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


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

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.

Copyright © 2012 by American Society of Mechanical Engineers
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Grahic Jump Location
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.

Grahic Jump Location
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).

Grahic Jump Location
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).

Grahic Jump Location
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))

Grahic Jump Location
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).

Grahic Jump Location
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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