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 International Conference on Complex Systems (ICCS2007)

Classification of Instabilities and Pattern Formation in Fractional Reaction-Diffusion Systems

Vasyl Gafiychuk
Institute of APMM, National Academy of Sciences of Ukraine

Bohdan Datsko
Institute of APMM National Academy of Sciences of Ukraine

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     Last modified: October 22, 2007

The key models that have been used in the study of self-organization phenomena, not only in physics, chemistry, or biology, but also in nonlinear science, are the phenomenological models like the Brusselator and Oregonator, the model of Gierer and Meinhardt, the Bonhoeffer–van der Pol model, etc. Recent studies of fractional reaction diffusion (FRD) models demonstrated the possibility of emerging the new phenomena in these systems [1–4]. The analysis of the structures in FRD systems described by these models evolves both from the standpoint of the qualitative analysis and from the computer simulation. Namely, these two problems are the goal of our present investigation.

Linear stage of the two component fractional reaction-diffusion system stability is studied. All limits of system stability are investigated. It is shown that by certain value of the fractional derivatives index the system becomes unstable towards perturbations of finite wave number for a certain value of fractional derivative index. As a result, inhomogeneous oscillations with this wave number become unstable and lead to nonlinear oscillations which result in spatial oscillatory structure formation [4]. Computer simulation of the FRD systems for the models of Brusselator, Oregonator, Gierer and Meinhardt and the Bonhoeffer–van der Pol model are performed. On the basis of qualitative analysis resulting pattern formations are investigated and particular phenomena are discussed.

1. V.Gafiychuk, B.Datsko, Physica A. vol. 365, 300–306, 2006.
2. V.Gafiychuk, B.Datsko, V.Meleshko. nlin.AO/0611005.
3. V.Gafiychuk, B.Datsko, and V.Meleshko, nlin.PS/0702013.
4. V.Gafiychuk, B.Datsko, Physical Review E 75, 055201-1, 2007.

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