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Nonlinear Phenomena in Complex Systems, 2000, Vol.3, No.4

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

Volume 3, Number 4, 2000
CONTENTS
Adam Gadomski
Kinetic Approach to the NucleationandGrowth Phase Transition in Complex Systems. pp. 321352
Summary: This study deals mostly with kinetic anomalies occuring in nucleationandgrowth phenomena in complex systems,
e.g. polycrystals, partly ordered alloys, quasicrystalline assemblies or mesomorphs. The main kinetic phenomenological
approach utilized in a versatile way is in fact an anomalous random walk approximation, though the process is not thought to
be realized in a position space (a most expected case) but in the space of grains (clusters) sizes. Two effective descriptions of
the processes are discussed. The first, in which a supreme role of capillary forces as the leading kinetic mechanism is
proposed. The second, for which the Fick's law is fulfilled, and the effective area of the clusters, like in the diffusionlimited
cluster cluster aggregation, is of prior importance. A novel contribution to the kinetic problem mentioned is offered which
seems to be very suitable for revealing kinetic anomalies in such systems. It relies on assuming that the systems under study are
not only statistically selfsimilar when looking at their distribution over the available physical space, but that the processes
proceed also in a selfsimilar manner when carefully inspecting their time behaviour. Therefore, the basic kinetic coefficients
characterizing the system's behaviour are often assumed to be inverse power law timedependent, which is by the way the
main assumption of the socalled dispersive or longtailed kinetics frequently applied to, e.g. reactive and fluctuating
softmatter systems, like model biomembranes or polymers. The presented description, which unfortunately offers no explicit
microscopic insight, is compared to the standard approaches of theoretical analysis of heterogeneous phase transformations as
the AvramiKolmogorov (MehlJohnson) concept, utilized mostly in metallic polycrystals, or to some extent, the
MullinsSekerkalike instability mechanism applied to biopolymers. The study is completed by a brief consideration of the
propagation of mechanical stresses in a polycrystal along crystalline boundaries and some orderdisorder effects manifested,
for example, in lipid mesomorphs. A critical comparison with other available kinetic approaches has been done as well.
Key words: phase transitions; nucleationandgrowth, anomalous kinetics, complex systems, power laws.
Claude Froeschle and Elena Lega
An Example of Interplay Between Theoretical Results on the Structure of Dynamical Systems and Numerical
Experiments Based on Mapping Modeling. pp. 362375
Summary: Starting from a recent mathematical result concerning the structure around invariant KAM tori we show how
numerical experiments using mapping as model problem allow to "visualize" and extend to a physical domain results which
could remain only of mathematical interest and somehow far from the real application to dynamical systems. The feasibility of
the numerical experiments mainly depend on the sensitivity and on the computational cost of tools of detection of weak chaos.
We present the most important tools and deeply compare each other in the aim of giving a global description of the actual state
of numerical methods of analysis.
Key words: chaos, dynamical systems, mapping modeling, numerical experiment.
Luis Seidel, Rosa M. Benito, and F. Borondo
Energy Level Statistics in a Molecular System with Three Degrees of Freedom: HCN/HNC. pp. 376379
Summary: Calculations of wavefunctions and energy levels for the vibrational dynamics of the isomerizing molecule
HCN/HNC in Jacobi coordinates for three degrees of freedom are carried out, using a recent \emph{ab initio} potential
energy surface. 800 energy levels have been converged using a DVR method (Discrete Variable Representation) that is well
suited for large amplitude motions and highly excited vibrational levels. The classical dynamics of the system has been studied
for a 2D model and shows a mixed phase space structure with regular and chaotic regions. We have also carried out an
statistical study of the energy level distribution. We have calculated the cumulative probability distribution of the level spacings
$W(S)$. This function has been fitted to a Brody distribution, which is intermediate between the wellknown Poisson and Wigner distributions.
Key words: quantum chaos, level statistics, DVR method.
N. Buric and N. Vasovic
Variations of the Stability Due to the TimeDelay in the Immune Response. pp. 380388
Summary: A phenomenological model of the immune response, given by two differential equations with timedelay, is
analyzed. The model has very simple formulation in terms of only two relevant quantities and a small number of independent
parameters. However, the timedelay introduces the possibility of a quite complex dynamics, with various types of attractors,
depending on the values of the parameters and the timedelay. In particular, we analyzed the conditions on the parameters such
that a destabilizing Hopf bifurcation of the fixed points due to variations of the timedelay is possible. Chaotic solutions of the
model are briefly described. We argue that the delaydifferential equations are very natural mathematical framework for
phenomenological models of the immune response.
Key words: immune response, delaydifferential equations, stability.
Yu.L. Klimontovich
To Thermodynamics and Kinetics of Second Order Phase Transitions. pp. 389406
Summary: Thermodynamic and fluctuational characteristics for all values of temperature have been calculated for the simplest
model of ferroelectrics in the critical region of the second order phase transition. The kinetic equation for the local density
function for the vector of electric polarization has been used as a basis for the calculations. The selfconsistent equations for the
first and second moments have been used for the description of mono and polydomain states appropriately. The obtained
theoretical results have been compared with experimental data.
Key words: kinetics, phase transition, critical point, ferroelectrics.
Chattopadhyay, N. Bairagi, and R. R. Sarkar
A PredatorPrey Model with Some Cover on Prey Species. pp. 407420
Summary: A predatorprey model with a constant number of refuge is proposed and analyzed. The permanence of the
solutions are studied and pointed out the role of prey refuge for survival of the species. We study the global stability of the
system around the positive interior equilibrium point. It is observed that global stability of the system around positive interior
equilibrium does not necessarily imply the persistence of the species. The marginal effects in predator and prey stock level as
well as the resource rent of each species are discussed. Two other reasonable possibilities such as constant proportion of
refuge and environmental stochasticity in both the population are also considered.
Key words: predatorprey system, refuge, permanence, global stability, environmental stochasticity.
Mitja Perus
Similarities in Mathematical Models of Information Processing in Neural and Quantum Networks. pp. 421425
Summary: A comparison of the mathematical formalism of associative neural network theory and quantum theory is made.
The analogies presented can help us in searching for similar (neurallike, but more profound) information processing capabilities of quantum systems.
Key words: neural network, information processing, quantum systems.