Spatio-temporal structures

Recently there has been an increasing interest in self-oscillatory phenomena and also in formation of spatio-temporal structure, accompanied by the rapid development of theory concerning dynamics of such systems under nonlinear, nonequilibrium conditions. The discovery of model chemical reactions to produce self-oscillations and spatio-temporal structures has accelerated the studies on nonlinear dynamics in chemistry. The Belousov-Zhabotinskii(B-Z) reaction is the most famous among such types of oscillatory chemical reactions, and has been studied most frequently during the past couple of decades [1,2]. The B-Z reaction has attracted much interest from scientists with various discipline, because in this reaction, the rhythmic change between oxidation and reduction states can be easily observed in a test tube. As the reproducibility of the amplitude, period and some other experimental measures is rather high under a found condition, the mechanism of the B-Z reaction has been almost fully understood until now. The most important step in the induction of oscillations is the existence of auto-catalytic process in the reaction network. 

From the point of view of the Hanusse theorem just discussed, a system with two intermediate stages and mono- and bimolecular reactions are not capable to reveal any temporary, spatial and spatio-temporal structures. However, results obtained in the past few years permit reconsideration of such an absolute point of view. 

Therefore, the joint correlation functions Xvjl (a t)> being at least potentially observable, are more a theoretical than an experimental tool for the description of interacting particles in condensed media. Both these joint functions and macroscopic concentrations nv t) determine the lowest level to characterize the spatio-temporal structure of a system. 

Higher + Electronic + Interaction with the environment Spatio-temporal structure (flexibility, conformation) Electronic properties (electron distribution, polarizability, ionisation) Solvation, hydration, partitioning, intermolecular interactions Conformational energy diagrams, computer display Molecular orbitals, electrostatic potential maps Computer display... 

S. Alonso, F. Sagues, and J. M. Sancho. Excitability transitions and wave dynamics under spatio-temporal structured noise. Ph ys. Rev. E, 65 066107, 2002. 

Fig. 10.7. Self-organisation and spatio-temporal structure formation. The figures show contour jjlots of the degree of condensation (left plot) and the dust temperature (right ])lot) in tile a-a-plaiie at three selected time steps. The white circle In the center of eacli figure mark the star.

Nonlinear dynamics of complex processes is an active research field with large numbers of publications in basic research and broad applications from diverse fields of science. Nonlinear dynamics as manifested by deterministic and stochastic evolution models of complex behaviour has entered statistical physics, physical chemistry, biophysics, geophysics, astrophysics, theoretical ecology, semiconductor physics and -optics etc. This research has induced a new terminology in science connected with new questions, problems, solutions and methods. New scenarios have emerged for spatio-temporal structures in dynamical systems far from equilibrium. Their analysis and possible control are intriguing and challenging aspects of the current research. 

As temporal oscillation was the subject of intensive research in the nineteen-seventies, it is reasonable to predict that the understanding of the mechanisms for the formation of chemical spatial and spatio-temporal structures will be one of the most challenging areas of chemical kinetics. 

The term pattern is a rather vague notion to denote the arrangement of certain abstract elements. According to our subjective feeling there is no clear distinction between the terms structure and pattern . Temporal, spatial and spatio-temporal structures in thermochemical systems might be the result of the interaction of chemical components by diffusion and chemical reactions. 

Growth phenomena (e.g. cluster formation) are adequately described by cellular automata (e.g. Wolfram, 1986). Cellular automata are examples of systems constituting many identical components. While the components are simple, their collection is capable of complex dynamic behaviour. Cellular automata provide an alternative approach to spatio-temporal structure formation, as opposed to models based on differential equations. 

The equations describing the spatio-temporal structure of the stimulus-response pathways could be written ... 

According to the conditions ascribed, various dynamical behaviors are observed in the interface and in its vicinity longitudinal or transveral deformations, dissipative spatio-temporal structures, motion in toto, interfacial turbulence, spontaneous emulsification. 

Spiral waves in an oscillating system are spatio-temporal structures with equiphases - lines of constant phase - describing particular geometrical... 

Fig. 14. Spatio-temporal structures appearing in a compartmentalized network. The variable Re Wj is represented as gray shades at a given time. Waves are propagating from one compartment into the others passing through the three windows and h. The parameters of Equation (1) are a = 0, / = 1, 7 = 2.22 and Aoj = 0.6.

Concerning open spatial reactors, a first approach was provided by the so-called Couette flow reactor . In this quasi-one-dimensionalreactor, the transport is ensured by turbulent diffusion. Fresh reagents permanently renewed at each end allow the system to be maintained at a controlled distance from equilibrium. When operated in this reactor, the CIMA reaction lead to various spatio-temporal structures (oscillating fronts) as well as to a nontrivial stationary spatial structure (three stationary fronts) [58, 59]. However, this is... 

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