Chemical waves spiral patterns

All the complex behavior described so far in this Chapter arises from the diffusive coupling of the local dynamics which in the homogeneous case have simple fixed points as asymptotic states. If the local dynamics becomes more complex, the range of possible dynamic behavior in the presence of diffusion becomes practically unlimited. It is clear that coupling chaotic subsystems could produce an extremely rich dynamics. But even the case of periodic local dynamics does so. Diffusively coupled chemical or biological oscillators may become synchronized (Pikovsky et ah, 2003), or rather additional instabilities may arise from the spatial coupling. This may produce target waves, spiral patterns, front instabilities and several different types of spatiotemporal chaos or phase turbulence (Kuramoto, 1984). 

Fig. 1. Target and spiral patterns of chemical waves in a thin layer (thickness, 0.7 mm) of an excitable Belousov-Zhabotinsky solution at an early (left) and a later stage (right). The reaction mixture contains CH2(COOH)2, NaBrOs, NaBr, H2SO4, and ferroin. Initial concentrations as given in [10].

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. 

A typical feature of a non-potential systems is the non-stationary oscillatory behavior that usually manifests itself in the propagation of waves. We have shown that the nonlinear evolution of waves near the instability threshold is described by the complex Ginzburg-Landau (CGL) equation. This equation is capable of describing various kinds of instabilities of wave patterns, like the Benjamin-Feir instability. In two dimensions, the CGL equation describes the formation of spiral waves that are observed in many biological and chemical systems characterized by the interplay of diffusion and chemical reactions at nano-scales. 

Figure 1 (a) A spiral wave formed in a thin gel layer of the Belousov-Zhabotinsky reaction (from Belmonte and Flesselles, Ref. 5. (b) Formation of a labyrinthine pattern in the bistable region of the iodine-ferrocyanide-sulfite chemical reaction in a gel reactor (from Lee and Swinney, Ref. 6). 

In addition to excitability, the FHN model can also possess oscillatory solutions. Figure 5 a) shows the configuration of the nullclines for a different choice of parameters. Now the system has an imstable fixed point or steady state and the stable attracting state is a limit cycle oscillation shown as the closed loop surrounding the unstable fixed point in the (Mt )-plane of the figure. Chemical patterns such as spiral waves can form in oscillatory as well as excitable systems and we shall have occasion to discuss some aspects of patterns in oscillatory media below. 

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