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

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

Volume 2, Number 2, 1999
CONTENTS
Srinivas Jammalamadaka, Jцrg Main, Gьnter Wunner
The Effect of Scars on the Statistics of Transition Probabilities of Classically Chaotic Quantum Systems. pp. 15Summary: We study the statistical properties of generalized intensities (squared matrix elements of Hermitian operators) for the hydrogen atom in strong magnetic fields in a
range of parameters where the classical analogue of the system exhibits completely chaotic dynamics. In this way we extend
previous work by Prosen and Robnik on the statistics of generalized intensities in billiard systems in the transition region with mixed classical dynamics. We
observe deviations from the statistics found in that work, and demonstrate that these are due to the effect of scarring of wave functions by unstable periodic orbits.
V.P. Gribkovskii, S.V. Voitikov, M.I. Kramar, G.I. Ryabtsev, and R. Kragler
Amplified Luminescence as a Source of Nonlinearity in Laser Diodes. pp. 610
Summary: Nonlinear properties of amplified luminescence in semiconductor bulk and quantumwell (QW) lasers
have been investigated and modeled. Amplified luminescence depends strictly on carrier concentration and temperature
representing by itself an additional relatively powerful source of nonlinearity that can make laser dynamics much more
complicated. In general, the rate of amplified luminescence still keeps its quadratic character as that of spontaneous
recombination. In QW lasers, in the vicinities of laser mode hopping the amplified luminescence exhibits local peaklike
carrier dependence and may affects abruptly on a bifurcation of mode hopping with the injection level increasing.
H.V. Grushevskaya
Chaos in NearHamiltonian Systems with Singular Perturbation: Applications to Oscillatory Model of HodgkinHuxley Neuron. pp. 1124
Summary: Neuron's oscillatory model is transformed to the form of nearHamiltonian system with singular
coefficients. We demonstratethat singular coefficients can be considered as a perturbation of the system by virtue of discrete
sequence of kicks. The linearization of the perturbation in the vicinity of periodical solution for a given kick allows us to
rewrite the system as a matrix Schrцdinger equation for twolevel quantum system in a resonant quasimonochromatic field.
As a result, the original system demonstrates chaotic behavior through the cascade of period doubling bifurcations. The
bifurcations in this case correspond to the series of quantum nutation of nutations.
A.V. Kartynnick, A.D. Linkevich
Global Attractors, Attracting Regions and Regions of Visiting for Dynamical Systems. pp. 2530
Summary: The phase space of an autonomous dissipative nonlinear dynamical system is investigated. It is
proved that under certain conditions such a system has an attracting region A with a basin of attraction around it, i.e. any
trajectory starting somewhere inside this basin reaches the region A after a finite time interval and never leaves A.
In certain cases, the region A turns into a global attractor of the system when the basin of attraction
coincides with the whole remaining part of the phase space and all attractors of the system lie inside the region A.
Another theorem yields sufficient conditions under which a system has a region of visiting G in the phase space,
i.e. any trajectory starting outside G reaches this region after a finite time interval (and further can leave G
in contrast to the case of an attracting region). Several examples are considered.
V.A. Gaisyonok, G.G. Krylov
On the Green Function for the Restricted Rotational Diffusion Model. pp. 3134
Summary: The explicit expression for the Green function of the restricted rotational diffusion equation has been
constructed based on singular perturbations approach to boundary problems.
Alexander Rauh
Remarks on Perturbation Theory for Hamiltonian Systems. pp. 3543
Summary: A comparative discussion of the normal form and action angle variable method is presented in a
tutorial way. Normal forms are introduced by Lie series which avoid mixed variable canonical transformations. The main interest
is focused on establishing a third integral of motion for the transformed Hamiltonian truncated at finite order of the
perturbation parameter. In particular, for the case of the action angle variable scheme, the proper canonical transformations are
worked out which reveal the third integral in consistency with the normal form. Details are discussed exemplarily for the
HйnonHeiles Hamiltonian. The main conclusions are generalized to the case of n perturbed harmonic oscillators.
H. Rehfeld, H. Alt, C. Dembowski, H.D. Grдf, R. Hofferbert, H. Lengeler, A. Richter
Wave Dynamical Chaos in Superconducting Microwave Billiards. pp. 4448
Summary: During the last few years we have studied the chaotic behavior of special formed Euclidian
geometries, socalled billiards, from the quantum or in more general sense "wave dynamical'' point of view. Due to the
equivalence between the stationary Schrцodinger equation and the classical Helmholtz equation in the twodimensional case (plain
billiards), it is possible to simulate "quantum chaos'' with the help of macroscopic, superconducting microwave cavities. Using
this technique we investigated spectra of three billiards of the family of Pascal's Snails (RobnikBilliards) with a different
chaoticity in each case in order to test predictions of standard stochastical models for classical chaotic systems.
Marko Robnik, Luca Salasnich and Marko Vranicar
WKB Corrections to the Energy Splitting in Double Well Potentials. pp. 4962
Summary: By using the WKB quantization we deduce an analytical formula for the energy splitting in a
doublewell potential which is the usual Landau formula with additional quantum corrections. Then we analyze the accuracy of our formula for
the double square well potential, the inverted harmonic oscillator and the quartic potential.
Vassilios M. Rothos and Tassos C. Bountis
NonIntegrability and Infinite Branching of Solutions of 2DOF Hamiltonian Systems in Complex Plane of Time. pp.6371
Summary: It has been proved by S.L. Ziglin, for a large class of 2degreeoffreedom (d.o.f) Hamiltonian systems,
that transverse intersections of the invariant manifolds of saddle fixed points imply infinite branching of solutions in the
complex time plane and the nonexistence of a second analytic integral of the motion. Here, we review in detail our recent
results, following a similar approach to show the existence of infinitelysheeted solutions for 2 d.o.f. Hamiltonians which
exhibit, upon perturbation, subharmonic bifurcations of resonant tori around an elliptic fixed point. Moreover, as shown recently,
these Hamiltonian systems are nonintegrable if their resonant tori form a dense set. These results can be extended to the case
where the periodic perturbation is not Hamiltonian.
Aneta Stefanovska, Saso Strle, Maja Bracic and Hermann Haken
Model Synthesis of the Coupled Oscillators Which Regulate Human Blood Flow Dynamics. pp.7287
Summary: A model synthesis for the system that regulates blood flow is presented. The model consists of coupled
oscillators which present subsystems involved in the regulation of one passage of blood through the cardiovascular system. It is
based on a priori physiological knowledge and observations of the system's functions by measurements and linear
and nonlinear analysis of measured time series. Furthermore, the blood flow through the system of closed tubes consisting of blood
vessels is described by wave equations.
V.V. Belov, G.S. Bokun, V.S. Vikhrenko
Manybody Correlations in Equilibrium Lattice Systems. pp. 8892
Summary: The procedure of successive approximations beyond the quasichemical one for lattice gas
systems of arbitrary density is suggested. Even at the lowest order approximation the calculation results for the phase diagram
of the system of particles with attractive nearest neighbor interaction are in good agreement with Monte Carlo simulation.
E.A. Nikiforov and R.M. Yulmetyev
First Experimental Observation of Memory Effects in Biological Complex System by NMR Method. pp. 9395
Summary: Direct registration of memory effects in biological complex system (plant tissue) has been carried out
for the first time. Measurements of temperature dependence of the proton spinlattice relaxation time in rotation frame allow to
observe quasiMarkovian scenario of NMR spin relaxation in biological system.
Sergey Sadov
Lissajous Solutions of the Satellite Oscillation Equation: Stability and Bifurcations via Higher Order Averaging. pp.96100
Summary: The equation of plane oscillations of a satellite is a Hamiltonian system with one degree of freedom
and 2pperiodic dependence on time and on x_{1}. It contains two parameters $e\eps$ and e, one of which ($\eps$) is small. The unperturbed system is linear. Solutions that correspond to cycles of the Poincar\'e map on a cylinder $(x_{1}\mpi,\,x_{2})$
are called Lissajous solutions. Their stability and bifurcations with parameter e changing are studied by the averaging
method. It is shown how degenerate cases, where calculation of higher order terms is needed, arise in a natural way. Sufficient
truncations of the normal form for those cases are described.
Andreas Ruffing
Discretized Schroedinger Eigenfunctions and qHypergeometric Series on Deforming Geometric Progressions. pp.101116
Summary: Discretizations of the Schroedinger equation are introduced on geometric progressions $\R_{q} :=
\{+q^{n}, q^{n} \vert n \in \Z\}, q > 1$. The symmetries of the geometric progressions are elaborated. We investigate the
influence of this discretization to qdeformations of Hermite polynomials. The limits $q \rightarrow 1$ and $q
\rightarrow \infty$ as deformations of $\R_{q}$ are considered. The first limit, $q \rightarrow 1$, is related to an
approximation theoretical problem for step functions in ${\cal{L}}^{2}(\R)$. The second limit, $q \rightarrow \infty$, is
related to the theory of topological deformations on compact Riemann surfaces. Both limits are related to each other. Proceeding into this direction, one obtains the fascinating fact that quantum group structures can be related to topological
degeneration effects. The results finally contribute to a better mathematical understanding of quantum models with dilatative
supersymmetry.