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.

References

References
1.
Agrawal
,
O. P.
,
Tenreiro Machado
,
J. A.
, and
Sabatier
,
J.
, 2007,
Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering
,
Elsevier
,
Amsterdam
.
2.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
, 2006,
Theory and Applications of Fractional Differential Equations
,
Elsevier
,
Amsterdam
.
3.
Arteshock, Uchaikin
,
V. V.
, 2008,
Fractional Derivative Method
,
Arteshock
,
Ulyanovsk
(in Russian).
4.
Henry
,
B. I.
, and
Langlands
,
T. A. M.
, 2006, “
Anomalous Diffusion with Linear Reaction Dynamics From Continuous Time Random Walks to Fractional Reaction-Diffusion Equations
,”
Phys. Rev. E
,
74
,
031116
.
5.
Gafiychuk
,
V.
, and
Datsko
,
B.
, 2007, “
Stability Analysis and Oscillatory Structures in Time- Fractional Reaction-Diffusion Systems
,”
Phys. Rev. E
,
75
,
055201
-
4
.
6.
Abad
,
E.
,
Yuste
,
S. B.
, and
Lindenberg
,
K.
, 2010, “
Reaction-Superdiffusion and Reaction- Superdiffusion Equations for Evanescent Particles Performing Continuous-Time Random Walks
,”
Phys. Rev. E.
,
81
,
031115
.
7.
Kochubei
,
A. N.
, 2011, “
Fractional-Parabolic Systems
,”
Potential Anal.
, (to be published).
8.
Haubold
,
H. J.
,
Mathai
,
A. M.
, and
Saxena
,
R. K.
, 2011, “
Further Solutions of Fractional Reaction-Diffusion Equations in Terms of the H-Function
,”
J. Comput. Appl. Math.
,
235
, pp.
1311
1316
.
9.
Mendez
,
V.
,
Fedotov
,
S.
, and
Horsthemke
,
W.
, 2010,
Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities
,
Springer
,
New York
.
10.
Weiss
,
M.
, and
Nilsson
,
T.
, 2004, “
In a Mirror Dimly: Tracing the Movements of Molecules in Living Cells
,”
Trends Cell Biol.
,
14
, pp.
267
273
.
11.
Zelenyi
,
L. M.
, and
Milovanov
,
A. V.
, 2004, “
Fractal Topology and Strange Kinetics: From Percolation Theory to Problems in Cosmic Electrodynamics
,”
Phys. Usp.
,
47
, pp.
809
852
.
12.
Nicolis
,
G.
, and
Prigogine
,
I.
, 1997,
Self-Organization in Non-equilibrium Systems
,
Wiley
,
New York
.
13.
Kerner
,
B. S.
, and
Osipov
,
V. V.
, 1994,
Autosolitons
,
Kluwer
,
Dordrecht
.
14.
Purwins
,
H.-G.
,
Bodeker
,
H. U.
, and
Amiranashvili
,
S.
, 2010, “
Dissipative Solitons
,”
Adv. Phys.
,
59
, pp.
485
701
.
15.
Adamatzky
,
A.
,
de Lacy Costello
,
B.
, and
Asai
,
T.
, 2005,
Reaction-Diffusion Computers
,
Elsevier
,
Amsterdam
.
16.
Henry
,
B. I.
, and
Wearne
,
S. L.
, 2000, “
Fractional Reaction-Diffusion
,”
Phys. A
,
276
, pp.
448
455
.
17.
Seki
,
K.
,
Wojcik
,
M.
, and
Tachiya
,
M.
, 2003, “
Fractional Reaction-Diffusion Equation
,”
J. Chem. Phys.
,
119
,
2165
.
18.
Langlands
,
T.
,
Henry
,
B. I.
, and
Wearne
,
S. L.
, 2007, “
Turing Pattern Formation with Fractional Diffusion and Fractional Reactions
,”
J. Phys. Condens. Matter
,
19
,
065115
.
19.
Gafiychuk
,
V.
,
Datsko
,
B.
,
Meleshko
,
V.
, and
Blackmore
,
D.
, 2009, “
Analysis of the Solutions of Coupled Nonlinear Fractional Reaction-Diffusion Equations
,”
Chaos, Solitons Fractals
,
41
, pp.
1095
1104
.
20.
Gafiychuk
,
V.
,
Datsko
,
B.
, and
Meleshko
,
V.
, 2008, “
Mathematical Modeling of Time Fractional Reaction-Diffusion Systems
,”
J. Comp. Appl. Math.
,
220
, pp.
215
225
.
21.
Podlubny
,
I.
, 1999,
Fractional Differential Equations
,
Academic Press
,
New York
.
22.
Pierre
,
M.
, 2010, “
Global Existence in Reaction-Diffusion Systems with Control of Mass: a Survey
,”
Milan J. Math.
,
78
, pp.
417
455
.
23.
Nicolis
,
G.
, and
Prigogine
,
I.
, 1989,
Exploring Complexity: An Introduction
,
Freeman & Co
,
New York
.
24.
Matignon
,
D.
, 1996, “
Stability Results for Fractional Differential Equations with Applications to Control Processing
,”
Comput. Eng. Syst. Appl.
,
2
, pp.
963
970
.
25.
Lubashevsky
,
I.
, and
Gafiychuk
,
V.
, 1994, “
Projection Dynamics of Highly Dissipative Systems
,”
Phys. Rev. E.
,
50
, pp.
171
181
.
26.
Poschl
,
G.
, and
Teller
,
E.
, 1933, “
Bemerkungen zur Quantenmechanik des anharmonischen Oszillators
,”
Z. Phys.
,
83
, pp.
143
151
.
27.
Sattinger
,
D. H.
, 1973,
Topics in Stability and Bifurcation Theory
,
Springer
,
New York
.
You do not currently have access to this content.