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Research Papers

# Different Types of Instabilities and Complex Dynamics in Reaction-Diffusion Systems With Fractional Derivatives

[+] Author and Article Information
Vasyl Gafiychuk

SGT Inc., 7701 Greenbelt Rd Suite 400, Greenbelt, MD, 20770; NASA Ames Research Center, Moffett Field, CA, 94035-1000

Bohdan Datsko1

Institute of Applied Problems of Mechanics and Mathematics, NAS of Ukraine, Naukova Street 3B, Lviv, 79053, Ukraineb_datsko@yahoo.com

1

Corresponding author.

J. Comput. Nonlinear Dynam 7(3), 031001 (Mar 19, 2012) (10 pages) doi:10.1115/1.4005923 History: Received August 16, 2009; Revised January 07, 2012; Published March 13, 2012; Online March 19, 2012

## Abstract

In this article we analyze conditions for different types of instabilities and complex dynamics that occur in nonlinear two-component fractional reaction-diffusion systems. It is shown that the stability of steady state solutions and their evolution are mainly determined by the eigenvalue spectrum of a linearized system and the fractional derivative order. The results of the linear stability analysis are confirmed by computer simulations of the FitzHugh-Nahumo-like model. On the basis of this model, it is demonstrated that the conditions of instability and the pattern formation dynamics in fractional activator- inhibitor systems are different from the standard ones. As a result, a richer and a more complicated spatiotemporal dynamics takes place in fractional reaction-diffusion systems. A common picture of nonlinear solutions in time-fractional reaction-diffusion systems and illustrative examples are presented. The results obtained in the article for homogeneous perturbation have also been of interest for dynamical systems described by fractional ordinary differential equations.

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## Figures

Figure 1

The schematic view of the marginal curve of α0 (solid line) and the parabola detF=tr2F/4 (pointed line) - (a). The position of eigenvalue λ corresponds to the marginal value of α0 in the coordinate system (Reλ,Imλ) - (b). Shaded domains correspond to the instability region.

Figure 2

Instability domains in coordinates (u¯1,τ1/τ2) for a fractional order reaction-diffusion system with sources W1=u1-u13-u2, W2=-u2+βu1+A for different values of α0=0.25,0.5,0.75,1.0,1.25,1.5,1.75. The results of computer simulation obtained at l1=l2=0 for: β=1.01 - (a), (b) and β=10.0 - (c), (d).

Figure 3

Instability domains and the eigenvalues (Reλ - black lines, Imλ - gray lines) for k=0,β=1.05 and different proportions of τ1/τ2=0.5-(i),1.0-(ii),2.0-(iii) - (a). The null-clines for β=2.1,A=-0.5 and eigenvalue spectrum for different values of k (k=0 - hair-lines, k=1 - dash lines, k=2 - thick lines) - (b). The eigenvalues are presented for the following parameters: l12/l22=0.025,β=2.1,τ1/τ2=0.1-(iv), l12/l22=0.1,β=1.01,τ1/τ2=0.6-(v), l12/l22=2.1,β=1.01,τ1/τ2=3.5-(vi).

Figure 4

The view of the surface T in coordinates (l1/l2,τ2/τ1) for β=2 and different values of u1 (u1=0.1 - (a), u1=5.0 - (b), u1=1.25 - (c), u1=1.5 - (d))

Figure 5

Dynamics of pattern formation for u1 variable. The results of computer simulations of the system at parameters: α=0.8, A=-0.25, β=2.1, l12=0.025, l22=1, τ1/τ2=0.1 - (a); α=0.8, A=-0.55, β=2.1, l12=0.025, l22=1, τ1/τ2=0.1 - (b); α=0.8, A=-0.4, β=2.1, l12=0.025, l22=1, τ1/τ2=0.1 - (c); α=0.8, A=-0.45, β=2.1, l12=0.025, l22=1, τ1/τ2=0.1 - (d); α=1.6, A=-0.01, β=1.05, l12=0.05, l22=1, τ1/τ2=1.45 - (e); α=0.7, A=-0.3, β=2.1, l12=0.05, l22=1, τ1/τ2=0.2 - (f).

Figure 6

Dynamics of pattern formation for u1 (left column) and u2 (right column) variables. The results of computer simulations of the systems at parameters: A=-0.01, α=1.8, β=1.01, l12=0.02, l22=1, τ1/τ2=3.5 - (a)-(b); A=-1.95, α=1.82, β=1.01, l12=0.1, l22=1, τ1/τ2=0.6 - (c)-(d); A=-0.01, α=1.75, β=10, l12=0.05, l22=1, τ1/τ2=0.05 - (e)-(f).

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