We present three examples how complex spatio-temporal patterns can be linked to hyperchaotic attractors in dynamical systems consisting of nonlinear biochemical oscillators coupled linearly with diffusion terms. The systems involved are: (a) a two-variable oscillator with two consecutive autocatalytic reactions derived from the Lotka–Volterra scheme; (b) a minimal two-variable oscillator with one first-order autocatalytic reaction; (c) a three-variable oscillator with first-order feedback lacking autocatalysis. The dynamics of a finite number of coupled biochemical oscillators may account for complex patterns in compartmentalized living systems like cells or tissue, and may be tested experimentally in coupled microreactors.

Gerold Baier

Sven Sahle

Spatio-temporal patterns with hyperchaotic dynamics in diffusively coupled biochemical oscillators

For the case of a reaction&#8211;diffusion system, the stationary states may be represented by means of a state surface in a finite-dimensional state space. In the simplest example of a single semi-linear model equation given. in terms of a Fredholm operator, and under the assumption of a centre of symmetry, the state space is spanned by a single state variable and a number of independent control parameters, whereby the singularities in the set of stationary solutions are necessarily of the cuspoid type. Certain singularities among them represent critical states in that they form the boundaries of sheets of regular stable stationary solutions. Critical solutions provide ignition and extinction criteria, and thus are of particular physical interest. It is shown how a surface may be derived which is below the state surface at any location in state space. Its contours comprise singularities which correspond to similar singularities in the contours of the state surface, i.e., which are of the same singularity order. The relationship between corresponding singularities is in terms of lower bounds with respect to a certain distinguished control parameter associated with the name of Frank-Kamenetzkii.

Dirk  Meink&#246;hn

Characteristic phenomena in combustion

This study augments the traditional linear cobweb model with lower and upper bounds for variations of output. Its purpose is to detect the relationship between the output 
constraints and the dynamics of the modified model. Due to the upper and lower bounds, 
a transitional function takes on a tilted <mml:math alttext="$z$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi></mml:math>-profile having three piecewise segments with 
two turning points. It prevents the price (or quantity) dynamics from explosive 
oscillations. This study demonstrates, by presenting numerical examples, that the 
modified cobweb model can generate various dynamics ranging from stable periodic 
cycles to ergodic chaos if a product of the marginal propensity to consume and the 
marginal product is greater than unity.

Akio Matsumoto

Ergodic cobweb chaos

A new method to stabilize dynamical systems by forcing the system variables into the desired unstable stationary point is proposed. The key conception of the method is based on parametric perturbation. This means that the equations of motion are influenced by continuous variation of some selected parameters. Thus &#8211; without using any external forces &#8211; the motion of the system approaches the chosen unstable stationary point. The variation of the parameters will vanish after the successful stabilization. Therefore, the system and its parameters are changed during the control process only. The algorithm is applied to an urban system within a metropolitan area obeying a Lorenz-type dynamics as well as to the H&#233;non attractor as an example for a discrete scenario.

G&#252;nter  Haag

Tilo  Hagel

Timm  Sigg

Active stabilization of a chaotic urban system

This paper discusses the significance of applying nonlinear theory to examine the complexity of social and economic evolution. First, we generally examine possible implications of nonlinear theory for analyzing the complexity of human societies. Second, we select two socio-economic models to illustrate our viewpoints.

&#197;ke E.  Andersson

Wei-Bin  Zhang

Nonlinearity in social dynamics &#8211; order versus chaos

The universality of oscillation theory laws is discussed. It is suggested that all models of concrete systems be separated into four categories: models &#8211;  &#8220;portraits&#8221; of investigated systems, models of the type of &#8220;black box&#8221;, aggregating models, and models of certain phenomena which can occur in real systems. As an example of the model of the fourth type, the equation of oscillations of a pendulum with a randomly vibrating suspension axis is considered for the purpose of clarifying the mechanism of the transition to turbulence.

Polina S. Landa

Universality of oscillation theory laws. types and role of mathematical models

The article consists of two parts. In the first section the concepts of sociodynamics are briefly explained. Sociodynamics is a general modelling strategy for the quantitative description of dynamic processes in the human society. The central concepts of sociodynamics include transition rates depending on dynamic utilities and the master equation for the probability distribution over macrovariables. From it the set of nonlinear coupled differential equations for the quasi-meanvalues of the macrovariables can be derived.In the second part this modelling strategy is applied to two &#8220;windows of perception&#8221; of the evolution of settlement structures: one model refers to the relatively fast urban evolution on the microscale and the other refers to the relatively slow regional evolution on the macroscale.The micromodel considers the urban structure as a system of sites on which different kinds of buildings (dwellings, schools, stores, service-stations, factories...) can be erected. The step by step evolution of the city configuration is treated as a stochastic process guided by utility considerations. The sociodynamic formalization of this concept leads to equations for the evolution of the urban city configuration. Numerical simulations illustrate this urban &#8220;microdynamics&#8221;.The macromodel treats the settlement formation in a region on a more global scale. The evolution of the density of economically active populations who produce and consume goods is considered. The driving force of density changes is the spatial difference of incomes motivating the individuals to migrate to locations of optimal income. This nonlinear process leads to the self-organisation of spatially heterogeneous population distributions forming the settlements. Their micro-structure can thereupon be treated by the micromodel.

Wolfgang  Weidlich

Sociodynamics applied to the evolution of urban and regional structures

Vladimir Gontar

Michael Sonis

Editorial

About this journal

Discrete Dynamics in Nature and Society publishes research that links basic and applied research relating to discrete dynamics of complex systems encountered in the natural and social sciences.

Journal profile

Meet the editorial board

Chief Editor, Dr&nbsp;Renna, is an associate professor at the University of Basilicata, Italy. His research interests include manufacturing systems, production planning and enterprise networks.&nbsp;

Editor spotlight

Submit to Special Issue

We currently have a number of Special Issues open for submission. Special Issues highlight emerging areas of research within a field, or provide a venue for a deeper investigation into an existing research area.

Special Issues

Author guidelines

Editorial board

Databases and indexing

Discrete Dynamics in Nature and Society - Table of contents 1997

Discrete Dynamics in Nature and Society | Hindawi

title

Discrete Dynamics in Nature and Society publishes research that links basic and applied research relating to discrete dynamics of complex systems encountered in the natural and social sciences.

description

https://images.hindawi.com/journals/ddns/ddns.banner.jpg

Discrete Dynamics in Nature and Society