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

/ Publications / Scientific Journals
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

## CONTENTS

Kinetic Approach to the Nucleation-and-Growth Phase Transition in Complex Systems. pp. 321--352

Summary: This study deals mostly with kinetic anomalies occuring in nucleation-and-growth 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 diffusion-limited 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 self-similar when looking at their distribution over the available physical space, but that the processes proceed also in a self-similar 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 time-dependent, which is by the way the main assumption of the so-called dispersive or long-tailed kinetics frequently applied to, e.g. reactive and fluctuating soft-matter 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 Avrami-Kolmogorov (Mehl--Johnson) concept, utilized mostly in metallic polycrystals, or to some extent, the Mullins-Sekerka-like 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 order-disorder effects manifested, for example, in lipid mesomorphs. A critical comparison with other available kinetic approaches has been done as well.
Key words: phase transitions; nucleation-and-growth, 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. 362--375

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. 376--379

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 2-D 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 well-known Poisson and Wigner distributions.
Key words: quantum chaos, level statistics, DVR method.

N. Buric and N. Vasovic
Variations of the Stability Due to the Time-Delay in the Immune Response. pp. 380--388

Summary: A phenomenological model of the immune response, given by two differential equations with time-delay, is analyzed. The model has very simple formulation in terms of only two relevant quantities and a small number of independent parameters. However, the time-delay introduces the possibility of a quite complex dynamics, with various types of attractors, depending on the values of the parameters and the time-delay. In particular, we analyzed the conditions on the parameters such that a destabilizing Hopf bifurcation of the fixed points due to variations of the time-delay is possible. Chaotic solutions of the model are briefly described. We argue that the delay-differential equations are very natural mathematical framework for phenomenological models of the immune response.
Key words: immune response, delay-differential equations, stability.

Yu.L. Klimontovich
To Thermodynamics and Kinetics of Second Order Phase Transitions. pp. 389--406

Summary: Thermodynamic and fluctuational characteristics for all values of temperature have been calculated for the simplest model of ferro-electrics 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 self-consistent equations for the first and second moments have been used for the description of mono- and poly-domain 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 Predator-Prey Model with Some Cover on Prey Species. pp. 407--420

Summary: A predator-prey 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: predator-prey system, refuge, permanence, global stability, environmental stochasticity.

Mitja Perus
Similarities in Mathematical Models of Information Processing in Neural and Quantum Networks. pp. 421--425

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 (neural-like, but more profound) information processing capabilities of quantum systems.
Key words: neural network, information processing, quantum systems.

Designed and maintained by Dr. Nikolai N. Kostyukovich. Last updated: January 23, 2001
Created with assistance of Dr. Leonid F. Babichev