Oscillating reactions diffusion

Homo J, Gonzalez CF, Hayas A (1995) The network method for solutions of oscillating reaction-diffusion systems. J Comput Phys 118 310-319... 

When an oscillating reaction-diffusion system is near a direct Hopf bifurcation, the dynamics may be described by the CGL Equation (1). The analytic study of spiral waves in the framework of the CGL equation has been carried out by Hagan [8]. This author searched for solutions of Equation (1) by considering the following ansatz ... 

The reaction involving chlorite and iodide ions in the presence of malonic acid, the CIMA reaction, is another that supports oscillatory behaviour in a batch system (the chlorite-iodide reaction being a classic clock system the CIMA system also shows reaction-diffusion wave behaviour similar to the BZ reaction, see section A3.14.4). The initial reactants, chlorite and iodide are rapidly consumed, producing CIO2 and I2 which subsequently play the role of reactants . If the system is assembled from these species initially, we have the CDIMA reaction. The chemistry of this oscillator is driven by the following overall processes, with the empirical rate laws as given ... 

It is possible to show that when the different parts of a system are connected by nonlinear interactions, one can again obtain oscillation in concentrations, patterns of chemical substances in space, and wave propagation. These phenomena are important in some biological problems when the reaction-diffusion mechanisms cannot give an adequate description of the system. Morphogenetic fields and neural networks are examples of such systems. 

Chemical reactions with autocatalytic or thermal feedback can combine with the diffusive transport of molecules to create a striking set of spatial or temporal patterns. A reactor with permeable wall across which fresh reactants can diffuse in and products diffuse out is an open system and so can support multiple stationary states and sustained oscillations. The diffusion processes mean that the stationary-state concentrations will vary with position in the reactor, giving a profile , which may show distinct banding (Fig. 1.16). Similar patterns are also predicted in some circumstances in closed vessels if stirring ceases. Then the spatial dependence can develop spontaneously from an initially uniform state, but uniformity must always return eventually as the system approaches equilibrium. 

Kay, S. R. and Scott, S. K. (1988). Multiple stationary states, sustained oscillations and transient behaviour in autocatalytic reaction diffusion equations. Proc. R. Soc., A418, 345-64. 

Scott, S. K. (1987). Isolas, mushrooms and oscillations in isothermal, autocatalytic reaction-diffusion equations. Chem. Eng. Sci., 42, 307-15. 

In the various situations we have seen before, allowing a finite decay rate for the catalyst B has had significant results. The concentrations of A and B are then decoupled and this has allowed oscillations, isolas, and mushrooms. In the present case of reaction-diffusion waves, the uncoupling is again an important step upwards in complexity, sufficiently so as to prevent any completely general form of analysis. 

Saul, A. and Showalter, K. (1985). Propagating reaction-diffusion fronts. In Oscillations and traveling waves in chemical systems, (ed. R. J. Field and M. Burger), ch. 11, pp. 419-39. Wiley, New York. 

Several groups have developed cellular automata models for particular reaction-diffusion systems. In particular, the Belousov-Zhabotinsky oscillating reaction has been examined in a number of studies.84-86 Attention has also been directed at the A + B —> C reaction, using both lattice-gas models 87-90 and a generalized Margolus diffusion approach.91 We developed a simple, direct cellular automaton model92 for hard-sphere bimolecular chemical reactions of the form... 

Many biochemical signaling processes involve the coupled reaction diffusion of two or more substrates. Metabolic biochemical pathways are mainly multicomponent reaction cycles leading to binding and/or signaling and are coupled to the transport of substrates. A reaction-diffusion model can also describe the diffusion of certain proteins along the bacterium and their transfer between the cytoplasmic membrane and cytoplasm, and the generation of protein oscillation along the bacterium (Wood and Whitaker, 2000). 

Here we also assume that the reaction term does not depend explicitly on the spatial coordinate, therefore the dynamics of the medium is uniform in space. It is easy to see that the spatially uniform time-periodic oscillation is a trivial solution of the full reaction-diffusion-advection system, so the question is whether this uniform solution is stable to small non-uniform perturbations and more generally, if there are any persistent spatially non-uniform solutions in which the spatial structure does not decay in time. 

Typically, large scale transport in flows with a finite correlation length of the velocity field is diffusive with an effective diffusion coefficient Deff (Sect. 2.2.2). Therefore the coarse grained structure of the oscillatory reaction in this flow should be similar to a onedimensional oscillatory reaction-diffusion system, i.e. propagating waves and no synchronization of the local oscillations on large scales. 

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. 

Over the past several years there have been many experimental and theoretical studies aimed at developing a better understanding of pattern formation in reaction-diffusion systems. The focus of recent studies has been on more complex behavior away from the onset of instability. For some parameter values, spatiotemporal chaos may occur near the boundary between the Turing region and the region of homogeneous oscillations (Figure 12). 

The proofs of Theorems 10.2, 10.3, and 10.4 are found in [348]. Equation (10.17) is of particular interest. Near the Takens-Bogdanov point, the frequency of the limit-cycle oscillations along the line of Hopf bifurcations, a = 0, is given by >h = see above. On the line of saddle-node bifurcations we have Aj = 0. An equation like (10.17) is expected from simple dimensional arguments. The only intrinsic length scales in reaction-diffusion systems come from the diffusion coefficients. The inverse time is determined by the rate coefficients of the reaction kinetics. Thus (10.17) provides an estimate of the intrinsic length of the Turing pattern near a double-zero point ... 

Over the four hour time period, the force produced can be seen to follow the oscillating reaction very closely, indicating that the diffusion through the gel is still quite fast relative to the timescale of the oscillations. The maximum force produced by the gel is 0.012N. The slight upward drift in the data is due to an increase in mass caused by a small mismatch between the flow rates of the pumps used. 

In a recent numerical study (17), spontaneous chemomechanicaJ pulsations were achieved by coupling a mechanically responsive sphere of gel with a chemical reaction which cannot produce oscillations by itself. The toy model used in this study is based on a two variables model of a quadratic autocatalytic reaction described by the following reaction-diffusion system dufdt = -h V u dv/dt = (12/7)w i +... 

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