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Vasyl GafiychukSGT 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, Ukraine

e-mail: [email protected]

Different Types of Instabilitiesand Complex Dynamics inReaction-Diffusion Systems With

Fractional DerivativesIn this article we analyze conditions for different types of instabilities and complexdynamics that occur in nonlinear two-component fractional reaction-diffusion systems. Itis shown that the stability of steady state solutions and their evolution are mainly deter-mined by the eigenvalue spectrum of a linearized system and the fractional derivativeorder. The results of the linear stability analysis are confirmed by computer simulationsof the FitzHugh-Nahumo-like model. On the basis of this model, it is demonstrated thatthe 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 amore complicated spatiotemporal dynamics takes place in fractional reaction-diffusionsystems. A common picture of nonlinear solutions in time-fractional reaction-diffusionsystems and illustrative examples are presented. The results obtained in the article forhomogeneous perturbation have also been of interest for dynamical systems described byfractional ordinary differential equations. [DOI: 10.1115/1.4005923]

1 Introduction

In recent years, there has been an increasing interest in the studyof dynamical mathematical models with fractional derivatives. Thisinterest is mainly determined by the attempts to understand phe-nomena in fractal, irregular and hereditary media. The last investi-gations have shown that anomalous behavior of many complexheterogeneous systems is better described by fractional differentialequations (see for example Refs. [13] and references therein).

Among the fractional differential models, much attention hasbeen given to the fractional reaction-diffusion systems (RDS)[49]. At the present time, fractional RDS (FRDS) describe a

large class of systems at different scales from the molecular [10]to the space ones [11]. Due to this fact, the development of thetheory of such systems is important both from the scientific per-spective and for its application in a set of new technologies. Onthe basis of mathematical modeling of the standard reaction-diffusion equations, a lot of amazing nonlinear self-organizationphenomena in physical, biological and chemical systems havebeen explained [1214]. Moreover, the investigations of thespatio-temporal order in such nonlinear systems and mechanismsof pattern formation are a top-priority theme of present researchstudies in many modern technological applications [14,15]. There-fore, the development of fractional models for such media, asgranular and porous materials, various materials with memory,complex chemical environments and biological tissues, whereself-organization phenomena are observed and diffusion hasessentially an anomalous character, represents a special interest.

A fractional reaction-diffusion equation is derived from a con-tinuous time random walk model when the transport is dispersive[16,17]. In Ref. [4], a general multi-species system undergoinganomalous sub-diffusion with linear reaction dynamics is consid-ered. The validity of such fractional reaction-diffusion (RD)model by comparing solutions with Monte Carlo simulationswas confirmed. The extension to nonlinear reaction terms isnon-trivial. Several models with temporal order derivatives oper-ating on both the Laplacian diffusion term and nonlinear reaction

kinetics from the law of mass action have been considered[18,19]. The results for front propagation in such models are alsoin reasonable agreement with Monte Carlo simulations [18].These papers provide a useful platform for developing robustmodels for multispecies fractional systems with nonlinear reac-tions (the overview of different platforms for modeling reaction-transport systems, including the ones with anomalous diffusion, ispresented in Ref. [9]).

For commensurate nonlinear time-fractional activator-inhibitorRDS, the basic results are obtained for classical reaction kinetics.In a series of articles [1820], through linear stability analysis andnumerical simulations, a stationary Turing pattern formation is

investigated. In Refs. [5,20] it is shown that the fractional deriva-tive index is an additional bifurcation parameter which switches thestable and unstable states of the system. Furthermore, in Ref. [5] itwas revealed that in FRDS, for a certain value of fractional deriva-tive index, a new type of bifurcation takes place and the system canbe unstable towards perturbations of finite wave number. In thesearticles, the limited cases are primarily investigated, when theFRDS demonstrate relatively simple nonlinear dynamics.

In the present article, we have focused on FRD model under con-ditions, when sufficiently complex spatial-temporal patterns arisein the system dynamics. By linear stability analysis and computersimulation, it was shown that fractional derivative order can changethe stability of steady state solutions and significantly enrich non-linear system dynamics. The diversity of observed nonlinear phe-nomena needs some classification in order for us to understand the

conditions for different pattern formation in such systems. In thestandard reaction-diffusion (RD) systems, such a classification hasbeen done a long time ago [13]. In our paper, we would like tomake the first attempt for such classification for fractionalRDS. Onthe basis of the time-fractional system with cubic nonlinearity, theconditions for different types of instability in detail are analyzedand an overall picture of different types of nonlinear solutions,depending on parameters of the system, is presented.

2 Formulation of the Problem

In a general case, a reaction-diffusion model can be describedby a system of m nonlinear partial differential equations of para-bolic type

1Corresponding author.Manuscript received August 16, 2009; final manuscript received January 7, 2012;

published online March 13, 2012. Assoc. Editor: Om Parkash Agrawal.

Journal of Computational and Nonlinear Dynamics JULY 2012, Vol. 7 / 031001-1CopyrightVC 2012 by ASME

Downloaded 21 Dec 2012 to 216.91.96.130. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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sut DDu fu; A (1)

in space-time domain XW subject to certain boundaryconditions

l@u

@n 1 lu on @XW (2)

and initial conditions

ur; 0 u0r for r 2 X (3)Here, X is a bounded space domain in Rp, p 2 f1; 2; 3g withsmooth boundary @X and W 0; T defines a time domain with0 < T < 1. Vector u u1r; t; ; umr; tT specifies thevariables of the model, positive diagonal matrices smm and Dmmrepresent the time and space scales of the system, respectively,A 2 R R is an external bifurcation parameter and vectorf f1; f2;:::; fmT defines the given smooth reaction-kineticsfunctions fiu1r; t; u2r; t;:::; umr; t : Rm !Rm Vector 1; ; m, diagonal matrices l diagl1; ; lm,1 diage1; ; em with ei 1, li 2 0; 1, i 2 C2@XWfor all 2 1; ; m and normal vector n to the boundary @X deter-mine the boundary conditions of the problem.

The starting point of our consideration is the fractional RDS

suat DDu fu; A (4)

Time derivatives uat @au1r; t

@ta ; ;@aumr; t

@ta

Ton the left-hand

side of the Eq. (4) instead of the standard time derivatives are theCaputo fractional derivatives in time [2,3,21] of the order0 < a < 2 and are represented as

@ac uir; t@ta

: 1Cn a

t0

uni r; s

t sa1n ds (5)

where n 1 < a < n; n 2 f1; 2g, i 2 f1; 2; ; mg, a 2 R.Introduction of time fractional derivatives into differential

equations appreciably expands the family of dynamical models. Inparticular, the time-fractional reaction-diffusion system sets a pos-sibility of continuous transition between the parabolic, elliptic andhyperbolical types of partial derivative equations, and we can con-sider the fractional RDS as a generalized system which is a matterof considerable theoretical interest [7,8].

For the study of the main characteristics of nonlinear dynamics,the common RD model Eq. (1) in the simplest but quite generalcase can be reduced to a system of two coupled one- dimensionalnonlinear equations of the activator-inhibitor type [12,13]:

s1@ac u1x; t

@ta l21

@2u1x; t@x2

Wu1; u2; A (6)

s2@ac u2x; t

@ta l22

@2u2x; t@x2

Qu1; u2; A (7)

Here u1x; t, u2x; t are the activator and inhibitor variables,0 x L, s1; s2; l1; l2, as defined above, are the characteristictimes and lengths of the system, correspondingly, and A is anexternal parameter. In the integer case a 1, the system (Eqs.(6) and (7)) is the basis model for investigation of self-organization phenomena in non-equilibrium media of different na-ture [1214]. The positive feedback on the activator variable u1,negative feedback on the inhibitor variable u2 and the differencein characteristic times and lengths lead to rich variety of qualita-tively different solutions [12,13]. In the present article we investi-gate the solutions of the system (Eqs. (6) and (7)) in a moregeneral case 0 < a < 2 and demonstrate the role of the frac-tional derivative order on their stability and evolution.

It should be noted, that the question of global existence ofa solution for nonlinear RD systems even in the case of a two-component ones with an integer derivative is an unresolvedproblem. The review of the main results on global existence ofsolutions in time for the family of m m RDS is well repre-sented in a survey [22]. These RDS must satisfy the two mainproperties: (a) the non-negativity of the solutions is preserved forall time, and (b) the total mass of the components is a prioribounded on all finite intervals. A lot of systems come naturallywith these two properties in applications. In our article, we

investigate RDS, which with a certain transformation can bereduced to these properties. It is classical that for such type ofsystem a local solution exists and may be extended on interval0; T. Moreover, for the system (Eq. (1)), the existence of spa-tially homogeneous and stationary solutions, which can beobtained from algebraic system fu; A 0 and realized at peri-odic or neutral boundary conditions, is suggested. If the algebraicsystem is consistent, it determines a set of solutions dependingon the bifurcation parameter. In this article, we consider that thealgebraic system has only one solution and in the next subsectionwe will study the stability of this stationary solution for general-ized system (Eq. (4)).

3 Linear Stability Analysis

Lets first consider the formulation which allows us to deter-

mine the stability of such stationary solutions. The system(Eq. (4)) can be presented in operator form

uat Nu; A (8)

where

Nu; A s1DDu fu; A (9)

Due to properties of the Caputo derivative, the stationary solutionur is a particular solution of the Eq. (8). The stability of thisstationary solution, as in the case of integer time derivative, canbe estimated on the basis of the principle of linear stability, whichfor fractional derivative evolutionary equation can be formulatedas:

Theorem 1. Lets assume ur is the stationary solution ofsystem Eq. (8) and ~ur; t ur; t ur is the vector of devia-tions from ur: j ~u j=j u j

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Wu1; u2; A 0; Qu1; u2; A 0 (11)

A simultaneous solution of the Eq. (11) leads to a spatially-homogeneous solution u u1; u2, which is realized at periodic

uijx0 uijL; @ui=@xjx0 @ui=@xjxL; i 1; 2 (12)

or neutral boundary conditions

@ui=@x

jx

0

@ui=@x

jx

L 0; i

1; 2 (13)

The algebraic system (Eq. (11)) can have a set of real roots, butwe consider the case when the system (Eqs. (6) and (7)) has onlyone stationary and spatially homogeneous solutions. The stabilityof such steady-state solution of the system (Eqs. (6) and (7)) cor-responding to the homogeneous equilibrium state Wu; A 0,Qu; A 0 can be analyzed by linearization of the systemnearby this solution u u1; u2T. As result, we can write theexpression for eigenvalues of the linearized system [9,12,13]

k1; 2 12trF6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitr2F 4 detF

p (14)

which mainly defines the stability and nonlinear dynamics of thesystem (Eqs. (6) and (7)). Here, the matrix

Fk Lju1; u2 a11 k2l21=s1 a12=s1

a21=s2 a22 k2l22=s2

(15)

is determined by a11 W0u1 , a12 W0u2

, a21 Q0u1 , a22 Q0u2

(all derivatives are taken at homogeneous equilibriumstates u u1; u2T, trFk a11 k2l21=s1 a22 k2l22=s2,detFk a11 k2l21a22 k2l22=s1s2 a12a21=s1s2, kpj=L, j 1; 2; ..

Fora 1, this stationary solution is unstable when the real partof any eigenvalue Rek > 0.. In reaction-diffusion systems, a stablehomogeneous equilibrium solution ux const usually changesspontaneously with the external parameters to the limit cycle byHopf bifurcation or stationary dissipative structures (DS) byTuring bifurcation. As a result, we obtain nonlinear dynamics

leading to spatially-homogeneous oscillatory solutions orspatially-inhomogeneous stationary DS. When conditions of bothinstabilities arise, we can expect more complex dynamics. In thecase of a fractional reaction-diffusion (FRD) system, the dynamicscan be much more complex [5,19,20].

For a 6 1 we can also apply an analogous procedure to thesystem (Eqs. (6) and (7)). Due to the property of Caputo deriva-tive [3,21], we have a similar linear system with the same right-hand side operator (Eq. (15)). The stability of this fractional linearsystem is determined by the next theorem of Matignon [24]:

Theorem 2. The linear autonomous system:

dautdta

: Au; u0 u0 (16)

with 0 < a < 1; u 2 Rn

and A 2 Rn

n

is asymptotically stable ifand only if

a 0

2 2p

arctanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4detF=tr2F 1p

; trF < 0

8>: (18)

In this case, the value of a is a certain additional bifurcationparameter which switches the stable and unstable states of the sys-tem (Figs. 1(a) and 1(b)). At a lowera :a < a0 2pjArgkij, thesystem has oscillatory modes, but they are stable. Increasing thevalue ofa > a0 2pjArgkij leads to oscillatory instability.

It is widely known for integer time derivatives [12,13] that thesystem (Eqs. (6) and (7)) becomes unstable according to eitherHopf (k 0)

trF > 0; detF0 > 0 (19)

or Turing (k0 6 0) bifurcationstrF < 0; detF0 > 0; detFk0 < 0 (20)

and these both types of instabilities are realized for positive feed-back a11 > 0) [12,13].

For fractional system, Hopf bifurcation is not connected withthe condition a11 > 0 and can take a place for a11 < 0 at a certainvalues of a [20]. Moreover, in fractional RD systems at a > 1when it is easier to satisfy conditions of Hopf bifurcation, wemeet a new type of instability [5]

trF < 0; 4detF0 < tr2F0; 4detFk0 > tr2Fk0 (21)

It is worth analyzing inequalities (Eq. (21)) in detail. Taking intoaccount the explicit form of Fk, the last two conditions can berewritten as:

a11s1 a22s22 > 4a12a21s1s2 (22)

4a12a21s1s2 > a11 k2l21s2 a22 k2l22s1 2 (23)The simplest way to satisfy the last condition is to estimate theoptimal value ofk k0:

k0 2 a12a21l21=s2 l22=s1

1=2(24)

Having obtained Eq. (24), we can estimate the marginal valueofa0

Fig. 1 The schematic view of the marginal curve of a0 (solidline) and the parabola detF tr2F=4 (pointed line) - (a). Theposition of eigenvalue k corresponds to the marginal value of a0in the coordinate system (Rek; Imk) - (b). Shaded domains cor-respond to the instability region.

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axis, the system is unstable with a wave numbers k 0, and out-side - it is stable. The left-hand side plots correspond toa < 1 andthe right- hand side plots corresponds to a > 1. Figures 2(a) and2(c) demonstrate that at a < 1 an increase of relation s1=s2 leadsto decrease of the instability domain, which at s1=s2 1 anda 1 vanishes completely. The situation changes fora > 1. Thesystem is unstable not only fors1=s2 < 1, but also for s1=s2 > 1.An increase in a makes the instability domain much wider withrespect to two coordinates (u1; s1=s2) and we obtain butterfly-likedomains fora > 1. This means that, with increasing s1=s2 (mov-

ing along the vertical axis), the system becomes stable in the cen-ter and unstable at the greater values ju1j. In such a case, theinstability domain becomes symmetric along the vertical axis withthe minimum point at u1 0. Figures 2(a), 2(b), 2(c) and 2(d)present the plots for different values ofb and show the same trendwith respect to a, but the region for a > 1 is much greater forlarger values ofb. At the same time, fora < 1 and large values ofb the instability domain shrinks very sharply in comparison to thesmall values ofb.

It is possible to understand the mechanism of instability fromthe plot of eigenvalue spectrum. Typical instability domains forthe same parameters as on Figs. 2(a) and 2(b) for k 0 are pre-sented on the Fig. 3(a). The horizontal lines i, ii, iii on the upperplot represent typical relations between s1 and s2, when in the sys-tem, depending on the value ofa, qualitatively different instabilityconditions can be realized. Eigenvalue spectrums for these typical

relations are presented on plots (i), (ii), (iii), correspondingly. Letus analyze each of the possible situations in more detail.

Case (i): full type instability domain.In this case, the instability domain consists of three sub-

domains with real and complex eigenvalues. The easiest way of

obtaining instability is realized at ju1j < uE1 when all eigenvaluesof the linearized system are real and positive (Fig. 3(a), (i)). Thisregion is represented by dark gray color. Positive eigenvaluesmean that the system is unstable practically for any value ofa > 0. Inside the domain juE1 j < ju1j < juC1 j the Hopf bifurcationtakes place for certain values ofa from the interval 0 < a < 2.Point D divides this region into two sub-domains whereRek > 0 (gray color) and Rek < 0 (light gray color). In thesub-domain Rek < 0 the system can be unstable for certain val-ues of a > 1, which can be determined by the condition

a > a0 2

pjArgkj. In turn, for Rek > 0,, the system can be sta-ble for a < 1 by the same reasoning. In other words, betweenpoints C and E we have eigenvalues with an imaginary part, andthe value ofa can change the stability of the FRD system. In thedomain ju1j > juC1 j the eigenvalues are pure real and negative and;as a result, the system is stable.

Case (ii): continuous instability domain with complexeigenvalues.

For system parameters corresponding to the case (ii) the realpart of eigenvalues becomes less than zero for all values u1. Atthe same time, for ju1j < juK1 j the roots are complex and accordingto the condition a > a0 2pjArgkj instability takes place fora > a0 > 1. For ju1j > juK1 j, the eigenvalues become real andnegative, and the system is stable.

Case (iii): separated instability domain with complexeigenvalues.

In this case at the central part of the plot (ju1j < juG1 j) theeigenvalues are pure real and negative, and as a results the systemis stable. ForjuG1 j < ju1j < juH1 j eigenvalues are complex with neg-ative real parts and like in the case (ii) instability takes place fora > a0 > 1. In other words, the instability domain consists of twosymmetrical regions of instability, separated by a stable region atthe center where the system is stable for any a. For ju1j > uH1eigenvalues are real and negative again and the system is stable.

The presented analysis for homogeneous perturbation with thewave number k 0 in a fractional RD system corresponds to aninvestigation of a point system with l1 0; l2 0. Because ofthis the same situations in the fractional systems of ordinary dif-ferential equations can be realized.

4.1.2 Turing Instability. Conditions for Turing bifurcation

(k6 0) can be analyzed on the basis of nonhomogeneous modes ineigenvalue spectrum of the system. Eigenvalues for different val-ues ofkare presented in Fig. 3(b). The top plot represents the null-clines of the system and determines the parameters b and A forplot on Fig. 3(b), (iv). In Fig. 3(b), eigenvalues for k 1 andk 2 and for comparison for k 0 are presented. It can be seenfrom the picture (iv) that at intersection of null-clines in the neigh-borhood of zero values of u1 nonhomogeneous modes have muchgreater values and we can expect a formation of stationary dissipa-tive structures. If the ratio l1=l2 is sufficiently small, Turing bifur-cation is dominant for all region ju1j < 1. Analyzing conditions,(Eq. (20)) we can conclude that they are practically the same forfractional and standard RDS. However, what is very important isthat the transient dynamics and the role of other modes in frac-tional systems can be principally different. For this reason the final

attractors in the fractional system can be also be different eventhough the linear conditions of instability look the same.The typical situation is presented in Fig. 3(b), (iv). Lets assume

the system parameters are close to the ones represented by point Pin the top plot in Fig. 3(b). For a given value Imk=Rek, we canexpect the next scenario of instability for homogeneous solutioncorresponding to null-cline intersection in point P. The mode withk 2 (Fig. 3(b), (iv)) is dominant and the stationary dissipativestructures with this wave number appear in the system for a widerange of values of 0 < a < 2. Decreasing the value ofa in point Por moving the null-cline intersection to the coordinate centerenhances this tendency. In turn, increasing the value ofa or mov-ing the null-cline intersection from the coordinate center leads toactivation of other modes, including k 0 and stimulates

Fig. 3 Instability domains and the eigenvalues (Rek - blacklines, Imk - gray lines) for k 0; b 1:05 and differentproportions of s1=s2 0:5 i; 1:0 ii; 2:0 iii - (a). Thenull-clines for b 2:1; A 0:5 and eigenvalue spectrum fordifferent values of k (k 0 - hair-lines, k 1 - dash lines, k 2- thick lines) - (b). The eigenvalues are presented for the follow-ing parameters: l21 =l

22 0:025; b 2:1; s1=s2 0:1 iv,

l21 =l22 0:1; b 1:01; s1=s2 0:6 v, l21 =l22 2:1; b 1:01; s1=s2 3:5 vi.



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oscillatory instabilities. Such a trend is quite general and bychanging the intersection point of null-clines or value of a we canstimulate a different scenario of pattern formation. If the influenceof eigenvalues for k 0 and k6 0 is comparable we can expectcomplex spatio-temporal dynamics.

4.1.3 Nonhomogeneous Hopf Instability. Above, we haveconsidered situations when a homogeneous solution is unstablefor Hopf or Turing bifurcations, which is typical for classicalRDS. In this subsection, the conditions for another type of insta-bility nonhomogeneous Hopf bifurcation, which takes placeonly in fractional RDS, will be demonstrated. The eigenvaluespectrum for such instability is presented in Fig. 3(b), (v, vi).From the plot we can see that outside the small domain in the cen-ter the system is stable fork 0. At the same time, the spectrumof the linearized system has the complex eigenvalues for k6 0 inwide range of null-cline intersections (Fig. 3(b), (v)). Moreover,at l1=l2 < 1 the plot consists of a separate domain for k 2 whereeigenvalues for other modes are pure real and negative. In thisdomain at some critical values a0 inhomogeneous oscillationswith this wave number can be expected.

Figure 3(b), (vi) presents the situation, where eigenvalues havenegative real part for all modes. In this case, standard RDS isstable for any values of parameter ju1j > 1. In turn, for fractionalRD system we can satisfy conditions for nonhomogeneous Hopfbifurcation (Eq. (21)) even for s1=s2 > 1 and l1=l2 > 1. This

situation can be predicted from a symmetrical view of expression(26) for the system under consideration

T 2ffiffiffib

p= 1 u21

l21s2 l22s1l21s2 l22s1

1 u21 s2=s1 1

!(33)

The plot of these surfaces, as a function of l1=l2 and s2=s1, for dif-ferent values ofu1 is presented in Fig. 4(a)4(d). The surfaces onFigs. 4(a) and 4(b). demonstrate that at small and large values ofu

1, the maximum of T is reached at the boundaries, where

l1=l2

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nonhomogeneous Hopf bifurcation at s1=s2 > 1 and l1=l2 > 1 canbe realized. In the central part of this plot we can see only pure realand negative eigenvalues for k 0. At the same time for k6 0eigenvalues are complex. Thus, the conditions for Hopf bifurcationcan be realized only for a nonhomogeneous wave number. As aresult, perturbations with k 0 relax to the homogenous state, andonly the perturbations with k 1 or k > 1 become unstable andthe system exhibits inhomogeneous oscillations.

4.2 Space Nonhomogeneous Solutions. Lets assume the

homogeneous distribution of the variables u1; u2 becomes unsta-ble due to Turing bifurcation according to wavelength

l1l21=2 $ L and the size of the domain L satisfies the inequal-ities l1 ( L( l2. Then the system under consideration has thestationary nonhomogeneous solution, which forA $ 0 and peri-odic or neutral boundary conditions can be represented as [13,25]

u1x % tanh L=2 xffiffiffiffiffiffiffiffiffi2l1l2

p !

; u2 % const; jj ( 1 (34)

or [12]

u1x%a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAAcp cosmpx=L; u2x%b ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAAcp cosmpx=L(35)

Fig. 5 Dynamics of pattern formation for u1 variable. The results of computer simulations of the system at parameters:a 0:8, A 0:25, b 2:1, l21 0:025, l22 1, s1=s2 0:1 - (a); a 0:8, A 0:55, b 2:1, l21 0:025, l22 1, s1=s2 0:1 - (b);a 0:8, A 0:4, b 2:1, l21 0:025, l22 1, s1=s2 0:1 - (c); a 0:8, A 0:45, b 2:1, l21 0:025, l22 1, s1=s2 0:1 - (d);a 1:6, A 0:01, b 1:05, l21 0:05, l22 1, s1=s2 1:45 - (e); a 0:7, A 0:3, b 2:1, l21 0:05, l22 1, s1=s2 0:2 - (f).



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where Ac corresponds to a critical value of bifurcation parameterA, a; b are constants, m 1; 2; 3::., mp=L $ l1l21=2. The lin-ear stability of the solutions Eqs. (34) or (35) can be determinedby Frechet operator which in this case is presented in the follow-ing form

L L11 L12L21 L22

" # l

21r2 1 u1x2 1

1 l22r2 1

" #(36)

For a standard system (a

1) fors1

s2 the system is asymptoti-cally stable and small perturbations d! 0. That means thatall eigenvalues of the Frechet derivatives have Rek < 0. For the

solution (Eq. (34)) operator L11 is a Schrodinger operator withPoschl-Teller potential [26], eigenfunctions of which are knownand the maximum eigenvalue is greater than zero. The stabilitycondition of the whole Frechet derivative L is due to the dampingproperties of second variable which lowers the maximum eigen-value. For the solution (Eq. (35)) operators L11 and L22 form twocoupled Mathieu equations. These equations have the real parts ofall eigenvalues less than zero if the bifurcation is supercritical andat least one real part of eigenvalues greater than zero if bifurcationis subcritical [27]. In general, finding the spectrum of the operator

(Eq. (36)) is a sufficiently complicated mathematical problem,which in special cases can be solved by using approximate meth-ods [13]. For the system with a 6 1 the stability conditions are

Fig. 6 Dynamics of pattern formation for u1 (left column) and u2 (right column) variables. The results of computer simula-tions of the systems at parameters: A 0:01, a 1:8, b 1:01, l21 0:02, l22 1, s1=s2 3:5 - (a)-(b); A 1:95, a 1:82,b 1:01, l21 0:1, l22 1, s1=s2 0:6 - (c)-(d); A 0:01, a 1:75, b 10, l21 0:05, l22 1, s1=s2 0:05 - (e)-(f).

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completely different and depend on the values a. This meansthat order of fractional derivative can also change the stability ofstationary spatially- nonhomogeneous solutions.

5 Spatio-Temporal Pattern Formation in Fractional

Van der Pol-Fitzhugh-Nahumo-Like System

The results of the numerical simulation of the fractional Vander Pol-FitzHugh-Nahumo model are presented on Figs. 5 and 6.From the pictures, we can see that the system demonstrates a rich

scenario of pattern formation: standard homogeneous oscillations,Turing stable structures, interacting inhomogeneous structuresand inhomogeneous oscillatory structures.

Computer simulations confirm, that the variations in any systemparameter which qualitatively changes the eigenvalue spectrumof the linearized system, can change also the system dynamics.Spatiotemporal dynamics of the FRD system mainly by the maxi-mum eigenvalues for the corresponding modes are determined. InFigs. 5(a) and 5(c) we can see a formation of stationary dissipa-tive structures as a result of influence of maximal unstable modepresented in Fig. 3(b), (iv) for a < 1. External parameters A; bdetermine the intersection point, the slope of the isoclines in thispoint and the power for each particular mode. In particular, forA 0:25, null-clines intersect at the point where the dominantvalue has an eigenvalue for mode with k 2 (thick lines onFig. 3(b), (iv)). This situation is preferable for Turing bifurcation.

ForA 0:55 null-clines intersect at the point where inhomoge-neous modes are not dominant and Hopf bifurcation takes place.The characteristic feature for these two limit cases is the instanta-neous formation of eitherdissipative structures (Fig. 5(a)) or ho-mogeneous oscillations (Fig. 5(b)). Increasing the influence ofHopf bifurcation mode when the Turing one is dominant, orincreasing the Turing bifurcation mode when Hopf is dominant,leads to more complicated transient dynamics (Figs. 5(c) and5(d)). When conditions of these two instabilities coincide, we canobtain either oscillatory inhomogeneous structures or spacemodulated homogeneous oscillations (Figs. 5(e) and 5(f)). More-over, when the real part of eigenvalues of linearized system isclose to zero, small variation ofa can change the type of bifurca-tion in the system. This trend is typical for any a 1.

For a > 1 the structure formation can be much more compli-

cated. Let us consider the bifurcation diagram presented in Figs.2(b) and 2(d). It was already noted that for a given value a a0the region inside the corresponding curve is unstable for wavenumbers k 0 and outside - it is stable. From the viewpoint ofhomogeneous oscillations, at s1=s2 > 1 the system is stable nearu1 0 (Fig. 2(b)). However, if we have l1

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[17] Seki, K., Wojcik, M., and Tachiya, M., 2003, Fractional Reaction-DiffusionEquation, J. Chem. Phys., 119, 2165.

[18] Langlands, T., Henry, B. I., and Wearne, S. L., 2007, Turing Pattern Formationwith Fractional Diffusion and Fractional Reactions, J. Phys. Condens. Matter,19, 065115.

[19] Gafiychuk, V., Datsko, B., Meleshko, V., and Blackmore, D., 2009, Analysisof the Solutions of Coupled Nonlinear Fractional Reaction-Diffusion Equa-tions, Chaos, Solitons Fractals, 41, pp. 10951104.

[20] Gafiychuk, V., Datsko, B., and Meleshko, V., 2008, Mathematical Modelingof Time Fractional Reaction-Diffusion Systems, J. Comp. Appl. Math., 220,pp. 215225.

[21] Podlubny, I., 1999, Fractional Differential Equations, Academic Press,New York.

[22] Pierre, M., 2010, Global Existence in Reaction-Diffusion Systems with Con-trol of Mass: a Survey, Milan J. Math., 78, pp. 417455.

[23] Nicolis, G., and Prigogine, I., 1989, Exploring Complexity: An Introduction,Freeman & Co, New York.

[24] Matignon, D., 1996, Stability Results for Fractional Differential Equationswith Applications to Control Processing, Comput. Eng. Syst. Appl., 2, pp.963970.

[25] Lubashevsky, I., and Gafiychuk, V., 1994, Projection Dynamics of HighlyDissipative Systems, Phys. Rev. E., 50, pp. 171181.

[26] Poschl, G., and Teller, E., 1933, Bemerkungen zur Quantenmechanik desanharmonischen Oszillators, Z. Phys., 83, pp. 143151.

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