НАЦИОНАЛЬНАЯ АКАДЕМИЯ НАУК БЕЛАРУСИ

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НЕЛИНЕЙНЫЕ ЯВЛЕНИЯ В СЛОЖНЫХ СИСТЕМАХ Международный научный журнал Издается учреждением "Издательство «Адукацыя і выхаванне»", Минск, Беларусь

Volume 3, Number 2, 2000

СОДЕРЖАНИЕ

Summary: Stability and asymptotic stability of solutions of impulsive partial differential-functional equations of first order are studied via Lyapunov functions.
Key words and phrases: stability of solutions, impulsive partial differential-functional equations, Lyapunov functions.

M. Morandi Cecchi, L. Salasnich
Non-Hyperbolic Dynamics: a Family of Special Functions. pp. 116--121

Summary: We study the iterative dynamics of a family of special functions from R² into R² with a non-hyperbolic fixed point in the origin. The characterization by the eigenvalues is analyzed and discussed.
Key words: iterative dynamics, discrete dynamics, basin of attraction, non-hyperbolic fixed point.

A.T. Vlassov
Bifurcations in Dynamical Systems with Inputs. pp. 122--131

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. 132--134

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. 135--152

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 data-tuned 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 Gram--Schmidt 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 entropy-like functionals can be introduced as quantities associated with an arbitrary Hermitian operator and any partition of its spectrum. We define also entropy-like functionals connected with expansions of a signal vector over orthonormal basis functions and over generalized frames.
Key words: Hilbert space, prediction of time series, Gram--Schmidt orthogonalization, entropy-like functional.

Claude Froeschle, Elena Lega
Reestablishing Equilibrium Paths in a Neo-Austrian Model: Analysis of Transitory Regimes. pp. 153--170

Summary: Using a neo-austrian 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. 171--173

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. El-Hakim
An Experimental Determination of the Critical Exponents at the Metal-Insulator Transition in Germanium, ”Doped” by Radiation Defects. pp. 174--176

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 1.5--300 K in germanium, disordered by large uences of the fast reactorneutrons, with the radiation defect concentration making it an insulator near the metal-insulator 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 Self-Ionization Process for Atomic Ions in a Magnetic Field. pp.177--180

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 self-ionization 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. 181--187

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. 188--191

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. 192--207

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. 208--213

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 predator-prey model yields prime-periodic preys. Moreover, this result will be used as a number-theoretical tool, namely to generate large prime numbers. Furthermore, we will demonstrate how a spatio-temporal extension of the model renders spiralwaves being reminiscent of those observed in excitable systems, host-parasitoid systems and prebiotic evolution.