Turing Structures in the CIMA-Reaction
The two species reaction-diffusion model developed by Lengyel and Epstein represents a key to understanding the recently observed Turing structures in the chlorite-iodide-malonic acid (or CIMA) reaction. We have studied the emergence, growth, competition and final stabilization of various concentration patterns through numerical simulations of this model. Using the supply of iodine as a bifurcation parameter, the regularity of the hexagonal patterns that develop from the noise inflected homogeneous steady state has been examined. In the region where they are both stable, the competition of Hopf oscillations and Turing stripes has been studied by following the propagation of a front connecting the two modes. Finally, examples were obtained for the types of structure that can develop when a gradient in feed concentration is applied to the system.
(P. Borckmans, G. Dewel), O. Jensen and E. Mosekilde
Wave-Splitting in the Gray-Scott Model
The Gray-Scott model describes a chemical reaction in which an activator grows autocatalytically on a continuously fed substrate. For certain feed rates and activator life times the model allows for the coexistence of two different homogeneous steady states, the red state in which the conditions correspond to those of the pure substrate, and the blue state where the activator concentration is relatively high and limited by substrate depletion. The blue state may again undergo a Hopf bifurcation and a subcritical Turing bifurcation. Interactions between these instabilities can lead to a variety of different phenomena including Turing-Hopf mixed modes and spatio-temporal chaos. We have studies some of these phenomena through numerical simulation. Special emphasis was given to the propagation, collision, and splitting of traveling pulses that can develop in response to strong local perturbations of the red state.
(G. Dewel, P. Borckmans), W. Mazin, E. Mosekilde and K.E. Rasmussen
Generalized Amplitude Equations
Chemical reactions taking place in spatially extended media can develop waves in which the concentrations of the reacting species vary in time and space. Such system can be described by a reaction-diffusion equation, a partial differential equation in the concentration vector. However, the dimension of the concentration space is often high, and the numerical solution of a chemical reaction-diffusion equation can be quite time consuming (even impossible). Close to a Hopf bifurcation, the equation can be reduced to a scalar equation in a complex amplitude of local oscillations with parameters that can be determined experimentally by methods developed by the research group, namely the complex Ginzburg-Landau equation.
The validity of the complex Ginzburg-Landau equation has been tested. It may fail in practice for chemical systems with slow reactions. We have developed a method of deriving generalized amplitude equations using a normal form approach working from more general bifurcations. The method has been used to establish a generalized Ginzburg-Landau equation for oscillating systems with a slow transient motion toward the plane of oscillations. Numerical comparisons with the corresponding reaction-diffusion equation show that the new equation provides a much better description of chemical waves and chemical turbulence for the Belousov-Zhabotinsky reaction than does the Ginzburg-Landau equation.
M. Ipsen, F. Hynne and P.G. Sørensen
Glycolysis is a very basic part of the energy metabolism taking place in almost every living cell. We are conducting glycolysis in the laboratory on extracts from yeast cells. We have developed an efficient method of producing the extract and found experimental conditions where the oscillations can be maintained in a flow reactor for a very long time. We have confirmed the chaotic oscillations observed previously. With a view to analyse the biological system by quenching experiments, we are searching for Hopf bifurcations and by simulation of models of the reaction.
P.G. Sørensen, F. Hynne, K. Nielsen, M. Hoffmann and M. Smrcinova
Universality in Two Classes of Reaction-Diffusion Systems
We are studying two types of reaction-diffusion systems. The first is the annihilating/scattering process , , , and . The second is the annihilation-fission system with and . We have shown that both these systems exhibit both real and imaginary noise in a Langevin-type description. The presence of real noise has recently been suggested (by Grinstein et al.) to give rise to new non-trivial behaviour in active-absorbing transitions (as in, for example, auto-catalytic chemical reactions). However, despite this suggestion, we have demonstrated that both of our systems lie in the same universality class as - a highly non-trivial result - and hence fail to exhibit the proposed novel behaviour. We are investigating the implications of this result for the model of Grinstein et al.
M.J. Howard and (U.C. Taüber)
Vortex Dynamics in Dissipative Systems
We derive the exact equation of motion for a vortex in the two- and three-dimensional complex Ginzburg-Landau equation. The velocity is given in terms of local gradients of the magnitude and phase of the complex field, for arbitrarily small inter-vortex distances. The results for vortices in a superfluid or a superconductor are recovered as special cases.
(O. Törnkvist) and E. Schröder