Spatio-temporal coupling

The local dynamics of tire systems considered tluis far has been eitlier steady or oscillatory. However, we may consider reaction-diffusion media where tire local reaction rates give rise to chaotic temporal behaviour of tire sort discussed earlier. Diffusional coupling of such local chaotic elements can lead to new types of spatio-temporal periodic and chaotic states. It is possible to find phase-synchronized states in such systems where tire amplitude varies chaotically from site to site in tire medium whilst a suitably defined phase is synclironized tliroughout tire medium 51. Such phase synclironization may play a role in layered neural networks and perceptive processes in mammals. Somewhat suriDrisingly, even when tire local dynamics is chaotic, tire system may support spiral waves... 

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. 

TNC.65. 1. Prigogine and R. Lefever, On the spatio-temporal evolution of cellular tissues, in Biological Structures and Coupled Flows, A. Oplatka and M. Balaban, eds.. Academic Press, New York, 1983, pp. 3-26. 

Thus, the additional approximations underlying the NEE are paraxiality both in the free propagator and in the nonlinear coupling, and a small error in the chromatic dispersion introduced when the background index of refraction is replaced by a constant, frequency independent value in both the spatio-temporal correction term and in the nonlinear coupling term. Note that the latter approximations are usually not serious at all. 

O. Beck, A. Amann, E. Scholl, J. E. S. Socolar, and W. Just Comparison of time-delayed feedback schemes for spatio-temporal control of chaos in a reaction-diffusion system with global coupling, Phys. Rev. E 66, 016213 (2002). 

We have also discussed the formation of spatio-temporal patterns in non-variational systems. A typical example of such systems at nano-meter scales is reaction-diffusion systems that are ubiquitous in biology, chemical catalysis, electrochemistry, etc. These systems are characterized by the energy supply from the outside and can exhibit complex nonlinear behavior like oscillations and waves. A macroscopic example of such a system is Rayleigh-Benard convection accompanied by mean flow that leads to strong distortion of periodic patterns and the formation of labyrinth patterns and spiral waves. Similar nano-meter scale patterns are observed during phase separation of diblock copolymer Aims in the presence of hydrodynamic effects. The pattern s nonlinear dynamics in both macro- and nano-systems can be described by a Swift-Hohenberg equation coupled to the non-local mean-flow equation. 

Real systems are open systems and may consist of numerous subsystems global system, human society and human body are typical examples. The nature of subsystems, variables, fluxes and forces, their coupling leading to cross-phenomena, temporal and spatio-temporal changes, pattern formation and self-organization would be discussed in the subsequent chapters. 

Non-linear phenomena such as temporal oscillations and chemical waves in the case of chemical reactions are governed by the autocatalytic reaction (positive feedback) and reaction where the product of autocatalysis is destroyed by some other species (negative feedback). Of course in the case of chemical waves (spatio-temporal oscillations), diffusion does play a role, and the concept of reaction diffusion equation is evoked to predict the dependence of velocity of chemical waves on different parameters. In this chapter, we propose to discuss electrical potential oscillations generated due to coupling of volume flux, solute flux and electric current through solid-liquid interface (membrane systems), liquid-liquid interface, solid-liquid-liquid interface (density oscillator) and liquid-liquid-vapour interface. 

Chemical wave patterns corresponding to moving surface concentration patches are the result of coupling of siuface diffusion, surface reconstruction, and surface reaction. Depending on the reaction condition, such spatio-temporal phenomena can also lead to an oscillatory behavior of... 

Coming back to limit cycle oscillations shown by systems of ordinary differential equations, this simple mode of motion still seems to deserve some more attention, especially in relation to its role as a basic functional unit from which various dynamical complexities arise. This seems to occur in at least two ways. As mentioned above, one may start with a simple oscillator, increase [x, and obtain complicated behaviors this forms, in fact, a modern topic. However, another implication of this dynamical unit should not be left unnoticed. We should know that a limit cycle oscillator is also an important component system in various self-organization phenomena and also in other forms of spatio-temporal complexity such as turbulence. In this book, particular emphasis will be placed on this second aspect of oscillator systems. This naturally leads to the notion of the many-body theory of limit cycle oscillators we let many oscillators contact each other to form a field , and ask what modes of self-organiza-tion are possible or under what conditions spatio-temporal chaos arises, etc. A representative class of such many-oscillator systems in theory and practical application is that of the fields of diffusion-coupled oscillators (possibly with suitable modifications), so that this type of system will primarily be considered in this book. 

People often speak of chemical turbulence whereby either of two distinct chaotic phenomena may be meant. One is the spatially uniform but temporally chaotic dynamics exhibited by the concentrations of chemical species, while the other involves spatial chaos too. For chemical turbulence in the latter sense, our attention is usually focused upon systems in which the local dynamics itself is non-chaotic, while such non-chaotic elements are coupled through diffussion to produce spatio-temporal chaos. In fact, if the local elements were already chaotic, the fields composed of them would trivially exhibit spatio-temporal chaos. Hence non-trivial chemical turbulence involving spatio-temporal chaos may be called diffusion-induced chemical turbulence. 

There may be an additional value in studying spatio-temporal chemical turbulence, in connection with its possible relevance to some biological problems. This is expected from the fact that the fields of coupled limit cycle oscillators (or nonoscillating elements with latent oscillatory nature) are often met in living systems. In some cases, such systems show orderly wavelike activities much the same as those observed in the Belousov-Zhabotinsky reaction. There seems to be no reason why we should not expect such organized motion to become unstable and hence show turbulent behavior. The recent work by Ermentrout (1982) who derived a Ginzburg-Landau type equation for neural field seems to be of particular interest in this connection. 

A direct simulation of the coupled system of the partial differential equations, Eq. 29, with appropriate boundary conditions (v = 0, n prescribed at the confining plates, etc.) is at the limits of the supercomputers of today. It will turn out that in the liquid crystal systems a rich scenario of patterns, including spatio temporal chaos, develops already near threshold so that perturbational calculations are useful. 

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