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Chaotic dynamics in Bonhoffervan der Pol fractional

reactiondiffusion system

B.Y. Datsko a,, V.V. Gafiychuk b,c

a Institute of Applied Problems of Mechanics and Mathematics of National Academy of Sciences, Naukova 3b, Lviv 79063, Ukraineb SGT Inc., 7701 Greenbelt Rd Suite 400, Greenbelt, MD 20770, USAc NASA Ames Research Center, Moffett Field, CA 94035-1000, USA

a r t i c l e i n f o

Article history:

Received 21 November 2009

Received in revised form

5 March 2010

Accepted 5 April 2010

Available online 9 April 2010

Keywords:

Fractional differential equation

Anomalous diffusion

Reactiondiffusion

Pattern formation

Pattern recognitionChaotic dynamics

Applications

a b s t r a c t

In this article we analyze the linear stability of nonlinear fractional reactiondiffusion

systems. As an example, the reactiondiffusion model with cubic nonlinearity is

considered. By computer simulation, it was shown that in such simplest system, a

complex nonlinear dynamics, which includes spatially non-homogeneous oscillations

and spatio-temporal chaos, takes place. Possible applications of the fractional reaction

diffusion system for signal processing and pattern recognition systems are presented.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

In the recent years, the study of fractional differential

equations has been driven by considerable interest both

from the theoretical and the applied points of view [15].

This interest is mainly determined by the attempts to

understand phenomena in fractal and irregular systems ofnature. Fractional derivatives are already widely used for

description of granular and porous media [6,7], many

aspects of signal processing [812], processes in living

tissue [13,14], polymers and amorphous materials [15,16]

and others. In the recent years, many scientists

have considered that diffusion in real systems of

nature has anomalous character [17,18]. Although in the

majority examples a degree of anomaly is insignificant,

a set of such complex systems, as composite or

amorphous materials, complex micro-emulsions, living

tissues require development of the models in which

substantial anomalousness of diffusion has to be taken

into account.

Investigation of the reactiondiffusion (RD) models in

the last decades has changed understanding of thenonlinear phenomena in many complex systems. On the

basis of mathematical modeling of classic reaction

diffusion, a set of nonlinear effects in the physical,

biological and chemical systems [1922] was explained.

Accordingly, studying the fractional RD system has

generated increasing attention among physicists and

mathematicians [2328]. At present time fractional reac-

tiondiffusion system (RDS) are generally used to describe

a large class of systems at different scales from

the molecular [17] to the space one [29]. In this case the

development of the theory of such systems could be

important for both the scientific perspective and its

application or implementation in a set of new devises.

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/sigpro

Signal Processing

0165-1684/$- see front matter & 2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.sigpro.2010.04.004

Corresponding author.

E-mail addresses: [email protected] (B.Y. Datsko),

[email protected] (V.V. Gafiychuk).

Signal Processing 91 (2011) 452460

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There are many promising approaches of using RD

systems and we would like to mention couple of them

just to show relevance of the proposed study to signal

processing and pattern recognition systems.

It was shown that a photosensitive BelousovZhabotinsky

system in water-in-oil micro-emulsion can store spatial

information, even without replenishment of reactants.New properties of utilizing of reactiondiffusion in micro-

emulsions systems for functional memory devices were

considered in [30,31]. The review of these phenomena

was considered in [22]. In the same time non-equilibrium

properties in micro-emulsions most probably posses

anomalous diffusion and can be described by fractional

derivatives [22].

The second application relates to pattern recognition

systems. The digital RDS presented as a discrete-time

discrete-space nonlinear reactiondiffusion dynamical

system was used to design of a fingerprint restoration

algorithm. This algorithm combines a ridge orientation

estimation technique using an iterative coarse-to-fineprocessing strategy and an adaptive digital RDS having a

capability of enhancing low-quality fingerprint images.

Signal processing algorithm for fingerprint image restora-

tion based on digital reactiondiffusion system was

considered in [32,33]. So as this approach is based on

computer simulation of RDS and computer simulation of

fractional RDS shows more diverse structure formation, it

is obvious that developed systems can be used for signal

processing algorithm either. Some preliminary results of

implication of fractional reactiondiffusion system for

fingerprint processing are presented in [34,35].

Interesting application of RDS could be developed

based on solid state electronic structures important forthe industrial applications [36]. This monograph brings

together results of experimental and simulated proto-

types of reactiondiffusion computing devices for image

processing, path planning, robot navigation, computa-

tional geometry, logics and artificial intelligence. On one

side classic RDS could be used in artificial intelligence to

control a minimally cognitive animate in distributed

controlling [37]. This approach is particularly appropriate

to a robot that is intended to remain in-silico, which

requires the RD controllers to maintain and use a

chemical memory. On the other side having fractional

controllers are widely used for industrial application

[8,9,12,38]. Synthesis of reactiondiffusion controllerswith implemented fractance can form fractional reac-

tiondiffusion media in-silico some nonlinear properties

of which are considered here.

RD equations with fractional order temporal operators

were used to model electronic properties of spiny

neuronal dendrites. These models predict that postsynap-

tic potentials propagating along dendrites with larger

spine densities can arrive at the soma faster and be

sustained at higher levels over longer times [23].

It should be noted that many processes in living media

are described by reactiondiffusion and use these proper-

ties to build and control structures on length scales from

microscopic to mesoscopic. Diffusion of molecules andreactions are fundamental to most cellular processes,

including enzymatic reactions, signaling, proteinprotein

interaction, as well as domain and pattern formation [17].

In this case studying diffusion of molecules in living cells

and proteins, interpreted with mathematical and physical

models, providing a glimpse into the world of molecules

[17,39]. Moreover, autowave phenomena in such complex

systems are experimentally revealed, as cells and living

tissues [41,42] give us additional reason for investigationof fractional RDS.

RD systems can provide also a versatile basis for new

applications in micro- and nanotechnology, where precise

control of RD processes in complex micro-geometries

makes it possible to fabricate small-scale structures,

devices, and functional systems which are not always

described by integer RDS [40].

It is worth to mention that FRDS under consideration with

diffusion coefficient equal to zero present systems of

fractional ordinary differential equations (FODE) widely used

for analysis of fractional order controllers [812,4346].

Introducing fractional operators to reactiondiffusion

systems substantially enriches dynamics of patternformation in such systems. In this article we will show

that even in the simplest model with cube nonlinearity,

a fractional RDS possesses an extraordinarily complex

dynamics, including space-temporal chaos. Chaotic

dynamics in the systems with fractional derivatives was

considered in some articles, including articles devoted to

fractional controllers (see please [4447] and reference

therein). However, all these articles are devoted to

ordinary fractional differential equations. The attention

is focused here on the formation of chaotic dynamics

in the fractional systems with partial derivatives. We will

try to recreate an integral picture of nonlinear solutions in

the dynamical systems with time fractional derivatives.

2. Mathematical model

The starting point of our consideration is the fractional

RDS with indices of different order

t1@a1n1x,t

@ta1 l2 @

2n1x,t@x2

Wn1,n2,A, 1

t2@a2n2x,t

@ta2 L2 @

2n2x,t@x2

Qn1,n2,A, 2

subject to Neumann:

@ni=@xjx 0 @ni=@xjx lx 0, i 1,2 3boundary conditions and with certain initial conditions.

Here n1(x, t), n2(x, t) are the activator and inhibitor

variables, 0rxrlx, t1, t2, l, L are the characteristic timesand lengths of the system, correspondingly, A is an

external parameter.

Time derivatives @ainix,t=@tai on the left-hand side ofEqs. (1),(2) instead of standard ones are the Caputo

fractional derivatives in time of the order 0oao2 andare represented as [48,49]

@a

@tani

t

:

1

Gm

aZt

0

nmi t

tta

1

mdt,

where Gq : R10 eppq1 dq is the well known EulersGamma function, m1oaom, m=1, 2. It should be

B.Y. Datsko, V.V. Gafiychuk / Signal Processing 91 (2011) 452460 453

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noted that Eqs. (1), (2) at a1=a2=1 correspond to the basicstandard reactiondiffusion system [1921].

Introduction of time fractional derivatives appreciably

expands the family of differential equations. In this case,

system (1), (2) sets a possibility of transitions between the

parabolic, elliptic and hyperbolical types of partial

derivative equations. In other words, system (1), (2)describes all spectrum of possible variants of types of

partial differential equations and describes all combina-

tions of subdiffusive and suboscillatory processes.

Due to the properties of the Caputo derivatives [48,49]

by certain substitution, the system can be transformed to

the set of differential equations with fractional derivative

index being the greatest common factor a of the valuesa1=pg, a2=rg simultaneously (for example a14a2), p, rAN[25]. Therefore, we obtain the system of p+r equations

tgu@gupx,t

@tg l2 @

2ux,t@x2

Wu,v,A, 4

tgv@gvrx,t

@tg L2 @

2vx,t@x2

Qu,v,A, 5

where derivatives on the right-hand side generate

recurrent equations for ui, i=p, p1,y,1 and vj, j=r,r1,y, 1.

tgu@gui1x,t

@tg uix,t, pZ i41, u1 u, 6

tgv@gvj1x,t

@tg vjx,t, rZj41, v1 v: 7

Such presentation of system (1), (2) makes it possible

to write down explicitly characteristic equation for any

relation between derivative orders [25].

3. Linear stability analysis

3.1. Standard reactiondiffusion systems

For standard RD system (a1=a2=1), it is convenient toanalyze null clines of system (1), (2): W=W(n1, n2, A)=0,

Q=Q(n1, n2, A)=0. Simultaneous solution of the

two equations W=Q=0 leads to homogeneous distribution

of n1 and n2. The eigenvalues l1,2 1=2tr F7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitr2 F4detF

p of the linearized right-hand side of system

(1), (2) play an important role in the system evolution.

Here, the matrix Fk a11k=t1 a12=t1a21=t2 a22k=t2

!, is deter-

mined by a11(k)=a11k2l2, a11 W0n1 , a12 W0n2 ,a21 Q0n1 , a22(k)=a22=k2L2, a22 Q0n2 (all derivatives aretaken at homogeneous equilibrium states (W=Q=0)),

k=(p/lx)j, j=1,2,y, tr F(k)a11(k)/t1+a22(k)/t2, det F(k)a11(k) a22(k)/t1t2a12(k) a21(k)/t1t2. In this case, wehave two types of bifurcations: for k =0 at conditions

tr F040, det F040: 8we have a Hopf bifurcation, and for ka0 at

tr Fo

0,

detF04

0,

detFk0o

0 9we have a Turing one. One can notice that condition (8)

may be rewritten as a114a22t1/t2 with the proper

critical frequency of oscillations offiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

detF0=tutvp

, and

condition (9)as a114a22l2=L22L=lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

detF0p

with

the proper critical wave number k0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

detF04p

=ffiffiffiffilL

p. The

instability conditions for these two types of bifurcation

will be realized due to positive feedback in the system

(a114

0) and at t1/t2-

0, l/L-

0 they approach toextremum points W(n1, n2, A)=0.

3.2. Fractional reactiondiffusion systems

In the case of fractional indices, we have additionally

taken into account relation between imaginary and real

parts of eigenvalues of the linearized system. In case

a1=a2=a (0oao2) for every point inside paraboladet F=tr2 F/4 there is a marginal value a0 2=pjArgli j2=pjArctgImli=Relij, obtained by theformula [24,50]

a0 2

parctan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4det F=tr2 F1

q, trF40,

2 2p

arctanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4det F=tr2 F1q

, trFo0:

8>>>: 10

which determines the stability domain of the system. In

other words, an order of fractional derivative is an

additional parameter of bifurcation, which determines

the type of bifurcation in the system. In case of fractional

derivatives, the Hopf bifurcation is not connected with

condition a1140 and may occur for a certain value a41even for a11o0 [24,50]. Moreover, for this case there is a

possible situation [24], when the next conditions arefulfilled

tr Fo0, detF040, detFk0o0: 11

and we can meet a new type of bifurcation which is not

inherent to standard RD systems [24]. In this case,

the system becomes unstable towards perturbations of

finite wave number for a given value of fractional

derivatives. As a result, inhomogeneous oscillations with

this wave number become unstable and lead to nonlinear

oscillations which result in spatial oscillatory structure

formation.

For the fractional RDS with arbitrary rational a, b,the linearization of equivalent system (4)(7) at the

equilibrium conditions described by vectors u u,0, . . . 0p1, v v,0, . . . 0r1 leads to characteristicequation det(JlI)= 0 which can be represented by(r+p)-degree polynomial [25]

lrp1r1 a22ktbv

lp1p 1 a11ktau

lr

1rpdet F 0: 12The roots of this polynomial will determine stability of

system (1), (2). In general case, the solution of such type of

equation can be obtained numerically. But in special

cases, we can analyze it analytically. One can see that inthis case we have to analyze eigenvalues with maximal

real part and maximal ratio of imaginary and real parts.

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4. Nonlinear dynamics and conditions of instability in

fractional model with cube nonlinearity

In this section, we consider the most famous and

simplest RDS with nonlinear source term

W n1n3

1n2 , 13for activator variable and with linear term

Q n2 bn1 A, 14for the inhibitor one [1921].

The homogeneous solution of system (1),(2) for

variables n1 and n2 can be obtained from the system of

equations W=Q=0, and for the determination of the

dependence n1 on external parameter A we can write

down a cubic algebraic equation

b1n1 n31=3A 0: 15With boundary conditions (3) system (1),(2) for given

nonlinearity (13), (14) can have one homogeneous

stationary solution n10,n20, which can be defined fromthe system of nonlinear algebraic equations (Fig. 1a).

4.1. Analysis of possible bifurcations

The conditions of Hopf bifurcation can be evidentlyexplained on the analysis of eigenvalues of the linearized

system. It is known, that in the standard RD model at

t1/t251 the Hopf bifurcation takes place. In Fig. 1b thereal and imaginary parts of eigenvalues of the linearized

system are presented at t1/t2=0.1 and l2/L2=1 for

different values of k. Due to linearization of the right-

hand side of the equation only, this presentation is true

for arbitrary values a1=a2=a. In the case of fractionalderivatives, we have to take into account relation between

imaginary and real parts, which follows from formula

(10). One can see from Fig. 1b that for k =0 we have real

positive eigenvalues on a certain interval and complex

conjugate eigenvalues with negative and positive realparts. For k =1 we have eigenvalues only with negative

real part. A subsequent increase of k leads to a situation

when eigenvalues become real and negative. Thus, space-

homogeneous solutions on an interval between points Rmand Rp (Fig. 1b) will be unstable practically for an arbitrary

value a and will grow, until nonlinearity will stop thisgrowth. On the intervals between points Cp and Rp(correspondently Cm and Rm) instability of the system

strictly depends on the value ofa and the Hopf bifurcationcan have a place for eigenvalues with negative and

positive real parts. Moreover, at complex eigenvalues,

we can always choose such value a at which the system

with eigenvalues with positive real part will be stable andat eigenvalues with negative real partunstable [24,50].

Turing bifurcation is realized at real eigenvalues and

conditions of instability are practically the same for both

fractional and standard RD systems. A difference in that

statement is that the order of fractional derivative

changes the region of parameters with Hopf bifurcation

independently from the conditions Turing bifurcation.

Due to the dominant development of oscillation with k =0

due to Hopf bifurcation, Turing pattern formation can not

be dominant. Real and imaginary parts of eigenvalues of

the linearized system are presented by plots in Fig. 2a at

t1/t2=1 and l2/L2=0.1 for k=0, 1 and 2, correspondingly.

It can be seen from Fig. 2a that for k =2 practically onthe entire interval, where dependence n2(n1) following

from W=0 is increasing (Fig. 1a), we have a maximal real

positive value of eigennumbers. Therefore, on this inter-

val, unstable modes exp(ikx) will grow exponentially with

these wave numbers until nonlinearity will stop this

growth.

At the same time, as follow from Figs. 1a, 2a on the

interval where dependence n2(n1) (W=0) descends, at

certain a41 it is possible to get either complex Hopfbifurcation for k =0 or specific for fractional system only a

k-mode Hopf bifurcation [24] for given wave numbers ka0.

In the system under consideration, we can choose the

parameters when we do not have Turing and Hopfbifurcations at all. Nevertheless, we obtain that conditions

for Hopf bifurcation can be realized for non-homogeneous

Fig. 1. Null-clines (W=Q=0)(a) and the eigenvalues (Re lblack lines,

Iml

gray lines) of the specific model for the case t15t2 and differentvalues ofk (k = 0hair-lines, k =1middle lines, k = 2heavy lines)(b).The eigenvalues are presented for a1=a2=a, and t1/t2=0.1, l/L=1.0,b=1.1.


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wave numbers. As it is seen from the Fig. 2a (heavy lines),

there are conditions where only instability according to

non-homogeneous wave numbers k=2 holds. We can seethat regions for k=1(gray middle lines) and k=2 (gray

heavy lines) overlap, but we have part of separate region

for k=2, and in this region only the perturbations with

k=2 become unstable and the system exhibits inhomo-

geneous oscillations.

4.2. Numerical schemes and computer simulation

System (1), (2) with nonlinearity (13), (14) and

corresponding initial and boundary conditions was inte-

grated numerically using the implicit scheme with respect

to time and centered difference approximation for spatialderivatives. The fractional derivatives were approximated

using the scheme on the basis of GrunwaldLetnikov

definition for 0oao2 [5,48,49]. In fact, according tofractional calculus [48,49] between the Caputo and the

RiemannLiouville derivatives we have next relation

C0D

atut, RL0 Datut,

Xmp 0

tpaGpa1

@p

@tpu0 , 16

where the operator in the RiemannLiouville senceRL0 D

atut, is equivalent to the GrunwaldLetnikov opera-

tor GL0 Datut, [48]

RL0 D

atut, GL0 Datut, lim

Dt-0Dta

Xt=Dtj 0

1ja

j

!utjDt,

17Because of the GrunwaldLetnikov operator is more

flexible for numerical calculations and can be approxi-

mated on the interval [0,t] with subinterval Dt as

GL0 D

atut, % X

t=Dt

j 0

caj utjDt,, 18

where caj

Dta1ja

j

!are the GrunwaldLetni-

kov coefficients [48,51], we used (16) approximation for

our computer simulation. It should be noted that solution

of the system using GrunwaldLetnikov derivative ap-

proximation (17) instead of Caputo one leads practically

to the same attractor. The only difference is contained in

transition dynamics due to influence of the last term in

Eq. (16).

In other words, for the system of m fractional RD

equations

tjC@ajujx,t

@taj dj @

2ujx,t@x2

fju1, . . . ,un, j 1,m, 19

where tj, dj, fjcertain parameters and nonlinearities ofthe RD system correspondingly, the scheme can be

represented as

ukj,idjDtajtjDx2

ukj,i12ukj,iukj,i 1Dtajtj

fjuk1,i, . . . ,ukn,i

DtajXmp 0

kDtpajGpaj1

@p

@tpu0j,i

Xkl 1

cajl

uklj,i ,

caj0

1, c

ajl

c

ajl

1 1

1aj

l , l 1,2, . . .

where ukj,i ujxi,tk ujiDx,kDt, m a.The applied numerical schemes are implicit, and for

each time layer they are presented as the system of

algebraic equations solved by NewtonRaphson techni-

que. Such approach makes it possible to get the system of

equations with band Jacobian for each node and to use the

sweep method for the solution of linear algebraic

equations. Calculating the values of the spatial derivatives

and corresponding nonlinear terms on the previous layer,

we obtained explicit schemes for integration. Despite the

fact that these algorithms are quite simple, they are very

sensitive to the value of step and require small steps of

integration. In contrast, the implicit schemes, in certainsense, are similar to the implicit Eulers method, and they

have shown very good behavior at the modeling of

Fig. 2. Eigenvalues (Re lblack lines, Imlgray lines) of the specific

model for the case l15l2, a1=a2=a(a) and l15l2, t15t2, a1=2a2(b).Different thicknesses of lines corresponds to different values of k

(k = 0hair-lines, k =1middle lines, k =2heavy lines). The eigenvalues

are presented for parameters t1/t2=1.0, l/L =0.1, b=1.1(a), t1/t2=0.1,l/L =0.1, b=1.1(b).


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fractional reactiondiffusion systems for different step

size of integration, as well as for nonlinear function and

order of fractional index. It should be noted, that implicit

numerical L1/L2-schemes with second order approxima-

tion (for more details see [5254]) also show good

behavior for numerical investigation of fractional RDS,

but for numerical calculations the scheme on the basis ofGrunwaldLetnikov is much more flexible.

4.3. Pattern formations

The characteristic dynamics of pattern formations

for mention above three types of bifurcation is presented

in Fig. 3. In this figure we can observe a stationary dissipative

structures(a), spatially homogeneous oscillations(b) and

inhomogeneous oscillatory structures(c), obtained by

computer simulations of system (1),(2) for the eigenvalues

presented in Figs. 1b, 2a.

Numerical simulations show that development of

Turing instability in the fractional system and general

dynamics of the RDS can differ from the standard case

a=1 and for this reason terminal attractors can differ fromone another, although the linear condition of instability

look the same. One of the reasons of such difference is the

relation between real and imaginary parts of eigenvaluesof the linearized system and overlapping of the domains

with different types of instability. Then, as a result of

competition of different types of bifurcation, it is possible

to see more complicated nonlinear dynamics. Especially

Fig. 3. Spatiotemporal structures obtained from computer simulation of

system (1), (2) with nonlinearities (13), (14) for a1=a2. Dynamics ofvariable n1 for: a1=1.2, a2=1.2, t1/t2=1.0, l/L=0.1, b=1.1, A= 0.1(a),a1=0.8, a2=0.8, t1/t2=0.1, l/L =1.0, b=1.1, A = 0.1(b), a1=1.85,a2=1.85, t1/t2=1.0, l/L=0.1, b= 2, A= 10.0(c).

Fig. 4. Influence of different derivative orders on the dynamics in pointsystem. Dynamics of variables n1 (black lines) and n2 (gray lines) at

l= L= 0 for: a1=1.5, a2=1.0(a), a1=1.0, a2=1.5(b). a1=1.75,a2=1.25(c). The other parameters are t1/t2=0.1, b=1.01, A= 0.3


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complex nonlinear dynamics of system (1), (2) with a

given nonlinearity is observed in the case of different

orders of fractional derivatives. Ratio of derivative orders

also plays a role of a new additional parameter, which

substantially influences the eigenvalues of the system and

changes an instability domain. Characteristic types of

dependence of eigenvalues at the same parameter b as inFig. 2a for a case a1=2a2 and t1/t251, l/L51 arepresented in Fig. 2b. As it is seen evidently from the

picture, an increase of the derivative order of activator

results in decreasing the domain of instability of homo-

geneous oscillations. At the same time, it results in the

increase of interval where complex bifurcation takes place

for the wave numbers different from zero. One can see

that these domains partly recover each other and at the

same parameters, the condition of instability for differ-

ent modes is fulfilled simultaneously. As a rule, pertur-

bation with k=0, becomes most competitive. However, in

domains, where classic Hopf bifurcation is absent,

inhomogeneous perturbations become unstable and

spatio-inhomogeneous oscillations appear in the system.

Overlapping of the domains results in co-operation of

different types of bifurcations and leads to more complex

transitional processes or steady-state spatio-temporal

dynamics (Fig. 5).

In addition, different orders of derivatives lead to non-

compensated behavior of activator and inhibitor variables.The characteristic type of dynamics of the point system

(spatial derivatives are absent) for different relation

orders of fractional derivatives with the same other

parameters is presented in Fig. 4.

Independence of fractional derivative orders and ratio

between them on internal bifurcation parameters of the

system can lead to condition when all type of bifurcations,

realized in fractional RD systems, are possible at the same

time. In this case by changing the orders of fractional

derivative we can change the type of bifurcation in the

system even at very insignificant deviation of derivative.

As a result, it is possible to choose such ratio of fractional

derivative orders that extraordinarily complex dynamicscan take place in the system, including spatio-temporal

chaos.

Fig. 5. Spatiotemporal structures obtained from computer simulation of

system (1),(2) with nonlinearities (13),(14) for a1aa2. Dynamics ofvariable n1 for: a1=1.5, a2=0.75, t1/t2=0.025, l/L=0.005, b=1.5,A= 0.3(a), a1=1.4, a2=0.7, t1/t2=0.2, l/L=0.025, b=1.5, A=0.05(b),a1=1.5, a2=1.0, t1/t2=1.0, l/L =0.05, b=1.1, A = 0.1(c).

Fig. 6. Chaotic pattern formation in system (1),(2) with nonlinearities(13),(14) for a1aa2. Dynamics of variables n1(a) and n2(b) for:a1=1.5, a2=0.75,t1/t2=0.1, l/L =0.1, b=1.1, A= 0.27. Dynamics ofvariable n1 for A =0.27(c). The other parameters are the same.


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Results of computer simulation of such dynamics are

presented on Figs. 6 and 7. This dynamics is inherent for

overlapping domains with complex eigenvalues for

different wave numbers (regions between points RP and

CP (RM and CM) on Fig. 2b. Spatio-temporal chaotic

dynamics is observed in the enough wide intervals of

relations of fractional derivatives (Figs. 6a, b, 7b).Moreover in the case of the symmetric null-cline

intersection we observe a principal different spatio-

temporal dynamics (Fig. 6c). The increasing of bifurca-

tion parameter A increases the influence of complex

bifurcation for k=0 (Figs. 1a, 2b) and leads to regulari-

zation of the inhomogeneous oscillations (Fig. 7a). When

the orders of derivatives become approximately equal to

each other, the instability domain of Hopf bifurcation will

expand and, as a result, the homogeneous relaxation

oscillations will arise in the system (Fig. 7c). Such

oscillations are characteristic for the standard equations

RD for given parameters of the system.

5. Conclusion

In this article we analyzed the linear stability of two

component reactiondiffusion system with time frac-

tional derivatives of different orders. By eigenvalue

analysis we investigated instability conditions for time

and spatial pattern formation for various values ofexternal parameters and indices of fractional derivatives.

Computer simulation confirms the eigenvalue analysis

and shows that system dynamics is mainly determined by

most unstable modes of linearized system.

By numerical simulations it was shown that even for

the simple fractional RD system with cube nonlinearity

a complex nonlinear dynamics, which includes spatially

non-homogeneous oscillations and spatio-temporal

chaos, takes place. In contrast to a standard reaction

diffusion system with integer derivatives, the fractional

RDS possesses new properties connected with values of

fractional derivative indices and a ratio between them.

A fractional derivative orders can change substantially aneigenvalue spectrum and significantly enrich nonlinear

dynamics in RD systems.

It should be noted, that many bright nonlinear

phenomena were explained on the basis classical RDS

with nonlinearities (13) and (14). Typical examples are

transition pulse in nervous fiber, cardiac muscle and solid

state neuristor. Such systems possess diverse physical

properties like spontaneous appearing stationary, pulse or

stochastic non-homogeneous distributions. In solid state

structures and gas-discharge systems with S- or N-shape

voltampere characteristic of cubic like nonlinearity the

domain of high current density and electrical field arise

[21,55]. Syntheses of layered structures in such activatorinhibitor system based on solid state or gas layers and

formation of separate activation and inhibition through

separated layers may be used for formation of the

hardware model considered above. As a result systems

with anomalous diffusion properties that are described by

reactiondiffusion equations can be created synthetically.

In this case, the corresponding layers have to be end-

owed with the properties inherent to fractional order

controllers.

Finally, we wish to remark here that spatio-temporal

pattern formation phenomena realized in fractional order

RDS have to stimulate experimental investigation of these

phenomena in distributive circuits created with the helpof modern solid state electronics.

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