that some species of cicadas (genus
Magicicada) appear every 7, 13 or 17 years and that these periods are prime numbers hasbeen regarded as a coincidence. Without
intending to argue in favour or against this statement, we will show here that a simple evolutionary predatorprey model yields
primeperiodic preys. Moreover, this result will be used as a numbertheoretical tool, namely to generate large prime
numbers. Furthermore, we will demonstrate how a spatiotemporal extension of the model renders spiralwaves being reminiscent of
those observed in excitable systems, hostparasitoid systems and prebiotic evolution.