Reaction-Diffusion (RD) models represent the chemical kinetic and transport aspect of molecules. These systems model the vast majority of chemical processes observed in nature. Experiments and computer simulations for RD systems show that pattern formation can lead to a rich variety of behaviors. One of the most interesting phenomena discovered in RD systems is self-replication, which has been observed in experiments as well as in computer simulations. It is well known that bi-stable RD systems which possess two locally stable solutions and one unstable solution can show< localized structures. Localized solutions are observed for systems with two stable fixed points and one unstable fixed point or one stable fixed point, a stable limit cycle, and one unstable limit cycle. Recently we have proposed a reaction-diffusion model which shows simultaneously a stable fixed point and a stable limit cycle. We have been able to obtain approximate analytical expressions for oscillating pulses in this RD model capturing all the essential ingredients of these breathing particle-like solutions (the figure hows a particle as a function of space and time). We have also observed, numerically, the coexistence of stable particle and hole solutions in this simple model.

We shall be interested in the following research topics:

(a) Complete phase diagram of the model.
(b) Coexistence of localized structures in the parameter space.
(c) Analytical approach to different kinds of oscillating holes.
(d) Numerical study of moving holes in the model.

People involved:

The research will be done by O. Descalzi (analytical work) and J. Cisternas (numerical work) with the collaboration of H. Brand (Bayreuth, Germany).

References:

[1] O. Descalzi, Y. Hayase and H.R. Brand, Analytical approach to localized structures in a simple reaction-diffusion system, Physical Review E 69, 026121 (2004).


[2] Y. Hayase, O. Descalzi and H.R. Brand, Coexistence of stable particle and hole solutions for fixed parameters for a simple reaction-diffusion system, Physical Review E 69, 065201(R) (2004).

[3] O. Descalzi, Y. Hayase and H.R. Brand, Oscillating localized structures in reaction-diffusion systems, International Journal of Bifurcation and Chaos 14, 4097-4104 (2004). Cover of the journal.

[4] Y. Hayase, O. Descalzi, and H.R. Brand, A simple two-component reaction-diffusion system showing rich dynamic behaviour: spatially homogeneous aspects and selected bifurcation scenarios, Physica A 356, 19-24 (2005).