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Proceedings of the ASME 2009 International Design Engineering Technical Conferences &Computers and Information in Engineering ConferenceIDETC/CIE 2009August 30-September 2, 2009, San Diego, USA

DETC2009/MSNDC-87644

DIFFERENT TYPES OF INSTABILITIES AND COMPLEX DYNAMICS INFRACTIONAL REACTION-DIFFUSION SYSTEMS

Vasyl Gafiychuk

Physics DepartmentNew York City College of Technology,NY 11201, U.S.A.

Email: vagaf@yahoo.com

Bohdan Datsko

Dapartmeny of Mathtematical modelingInstitute of Appl. Problems of Mechanics and MathematicsLviv, 79053, Ukraine

Email: b datsko@yahoo.com

ABSTRACTIn this article we analyze a influence of possible instabili-

ties on pattern formations in the reaction-diffusion systems with

fractional derivatives. The results of qualitative analysis are con-

firmed by numerical simulations. The main attention is paid to

two models: a fractional order reaction diffusion system withBonhoeffer-van der Pol kinetics and to Brusselator model.

1 INTRODUCTION

In many complex heterogenous systems, the description of a

media by fractional derivatives to represent anomalous behavior

is of crucial importance [15]. According to the traditional ap-

proach in the kinetic theory the diffusion length is determined by

the squareroot of the time characterizing the process. But the in-

vestigation of system with a certain intrinsic structure shows that

this length can have a lower or higher power depending on the na-

ture of the system. This feature can change the dynamics of the

system substantially. Such situation takes place in reaction dif-fusion system, where introduction of fractional derivatives leads

to a new nonlinear phenomena [615]. The diversity of phenom-

ena suggests that we want to classify them in order to understand

the conditions of pattern formation in such system. In standard

reaction diffusion (RD) system such classification is done a long

time ago [1618]. In this article using a computer simulation and

a linear stability analysis we would like to present general typical

Address all correspondence to this author.

spatio-temporal patterns revealed in fractional reaction-diffusion

(FRD) systems.

2 MATHEMATICAL MODEL

Let us consider the fractional RD system

cu

t = luxx +W(u,A), (1)

with two components u = (u1,u2)T,W = (W1,W2)

TR2, A - realparameter, W - smooth reaction kinetics functions ,WC1, andl are positive, diagonal matrices = diag[i], l = diag[l2i ] > 0,on the x (0,L) subject to (i) Neumann: ux|x=0,Lx = 0 or (ii) pe-riodic: u(t,0) = u(t,Lx), ux|x=0 = ux|x=Lx boundary conditionsand with the certain initial conditions.

Fractional derivatives cu

t on the left hand side of the equa-

tions (1), instead of the standard time derivatives, are the Caputo

fractional derivatives in time [19, 20] of the order 0 < < 2 andare represented as

cu

t =c u(t)

t:=

1

(m)t

0

u(m)()

(t )+1m d, (2)

where m1 < < m,m 1,2.

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Proceedings of the ASME 2009 International Design Engineering Technical Conferences &Computers and Information in Engineering Conference

IDETC/CIE 2009August 30 - September 2, 2009, San Diego, Cali fornia, USA

DETC2009-87644

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3 LINEAR STABILITY ANALYSISThe stability of the steady-state solutions of the system (1) corre-

sponding to homogeneous equilibrium state W(u0,A0) = 0 canbe analyzed by linearization of the system nearby this solution

u0 = (u01,u02)T.

3.1 Spectrum AnalysisThe system (1) can be transformed to the system of lin-

ear fractional ODEs with right hand side matrix F(k) =(a11 k2l21 )/1 a12/1

a21/2 (a22 k2l22 )/2

, diagonal form of which is

given by eigenvalues 1,2 =12

(trF

tr2F4det F) (coeffi-cients ai j represent Jacobian). For : 0 < < 2 for every pointinside the parabola det F = tr2F/4, we can introduce a marginalvalue : 0 =

2 |Arg(i)| given by the formula [10,11]

0 =

2 arctan

4det F/tr2F1, trF> 0,

2 2 arctan

4det F/tr2F1, trF< 0. (3)

The value of is a certain bifurcation parameter which switchesthe stable and unstable states of the system. At lower : 0 =2 |Arg(i)| leads to

instability.

3.2 Classification of Instabilities in FractionalReaction-Diffusion Systems

It is widely known for integer time derivatives that the system

(1) becomes unstable according to either Hopf or Turing bifurca-

tions.

Let consider the conditions for Hopf bifurcation which are held

at k= 0 if

trF> 0, det F(0) > 0. (4)

For the integer time derivatives, this bifurcation is connected with

condition a11 > 0. In the case of fractional derivative index, Hopfbifurcation is not connected with the condition a11 > 0 and canhold at a certain value of when the fractional derivative indexis sufficiently large [12].

The conditions for the Turing instability are

trF< 0, det F(0) > 0, det F(k0) < 0. (5)

In this case, eigenvalues are real and at a11 > 0,a22 < 0,a12a21 1 when it is easier tosatisfy conditions of Hopf bifurcation. Moreover, here we meet

a new type of instability [10, 11] at

trF< 0, 4det F(0) < tr2F(0), 4det F(k0) > tr2F(k0). (6)

The analysis of the expressions (6) shows that at k= 0 we havetwo real and less than zero eigenvalues, and the system is cer-

tainly stable. If the last inequality takes place for a certain value

of k0 = 0, we can get two complex eigenvalues and a new typeof instability, connected with the interplay between the determi-

nant and the trace of the linear system, emerges. With such type

of eigenvalues, it is possible to determine the value of fractional

derivative index when the system becomes unstable [10].

4 A FRACTIONAL ORDER BONHOFFER-VAN DERPOL AND BRUSSELATOR MODELS

In this section we consider RD systems wit cubical (Bonhoeffer-

van der Pol model) and quadratic (Brusselator model) nonlinear-

ities (See, for example [1618])

Bonhoeffer-van der Pol model. In this case, the source term for

activator variable is nonlinear W1 = u1 u31 u2 and its linearfor the inhibitor one W2 =u2 +Bu1 +A. The homogeneous so-lution of variables u1 and u2 can be determined from the system

of equations W1 = W2 = 0. Simple calculation at homogeneous

state makes it possible to write conditions of different types ofinstability explicitly.

For the Brusselator type fractional RD system the source term

for activator variable is W1 = A (B + 1)u1 + u21u2 and for theinhibitor one W2 = Bu1 u21u2 . The homogeneous solution ofvariables u1 and u2 can be also determined from the system of

equations W1 = W2 = 0 and are: u1 = A, u2 = B/A.

4.1 Stability Domains and Nonlinear Solutions of TheSpecific Models

Hopf bifurcation. The typical stability domain for a

Bonhoeffer-van der Pol type RD system in the coordinates(u1,1/2) is represented on Fig.1 (a,b). It is easy to see thatif the value of1/2, in certain cases, is smaller than 1, the insta-bility conditions (trF > 0) lead to Hopf bifurcation for regularsystem ( = 1) [1618]. In this case, the plot of the domain,where instability exists, is shown on the Fig. 1 by the curve with

index 1. The linear analysis of the system for = 1 shows that,if 1/2 > 1, the solution corresponding to the intersections oftwo isoclines is stable. The smaller is the ratio of 1/2, thewider is the instability region. Formally, at 1/2 0, the insta-bility region for u1 coincides with the interval (1,1) where the

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(a)

(b)

Figure 1. Instability domains in coordinates (u1,1/2) for a frac-tional order Bonhoeffer-van der Pol model. The results of computer

simulation obtained for B = 1.05, l1 = l2 = 0 and different values

of 0 = 0.2,0.5,0.9,1.0 - (a); B = 1.05, l1 = l2 = 0 and 0 =1.0,1.1,1.5,1.8 - (b).

null cline W(u1,u2) = 0 has its increasing part. These resultsare very widely known in the theory of nonlinear dynamical sys-

tems [1618].

In the fractional differential equations the conditions of the in-

stability depend on the value of and we have to analyze thereal and the imaginary part of the existing complex eigenvalues,

especially the equation:

4det Ftr2

F =4((B

1) + u2

1

)

12 (1u

2

1

)

1 1

22

> 0. (7)

In fact, with the complex eigenvalues, it is possible to find out the

corresponding value of where the condition > 0 is true. Wewill show that this interval is not correlated with the increasing

part of the null isocline of the system. Indeed, omitting simple

calculation, we can write an equation for marginal values of u1

u412(1 +12

)u21 +212221

2(2B1) + 1 = 0, (8)

Figure 2. Instability domains and the eigenvalues (Re - black lines,Im - grey lines) for the Bonhoeffer-van der Pol model. The parametersof the specific systems are the same as on Fig.1. The eigenvalues are

presented for 1/2 = 0.5

and solution of this biquadratic equation gives the domain where

the oscillatory instability arises:

u21 = 1 +122

B12

(9)

This expression estimates the maximum and minimum values

of u1 where the system can be unstable at certain value of

= 0 as a function of 1/2. The typical stability domainfor a Bonhoeffer-van der Pol type RD system in the coordinates

(u1,1/2) for different values of is represented on Fig. 1 (a),(b). In the regions between these curves and horizontal axis, the

system is unstable with wave numbers k = 0, and outside it isstable. For example, the light zone on Fig.1(a) presents zone,

where instability exists for = 0.5 and grey a zone on Fig.1(b)presents a zone, where instability exists for = 1.5 The typicalview of the eigenvalues is given on the Fig.2. We see that at k= 0the system can have a substantial imaginary part and under con-

dition > 0 it is unstable. A detailed analysis of the nonlineardynamics can be found in the paper [12]. It should be noted that

for the system with integer derivatives instability conditions for

Hopf bifurcation are very strict (1 < u1 < 1, 1 2).It is possible to obtain solid understanding of the mechanism of

the instability from the plot of eigenvalues. The typical domain

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Figure 3. Instability domains and the eigenvalues (Re - black lines,Im - grey lines) for the Brusselator model. The parameters of the spe-cific systems are l1 = l2 = 0 and 0 = 0.0,0.5,0.9,1.0,1.1,1.5,2.0.The eigenvalues are presented for A0 = 1.

of such dependence is presented on (Fig. 2) for the same param-

eters as on (Fig. 1 a,b). The easiest way of getting instability is

realized at 1 < 2 (Fig. 2a). For |u1| < uE

1 and at |u1| > uC

1 thetwo roots are real. Moreover, inside the dark region they are pos-itive which means that the system is unstable practically for any

value of> 0. Inside the domain uE1 < |u1|< uC1 there is a certaindomain of(0 < < 2) where the Hopf bifurcation takes place.Point D divides the region into two domains where Re< 0 andRe > 0. In the domain Re < 0 system could be unstable ac-cording to greater value of > 1. In turn, for Re > 0, systemcould be stable at smaller value . In other words, between pointsCand E we have eigenvalues with imaginary part and value ofcan change a stability of the fractional RD system.

In case of Brusselator nonlinearity is widely known for integer

time derivatives that the system (1) becomes unstable accordingto the Hopf bifurcations at

trF = (B1A2) > 0, det F = A2 > 0. (10)

The intersections of the nullcilines at the decreasing part of

W(u,v) = 0 lead to the limit cycle formation for = 1. Inthis case, the real part of the eigenvalues passes through zero

at B0 = 1 +A20 (Fig. 2, D(B0,A0)). That means that at this con-

dition, the system is unstable according to the Hopf bifurcation

because it has two complex conjugate roots with the real part

equal to zero. As a result, the real part becomes less than zero

at B < B0 and greater than zero at B > B0 (Fig. 3). For frac-tional equations in the first case, the condition of the Hopf bifur-

cation (4) realizes at > 1 and in the second one at < 1. Theimaginary part of eigenvalues (ellipse like curve) determines the

region of values B, where the roots are complex. The intervalof localization of these roots depend on the intersection point

P0 = (B0,A0). The greater the value of (B0) at the given (A0) is,than the wider the range with complex conjugate roots. The way

how the eigenvalues build up instability domain is shown also in

Fig. 2(b). On the typical stability domain for Brusselator, we

presented marginal curves corresponding to different values of

. On Fig.1 and Fig.3 these curves are obtained as a result ofsolution of equation (3)

(B1A2)22 = 4A2 (B1A2)2, (11)

where 2 = tan2 (0/2) tan2 ((20)/2). Explicitly, wehave the parametric dependance ofA(B) at the given 2 (0)

A2 = B1 + 2/(1 + 2)

(B1 + 2/(1 + 2))2 (B1)2.(12)

At > 0, dynamical system is unstable and at < 0, is sta-ble. In other words, any value 0 from zero till 0 = 2 dividesthe region (B,A) into two domains. Inside each particular curvethe corresponding 0 stationary state is unstable for this value of0 and outside is stable (Fig. 1 (c)). The matter is that between = 0 and = 2, the roots are complex and there is a minimalvalue 0 where the system is unstable according to eigenvalueswith the complex conjugate roots. It is expected that in the do-

main inside this particular curves 0 limit cycle arises accordingto the any value of>0. Outside this parabola-alike curves thesystem is stable according to this value of 0. Of course it canbe unstable according to greater than 0. One of the lines as itwas mentioned above corresponds to = 1 (B = 1 +A2) separat-ing the instability domain by two sub-domains where instability

takes place at < 1 (grey domain of Fig. 1 (c)) and > 1.

Turing Bifurcation. Analyzing (5) we can conclude that these

conditions are practically the same for fractional derivatives and

a standard RD system. But what is very important the transient

processes and the dynamics of these systems are different, and

for this reason final attractors can often be different even though

the linear conditions of instability look the same. Namely to this

type of dynamics the next subsection is devoted. Here we would

like to direct your attention to specific phenomena which are im-

minent to fractional RD system only.

Interaction of the Turing and Hopf bifurcations. Now, let us

consider that the system parameters are close to the one repre-

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sented by point D on Fig.2. From the viewpoint of homogeneous

oscillations, system is stable. But if we have l1 l2 = 1, thesystem becomes unstable according to Turing instability. As a

result, for such ratio ofIm and Re we expect the formation of

oscillatory inhomogeneous structures. However, with a small in-crease of we have homogeneous oscillations and decrease ofleads to steady state structures. Such trend is sufficiently general

and if in standard system we have steady state solutions increase

of leads to non stationary structures.

Instability according to k= 0,Im= 0. Above we have consid-ered that the linearized system is unstable either Hopf or Turing

bifurcation. Below, we consider the case when we dont have

Turing or Hopf bifurcation. When the fractional derivative in-

dex becomes greater than some critical value 0, the instabilitycondition holds true. So, as this instability conditions are possi-

ble to realize for some interval kmin < k< kmax, this means that

only the perturbations with these wave numbers are unstable, andthey are unstable for oscillatory fluctuations. This situation is

qualitatively different from the integer RD system whether ei-

ther Turing (k= 0) or Hopf bifurcation (k= 0) takes place, andthis depends on which condition is easier to realize. In the sys-

tem under consideration, we can choose the parameter when we

dont have Turing and Hopf bifurcations (for k= 0) at all. Nev-ertheless, we obtain that conditions for Hopf bifurcation can be

realized for nonhomogeneous wave number

In Fig. 4 plot displays the parameter ranges for the stabil-

ity and existence of dynamical structures. In the largest white

domain the system is unstable according to wave numbers k= 0

[10, 11]. For the case k= 0, we find instability conditions fordifferent wave numbers k = 1,2,3... (Solution of the equality|Arg(i)| = /2 at a certain . We can see that these regionsoverlap and at the same parameters, the instability conditions for

different regimes are fulfilled simultaneously (Fig. 4). As it is

seen from the figure, there are conditions where only instability

according to non-homogeneous wave numbers holds. As a re-

sult, perturbations with k= 0 relax to the homogenous state, andonly the perturbations with a certain value of k become unstable

and the system exhibits inhomogeneous oscillations.

4.2 Pattern Formation

The results of the numerical simulation of the fractional RDS(1) are presented on Fig. 5,6. From the pictures we can see

that in such system we obtain a rich scenario of pattern forma-

tion: standard homogeneous oscillations (a), Turing stable struc-

tures (b), interacting inhomogeneous structures (c) and inhomo-

geneous oscillatory structures (d). We have obtained that the ra-

tio of characteristic times (Fig.5, a,b) and the order of fractional

qualitative derivative (Fig.6, a,b) transforms pattern formation

dynamics: homogeneous oscillations in the first limiting case and

stationary dissipative structures in the second one. Moreover, at

parameters, when real part of eigenvalues is close to zero, small

(a)

(b)

Figure 4. Two-dimensional bifurcation diagram domains of specificmodels for k= 0. The results of computer simulations at = 1.8, l1 =2, l2 = 1 for Brusselator model (a) and at = 1.85,B = 2, l1 =0.1, l2 = 1 for Bonhoeffer-van der Pol one (b). Shaded regions cor-

respond to instability with k= 0.

variation of changes the type of bifurcation.

Let us consider the bifurcation diagram presented in Fig. 1b. It

was already noted that the region inside the curve is unstable for

wave numbers k = 0 and outside it is stable. From the viewpoint

of homogeneous oscillations, the system is stable near u1 = 0. But if we have l1

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(a)

(b)

(c)

(d)

Figure 5. Dynamics of pattern formation in a fractional order theBonhoeffer-van der Pol system for u1 variable. The results of computer

simulations of the systems (1) at parameters: A = 0, = 1.8, B = 1.1,l1 = 0.05, l2 = 1, 1/2 = 2.5 (a); A = 0, = 1.8, B = 1.1,l1 = 0.05, l2 = 1, 1/2 = 7 (b); A = 0, = 1.8, B = 1.1,l1 = 0.05, l2 = 1, 1/2 = 3.5 (c); A = 50, = 1.85, B = 2,l1 = 0.1, l2 = 1, 1/2 = 12 (d);

(a)

(b)

(c)

(d)

Figure 6. Dynamics of pattern formation in a fractional order the theBrusselator model for u1 variable. The results of computer simulations

of the systems (1) at parameters: A = 1, B = 1.8, = 1.1, l1 = 0.1,l2 = 1 (a); A = 1, B = 1.8, = 0.9, l1 = 0.1, l2 = 1 (b); A = 1,

B = 1.8, = 1.3, l1 = 0.01, l2 = 1 (c); A = 7, B = 7, = 1.9,l1 = 3.5, l2 = 1 (d).

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Hopf bifurcation (k= 0) takes place, and this depends on whichcondition is easier to realize.

5 CONCLUSION

We have performed the stability analysis and computer simula-

tion of the solutions in fractional RD systems for each particular

limit. The obtained solutions have the form of homogeneous os-

cillations or inhomogeneous structures which can be stationary

or oscillatory.

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