Physica D 215 (2006) 38–45www.elsevier.com/locate/physd

Target patterns in two-dimensional heterogeneous oscillatoryreaction–diffusion systems

Michael Sticha,∗, Alexander S. Mikhailovb

a Centro de Astrobiologıa (CSIC-INTA), Ctra de Ajalvir km 4, 28850 Torrejon de Ardoz, Madrid, Spainb Department of Physical Chemistry, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany

Received 25 August 2005; accepted 17 January 2006Available online 3 March 2006

Communicated by C.K.R.T. Jones

Abstract

Properties of target patterns created by pacemakers, representing local regions with the modified oscillation frequency, are studied for two-dimensional oscillatory reaction–diffusion systems described by the complex Ginzburg–Landau equation. An approximate analytical solution,based on the phase dynamics approximation, is constructed for a circular core and compared with numerical results for circular and square cores.The dependence of the wavenumber and frequency of generated waves on the size and frequency shift of the pacemaker is discussed. Instabilitiesof target patterns, involving repeated creations of ring-shaped amplitude defects, are further considered.c© 2006 Elsevier B.V. All rights reserved.

Keywords: Pattern formation; Reaction–diffusion systems; Target patterns

1. Introduction

The target pattern is one of the typical traveling wavepatterns observed in reaction–diffusion systems. It consistsof concentric waves that are periodically emitted from asmall central region, called a pacemaker. Target patterns arefound in a wide range of spatially-extended systems of achemical, biological, or physical nature [1,2] and are believedto be generic for dissipative systems far from equilibrium.Fundamentally, target patterns are two-dimensional patternsalthough often one-dimensional approximations have proven tobe sufficient to grasp important features of the pattern. In thispaper, we focus on properties of pacemakers and target patternswhich are intimately related to the two-dimensional nature ofthe medium.

A simple theoretical explanation of target patterns inoscillatory systems is that their wave sources are formed byheterogeneities which modify the properties of the mediumsuch that the oscillation frequency is locally increased. Indeed,the great majority of target patterns observed in chemical

∗ Corresponding author. Tel.: +34 91 520 6409.E-mail address: stichm@inta.es (M. Stich).

0167-2789/$ - see front matter c© 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2006.01.011

reaction–diffusion systems are associated with the presence ofa heterogeneity, e.g. a dust particle, that locally modifies theproperties of the medium and gives rise to a heterogeneouspacemaker. Most basic features of such patterns already weredescribed by the beginning of the 1980s [3–6], but sincethen target patterns have received less attention than, e.g.,rotating spiral waves. Recently, renewed interest in targetpatterns and pacemakers has yielded considerable progress inthe understanding of these patterns [7–13].

As a theoretical model to study heterogeneous pacemakersin oscillatory reaction–diffusion systems, we choose thecomplex Ginzburg–Landau equation (CGLE) [6,14]. Themedium possesses a localized region where the oscillationfrequency is modified and which represents a pacemaker givingrise to a spatially extended target wave pattern. In a previouspublication, we focused on such pacemakers and their wavepatterns in one-dimensional media [10]. In addition, inwardtraveling wave patterns and localized wave sink patterns wereconsidered. In this article, we extend the discussion to two-dimensional media, study in detail the dependence of thewavenumber and frequency of the generated waves on thesize and frequency shift of the pacemaker and show numericalexamples for stable and unstable wave patterns.

M. Stich, A.S. Mikhailov / Physica D 215 (2006) 38–45 39

The article is organized as follows: in Section 2, weintroduce the CGLE with heterogeneity as the basic model.In Section 3, we derive the phase dynamics equation anddescribe the conditions of target pattern formation. In Section 4,exact analytic and numeric solutions are presented for two-dimensional circular and square pacemaker core profiles.Although parts of the analysis of Sections 3 and 4 have beenpresented elsewhere, we show that important features of two-dimensional target patterns have not been explored yet. InSection 5, we present numerical simulations for pacemakersand wave sinks, with special emphasis on concentric waveinstabilities.

2. The CGLE and target patterns

Reaction–diffusion systems can display various types ofoscillatory dynamics. However, close to a supercritical Hopfbifurcation, all such systems are described by the complexGinzburg–Landau equation (CGLE) [6,14], given by

∂t A = (1 − iω)A − (1 + iα)|A|2 A + (1 + iβ)∇2 A, (1)

where A is the complex oscillation amplitude, ω is the linearfrequency parameter, α is the nonlinear frequency parameter, βis the linear dispersion coefficient, and ∇

2=

∂2

∂x2 +∂2

∂y2 denotesthe two-dimensional Laplacian operator.

We assume that ω is changed by an amount 1ω within asmall region of size R, called core, centered around the originof the coordinate system. We set ω(x) = ω+δω(x) and discusstwo cases: a square profile

δω(x, y) =

0 for |x | > R, |y| > R,1ω for |x | ≤ R, |y| ≤ R,

(2)

and a circular profile

δω(r) =

0 for |r | > R,1ω for |r | ≤ R,

(3)

where r =

√x2 + y2. We simply write ω(x) and δω(x),

whenever we do not need to distinguish between the twoprofiles.

In contrast to spiral waves, the CGLE does not support stabletarget waves without assuming a frequency heterogeneity in themedium. Since furthermore most target patterns found in realreaction–diffusion systems are created by small impurities, themodel presented above is well suited for studying target patternformation in oscillatory media.

Target patterns are two-dimensional patterns. Although noreal reaction–diffusion system is strictly two-dimensional, theextension of the system in the third dimension is often smalland not essential to the pattern-forming processes. In terms ofspatial polar coordinates r and θ , perfectly regular target wavesin the CGLE are then described by

A(r, t) = ρtp(r) exp(−iψtp(r)− iωtpt), (4)

where ρtp is the amplitude, ωtp the frequency, and ψtp(r) thephase of the target waves. Even though A does not depend on

θ , it is a difficult task to obtain explicit expressions for thesequantities.

Far from the center of the pattern, both the direct influenceof the wave source and the curvature of the waves become smalland the circular waves can approximately be described as one-dimensional plane waves, given by

A(r, t) =

√1 − k2 exp(ikr − iωk t), (5a)

ωk = ω + α + (β − α)k2. (5b)

If β−α > 0, the frequency increases with the wavenumber andthe waves have positive dispersion. In the opposite case, wavedispersion is negative.

In the next sections, we derive expressions for the frequencyωk and the wavenumber k of the waves created by aheterogeneous pacemaker, i.e., for 1ω 6= 0, in the two-dimensional medium.

3. Pacemakers in the phase dynamics approximation

Although this section is largely based on the classicderivation by Kuramoto [6], we believe that its presence is veryimportant for the sake of completeness of the argument and theunderstanding of the following sections.

3.1. The phase dynamics approximation

Introducing phase φ and real amplitude ρ as A = ρ exp(−iφ) and substituting this into the CGLE, we obtain

∂tρ = (1 − ρ2)ρ + ∇2ρ − ρ(∇φ)2

+ βρ∇2φ + 2β∇φ∇ρ, (6a)

∂tφ = ω(x)+ αρ2+ (2/ρ)∇ρ∇φ

+ ∇2φ − (β/ρ)∇2ρ + β(∇φ)2. (6b)

If we allow only for smooth phase perturbations with largecharacteristic lengths, the amplitude ρ follows adiabatically thedynamics of the phase and we can use the phase dynamicsapproximation. Then, ρ2

≈ 1 − (∇φ)2 + β∇2φ in first

approximation and Eq. (6) reduce to a single equation for thephase,

∂tφ = ω(x)+ α + (β − α)(∇φ)2 + (1 + αβ)∇2φ. (7)

We assume that uniform oscillations are modulationally stable,meaning that 1 + αβ > 0 is satisfied (Newell criterion of theBenjamin–Feir instability).

After applying the Cole–Hopf transformation

φ =1 + αβ

β − αln Q (8)

to Eq. (7), the new variable Q obeys the linear equation

∂t Q =β − α

1 + αβ(ω(x)+ α)Q + (1 + αβ)∇2 Q, (9)

where we have required that α 6= β and Q > 0 over the wholespace–time domain. A general solution of this equation can be

40 M. Stich, A.S. Mikhailov / Physica D 215 (2006) 38–45

written as

Q(x, t) =

∑n

Cn exp[(β − α)(ω + α)

1 + αβt + (1 + αβ)λn t

]× Qn(x), (10)

where λn and Qn(x) are the eigenvalues and eigenfunctions ofthe eigenvalue problem

(∇2− U (x))Q = ΛQ, (11)

where

U (x) = −β − α

(1 + αβ)2δω(x). (12)

The term exp[(β−α)(ω+α)t/(1+αβ)] in the general solutioncorresponds to uniform oscillations with frequency ω + α andhas been separated from the eigenvalue problem for the sake ofsimplicity.

Eq. (9) is formally equivalent to a Schrodinger equationfor a quantum particle in a potential (12) with width 2R anddepth

Umax =(β − α)1ω

(1 + αβ)2. (13)

Since we have not yet specified the signs of 1ω and β − α,U (x) may represent an attractive (“well”) or repulsive (“step”)potential. Unlike the case of the Schrodinger equation forthe quantum case, the time-dependent problem (9) does notcontain “imaginary time” and therefore the temporal behaviorof Q(x, t) is not oscillatory but exponentially decaying intime. The eigenvalues Λ relate to the energy eigenvalues of thequantum case as Λ = −E , in particular the largest eigenvalueλ0 corresponds to the smallest eigenvalue E0, associated withthe ground state.

All eigenvalues λn are real. For long times, the solution withthe largest eigenvalue λ0 dominates the expansion. Then, wehave Q(r, t) = C0 exp[(β − α)(ω + α)t/(1 + αβ)] exp[(1 +

αβ)λ0t]Q0(r) and

φ(r, t) = (ω + α)t +1 + αβ

β − α[(1 + αβ)λ0t + ln Q0(r)]. (14)

Hence, the frequency of the waves is

ω(λ0) = ω + α +(1 + αβ)2

β − αλ0. (15)

Since ω + α is the frequency of the underlying oscillations,only the frequency contribution

Ω(λ0) =(1 + αβ)2

β − αλ0 (16)

is of major interest. Using the one-dimensional dispersionrelation (5b) as an approximation, we can also expressthe wavenumber k as a function of the largest eigenvalueλ0:

k(λ0) =1 + αβ

β − α

√λ0. (17)

3.2. Pacemakers and wave sinks

From Eq. (17) we see that in order to obtain a target pattern,we have to require λ0 > 0 which in the quantum case signifiesthat the ground state has a negative energy eigenvalue. Thismeans that the potential must admit bound states.

Whether the potential U (x) admits bound states alsodepends on the dimensionality of the system: the potentialnecessarily must be attractive, a condition that is also sufficientin one- and two-dimensional media [15]. According toEq. (12), this means that a heterogeneity acts as a wave sourceor pacemaker, creating an extended target pattern if and only if

(β − α)1ω > 0 (18)

is fulfilled. Otherwise, i.e., if

(β − α)1ω < 0, (19)

a heterogeneity cannot lead to an extended target wave patternand is called a wave sink. A wave sink does not entrain thesystem and creates a localized wave pattern. Nevertheless, awave sink can have a considerable impact on pattern formation,as shown in Section 5. The direction of propagation of the(extended or localized) waves is determined by the sign ofthe local frequency shift. For 1ω > 0, the wave is movingoutward, for 1ω < 0 inward. Therefore, a target pattern can beeither formed of outgoing waves for the classic case of positivedispersion β − α > 0 and 1ω > 0 or of ingoing waves, for thecase of negative dispersion β − α < 0 and 1ω < 0.

In other words, the direction of propagation of the waves isgiven by the local frequency shift alone, whereas the capacity ofproducing an extended target wave pattern additionally dependson wave dispersion, i.e., a global property of the medium. Thisfundamental difference can also be expressed in terms of thephase velocity ω/k (outward vs. inward traveling waves) andgroup velocity dω/dk (pacemaker vs. wave sink). For the one-dimensional case, these issues are discussed, e.g., in Ref. [10].

4. Pacemaker solutions for the circular and rectangularcore

In order to explicitly solve the eigenvalue problem (11) onthe two-dimensional spatial domain for a circular shape of thecore (Eq. (3)), we write Eq. (9) with the Laplacian operator inpolar form

∇2r =

∂2

∂r2 +1r

∂

∂r+

1

r2

∂2

∂θ2 . (20)

We can perform a separation of variables and yield aneigenvalue equation for the radial part(∂2

∂r2 +1r

∂

∂r− U (r)

)Q0(r) = λ0 Q0(r). (21)

The solution of this eigenvalue problem is known (for ourcontext see e.g. Ref. [16]) and therefore we can be brief here.

Inside the two-dimensional potential well, the eigenfunc-tions of the potential well are Bessel functions of first kind,

M. Stich, A.S. Mikhailov / Physica D 215 (2006) 38–45 41

Fig. 1. Wavenumber (a) and frequency (b) of target waves created by a two-dimensional pacemaker with circular core (solid line: analytic solution, dottedline: numeric solution) and rectangular core (dashed line). The solutions for theone-dimensional case are shown by dotted–dashed lines. The parameters areα = 0.5, β = 1.0, 1ω = 0.2, the maximum wavenumber is kmax = 0.632.

m-th order, Jm . Outside the well, the eigenfunctions are mod-ified Bessel functions of second kind, m-th order, Km . Sincewe are interested in the eigenfunction Q0 corresponding to thelargest eigenvalue λ0, i.e., the eigenfunction to the ground statein the quantum case, the eigenfunctions are of zeroth order, i.e.,J0 and K0, respectively.

At the boundary of the core, i.e., at r = R, the solutions andtheir derivatives have to match, yielding the following equation

uJ1(u)

J0(u)= v

K1(v)

K0(v), (22)

where u and v are given by u = R√

Umax − λ0 and v = R√λ0,

where Umax is given by Eq. (13). From Eq. (22), the largesteigenvalue λ0 can be determined as a function of the radius R(or other parameters).

In Fig. 1, the frequency and wavenumber of a pacemaker, asdetermined by Eqs. (16), (17) and (22), are shown as a functionof the radius of the heterogeneity. The basic finding is thatk(R) > 0 for all R > 0. This means that – similar to theone-dimensional case and in contrast to the three-dimensionalcase – there is no critical value for the radius above which waveemission sets in [6]. However, the wavenumber becomes verysmall already for considerable radii and dk/dR approaches zeroas R → 0. This is in contrast to the one-dimensional case wheredk/dR ∝ R for small R. In the analogy with the quantum case,this means that the ground state is bound only very weakly.

In order to clarify the behavior of k(R) for small R, wehave a closer look at the functions appearing in Eq. (22). WhileJ0(0) = 1, J1 ∝ r for r → 0. Therefore, the left hand side ofEq. (22) scales like ∝ r2 for r → 0. Since K0 ∝ − ln r as r →

0, we may doubt that Eq. (22) admits values for all r . However,K1 ∝ 1/r for small r and therefore the right hand side scaleslike ∝ −1/ ln r . Consequently, the curves representing theright and left hand sides of Eq. (22) intersect for arbitrarychoices of the radius. This result confirms the general findingthat the Schrodinger operator on a two-dimensional domainadmits bound solutions for arbitrarily weak potentials, much aslike as on one-dimensional domains [15]. The fact that dk/dRapproaches zero as R → 0 is due to the slow convergence tozero of the right hand side of Eq. (22).

In the limit of large cores, R → ∞, the eigenvalue λ0approaches Umax. According to Eqs. (16) and (17), this means

Fig. 2. Wavenumber of target waves for different values of1ω (a) and effectivecritical radius of the pacemaker core (b). In (a), 1ω increases from bottom totop as 1ω = (0.1, 0.2, 0.3, 0.4, 0.5). The parameters are α = 0.5, β = 1.0.

that Ω → 1ω and k → kmax =√1ω/(β − α) as R → ∞.

This behavior is confirmed by the curves displayed in Fig. 1.Although k(R) > 0 for all R > 0, target patterns with small

k are practically not observable, since the system size and hencethe largest possible wavelength is limited for any real systemor simulation. Furthermore, any other wave source or spiralwith a larger frequency (assuming positive dispersion) wouldeasily entrain such a pattern. Therefore, we can conclude thatfrom a practical point of view there exists an effective criticalvalue Rc for the core radius, similar to the case of three spatialdimensions [6].

To explore this issue in more detail, we display in Fig. 2(a)the wavenumber as a function of the radius R for differentfrequency shifts 1ω. We observe that the larger 1ω, thesmaller is the interval where k(R) becomes very flat, and thelarger is the asymptotic value of k. Still, the courses of thedifferent curves are qualitatively similar for all1ω, in particulardk/dR = 0 for R → 0 even for large 1ω.

One possibility to quantify the effective critical radius isto make a linear fit to the curve k(R) at the point whered2k/dR2

= 0 and to define the effective critical radius Rc asthe radius where the linear fit crosses k = 0. In Fig. 2(b) weshow Rc as a function of 1ω and we clearly see that Rc doesnot vanish as 1ω increases. Note that values of k larger thanunity do not correspond to possible wave solutions since thentheir amplitude ρ =

√1 − k2 becomes zero. Moreover, wave

solutions may become unstable before reaching the existenceboundary, as will be discussed below. Furthermore, we haveto recall that the phase dynamics approximation is only validfor smooth phase perturbations associated with small amplitudedeviations and hence small frequency shifts. Therefore, valuesof1ω > 0.5 (for the assumed values of α and β) should not beconsidered within this approximation.

In Fig. 3 we display the wavenumber and frequency as afunction of the local frequency shift1ω for different choices ofR. For large R (dotted line), the curves approach the respectivecurves for the maximum values for k and Ω (thin solid lines).While for intermediate R = 5 (dashed line) the core is ableto give rise to target waves with a large range of wavenumberand frequencies (depending on 1ω), for already a little smallercore size (solid line) wave emission is very weak and effectivelystrongly limited.

In the case of a rectangular pacemaker core, no analyticsolution is available and we have to determine the smallest

42 M. Stich, A.S. Mikhailov / Physica D 215 (2006) 38–45

Fig. 3. Wavenumber (a) and frequency (b) of target waves created by a two-dimensional pacemaker with circular core for R = 3 (solid line), R = 5 (dashedline), and R = 15 (dotted line). The maximum values for k and Ω are shownas thin solid lines. The parameters are α = 0.5, β = 1.0.

Fig. 4. Two-dimensional pacemaker (a) and wave sink (b). Shown is Re A fort = 200. The parameters are ω = 0, α = 0.5, β = 1, R = 4.8, 1ω = 0.3 (a),1ω = −0.3 (b), and Lx = L y = 120.

eigenvalue of Eq. (11) numerically. The result is shown in Fig. 1as a dashed line. We observe that the wavenumber as a functionof R for the rectangular case behaves very similar to the circularcase. We also show the solution for the circular core, whichhad been used to validate the numerical procedure. As the onlyremarkable difference to the analytic solution, we state that forsmall radii the wavenumber does not approach zero, but settlesto values around 0.005 which are due to finite size effects forthe numerical scheme.

5. Simulations of stable and unstable target patterns

Numerical simulations of Eq. (1) have been carried out toconfirm the existence of stable pacemakers and wave sinks,and to investigate the behavior of unstable target patterns.We use a backward differentiation formula with adaptive timestepper and a next-neighbor representation of the Laplacianin Cartesian form (spatial discretization dx = dy = 0.4).The boundary conditions are no-flux and initial conditionsalways consist of uniform oscillations. In the simulations,heterogeneities have a square profile with half-width R(Eq. (2)).

In Fig. 4, simulations showing typical examples for apacemaker and a wave sink in a medium with positivedispersion (β−α > 0) are shown. Here and in most figures, weshow Re A, the real part of the complex oscillation amplitudewhich best illustrates the wave pattern. In Fig. 4(a), we applya local frequency increase in the center and hence create apacemaker. The periodic wave structure typical for a target

Fig. 5. Two-dimensional pacemaker with a wave sink. (a) shows an image ofRe A, (b) an image of the |A| at t = 500. The dark spot in (b) denotes the wavesink, where |A| is decreased to its minimum value 0.67 (black does not denotezero here). The parameters are: ω = 0, α = 1, and β = 0. The size of themedium is Lx = L y = 100. The pacemaker is characterized by 1ω = −0.6and R = 1.6, and the sink by 1ω = 0.7 and R = 1.6.

Fig. 6. Two-dimensional pacemaker generating open wave ends at a wave sink.(a) shows an image of Re A, (b) an image of |A| at t = 500. The pacemaker(R = 3.2, 1ω = 0.6) is located near the lower right corner while the sink(R = 7.2, 1ω = −0.6) is located in the center. The dark spots in (b) denoteopen wave ends where the amplitude is vanishing (defects). The parameters areω = 0, α = −1, β = 0, Lx = L y = 100.

pattern is clearly seen. The waves run outwards. In Fig. 4(b), thefrequency is locally decreased and the heterogeneity serves as awave sink. Asymptotically, the core region also oscillates withthe frequency of uniform oscillations, however, with a constantphase difference between the core and the rest of the medium.Since the core lags behind the periphery, the waves run inwards.

In Fig. 5, a simulation for a medium with negative dispersion(β − α < 0) is shown. Then, the lowest frequency willdominate the system and hence at the pacemaker core thefrequency must be locally decreased (1ω < 0). Although inthis case the target waves run toward the pacemaker, it is stillthe localized frequency shift which entrains the medium. Thepacemaker is located close to the center of the medium (andclearly recognizable in Fig. 5(a) as being the center of the targetpattern).

To simultaneously study the effects of a pacemaker and awave sink, a small localized frequency increase has additionallybeen applied close to the right lower corner of the medium.The wave sink slightly compresses the waves locally and theamplitude decreases which can be seen in Fig. 5(b) as a darkspot.

In the next simulation, the extension of the wave sink isgreatly enlarged to find out how a strong wave sink behaves.The results are displayed in Fig. 6. It can be seen that the

M. Stich, A.S. Mikhailov / Physica D 215 (2006) 38–45 43

Fig. 7. Two-dimensional pacemaker with unstable target waves in a mediumwith negative dispersion. (a) shows an image of Re A, (b) an image of |A| att = 131. The black line in (b) denotes a circular defect line. A strong decreaseof |A| is also present close to the core. (c) shows a space–time diagram of Re Afor a cross section through the center of the system parallel to the x-axis for atime interval1t = 250. The parameters are ω = 0, α = 1, β = 0,1ω = −0.7,R = 1.6, and Lx = L y = 150.

target waves emitted by the pacemaker (lower right corner)break at the wave sink (here in the center of the system).These broken waves have open ends which, in the absenceof a pacemaker, would actually curl in and form spirals(corresponding simulations are not shown here). The open endsmove and constitute phase singularities which are associatedwith amplitude defects that can be easily detected in Fig. 6(b) asdark spots. Since the pacemaker periodically emits waves, thebroken waves do not curl in but are pushed away and annihilateat the no-flux boundary.

As the magnitude of the frequency shift 1ω of a pacemakerincreases, phase slips are expected to develop in the wavepattern (see, e.g., Ref. [10]). It is a well known fact that planewaves in the CGLE become unstable through the Eckhausinstability if the wavelength of the waves is too short. Then,perturbations increase that may either lead to a modulatedamplitude wave or to the breakdown of the wave that isassociated with a phase slip of 2π and a momentary drop ofthe amplitude to zero. It can be expected that regular targetwaves, at least far from the pacemaker, behave in a qualitativelysimilar way. A simulation confirming this conjecture is shownin Fig. 7. There, a strongly negative frequency shift in a mediumwith negative dispersion leads to the emission of unstable targetwaves. In analogy with the one-dimensional case [10], phaseslips are observed at a certain distance from the wave source.A phase slip is associated with an amplitude defect which hasa circular geometry (ring-shaped defect). This is clearly seenin the image of the amplitude |A| for such a time moment[Fig. 7(b)]. In addition, phase slips also occur close to the

Fig. 8. Two-dimensional pacemaker generating line defects. (a, b) t = 537.5;(c, d) t = 546.5; (e, f) t = 586.0. The left column displays Re A, the rightrow shows |A|. The dark lines in the right column correspond to line defectswhere phase slips occur (a, f), and the isolated dark dots to point defects (phasesingularities) (d). The parameters are ω = 0, α = 1, β = 0, 1ω = −0.7,R = 1.6, and Lx = L y = 100.

border of the pacemaker, which may be due to the strongcurvature of the waves in the center of the pattern and to thesquare shape of the core. This shows that the appearance ofphase slips is possible at different distances to the core. Bothtypes of phase slips can also be seen in the space–time diagramfor a cross section through the center of the system [Fig. 7(c)].Since the dispersion is negative, the target waves propagatetoward the center. The phase slips periodically occurring farfrom the core and at the core boundary can be clearly identified.

The defect lines in the simulation presented in Fig. 7show perfect circular symmetry, which is due to the initialcondition (uniform oscillations) and the centric position of thepacemaker core. Next, a simulation is shown for the case thatthis symmetry is broken due to an interaction of the waves withthe no-flux boundary. Fig. 8 shows several time moments ofthe evolution of a wave pattern consisting of unstable targetwaves where the defect line appears close to the boundaryof the system. As a result of the collision of the target wavewith the boundary, the phase slip occurs first for the section ofthe wave which is directed toward the center of the medium.

44 M. Stich, A.S. Mikhailov / Physica D 215 (2006) 38–45

Although this open defect line disappears after the phase slip,its open ends persist and form two point defects. At these pointdefects, a phase singularity is present, resembling the core ofa spiral. Indeed, if the pacemaker is removed from the system,the point defects persist and form the cores of two spiral waves.However, if the pacemaker is kept within the system, the part ofthe target wave pattern which is interacting with the boundaryalso undergoes a phase slip process. This behavior is repeatedperiodically, leading to alternating phase slips occurring in thecenter and at the border.

6. Discussion

In this article, we have studied in detail two-dimensionaltarget wave patterns created by heterogeneities in oscillatoryreaction–diffusion systems. We have derived an analyticalsolution for a circular core and compared it to numericalsimulations. Applying strong frequency shifts, wave breakupmay occur.

The exact analytical solution for the two-dimensional casewith circular core is based on the classic derivation for one- andthree-dimensional cases published in Ref. [6]. Nevertheless,the solution of the two-dimensional case always has beenunderrated and – to our knowledge – has never been discussedexhaustively. Therefore, we have presented in Section 4 adetailed study of the system. The first question that wasaddressed is how the wavenumber and frequency of a targetpattern change as the radius of the circular heterogeneity isvaried. On a general ground it was expected that – like inone-dimensional systems – any radius leads to wave emission,i.e., no critical radius as in three-dimensional systems isneeded for wave emission. We confirmed this result althoughwe also found evidence that for small radii the wavenumberand frequency of the waves actually become very small.This novel result leads us to the definition of an effectivecritical radius. This implies that the size of the radius is moredecisive than the frequency shift. In particular, a small radiuscannot be compensated by a large frequency shift. Furthermore,even if wave emission could be triggered, desynchronizationphenomena are likely to set in, as discussed below.

By performing a numerical evaluation of the eigenvalueproblem, we confirm the analytic solution of the circular coreprofile and extend the discussion to heterogeneities with squareprofiles. The fundamental result for such cores is that thewavenumber and frequency of the waves are quite similar,even quantitatively, to the case of a circular core. In particularwe also observe an effective critical radius which lead usto the conjecture that its existence may be generic for two-dimensional pacemakers. The fact that the area of a core withhalf width R is 4/π -times larger than the area of a circularcore with radius R, can serve as an argument explaining whythe wavenumber of a wave emitted by a square core is a littlesmaller than the one sent out by a circular core for the samevalue of R.

As a result of the derivation, we can give the conditionthat a given heterogeneity must fulfill in order to becomea pacemaker. If the condition is not met, the heterogeneity

constitutes as a wave sink. This condition is the same as for theone-dimensional case, presented in Ref. [10]. There, we alsodiscussed inward traveling waves (if 1ω < 0) and wave sinks(if (β − α)1ω < 0). Again, these features translate directlyto the two-dimensional case. Although localized wave patternsproduced by wave sinks are of less interest, we should mentionthat their properties are also given be the eigenvalue problemexplored in Section 4. Then, the heterogeneity plays a similarrole as a repulsive potential in the quantum case.

Wave sinks lead to localized wave patterns only in the simplecase of uniform oscillations. However, in the presence of a wavesource, we can observe that waves are locally compressed at thesink. If the sink is large or formed by a large frequency shift,waves may even break and lead to open wave ends. Dependingon the medium, the presence of other pacemakers, etc., thesewave sinks may be advected away or form spiral cores. Hence,wave sinks can be important for pattern selection.

When the local frequency shift in the pacemaker core is in-creased, wave regimes with a periodic formation of phase slipsare observed. In these regimes, the effective oscillation fre-quency close to the core becomes different from the frequencyof oscillations in the far field. The phase slips occur becausethe medium is no longer able to compensate the frequency shiftthrough propagation of waves. Hence, desynchronization takesplace and oscillations in the core region become decoupledfrom the rest of the medium. This behavior is associated withthe periodic appearance of circular defect lines. This behaviorcan be explained by an Eckhaus instability mechanism for thetraveling concentric waves. We also observe phase slips closeto the core. Obviously, then the desynchronization process isstrictly local and its onset can be related to curvature effects orthe presence of strong frequency shifts.

It is well known that pacemakers which emit waves withdifferent frequencies compete with each other. For a mediumwith positive dispersion, the pacemaker with the largestfrequency suppresses all other pacemakers. However, this isonly true as long as the waves are Eckhaus stable. If phase slipsdevelop, the frequency and wavenumber of the waves decreasein the far field of the pacemaker, which is therefore not able toentrain the medium. Far from the heterogeneity, it effectivelyappears as a pacemaker with a lower frequency. For a mediumwith negative dispersion, the argument is of course very similar.Then, the pacemaker with the lowest frequency would suppressall others.

For another modified version of the CGLE, propertiesof heterogeneous pacemakers have been studied by Hendreyand co-authors [7]. They assume that the heterogeneitynot only changes the frequency, but also the amplitude ofoscillations. This implies that the derivation within the phasedynamics approximation shown here is not applicable to theirmodel. Another major difference is that they have studied aheterogeneity with an exponentially decaying size. In spiteof these differences, some of the numerical results found intheir model are similar to the results presented here. Forexample, they also encounter periodically appearing phase slipsif the frequency shift is large enough, a pattern they callbreathing target. A hysteresis between normal and breathing

M. Stich, A.S. Mikhailov / Physica D 215 (2006) 38–45 45

target patterns is observed. A substantial part of their workis devoted to spirals and the selection mechanism betweenspiral waves and target patterns, which are topics that have notbeen addressed in the present work. It may well be possiblethat hysteresis and a complex interaction of spirals and targetpatterns can also be found for the model studied in this work.

The model investigated in this paper has also been used tostudy how target waves emitted by a pacemaker may suppressspatio-temporal turbulence observed in the two-dimensionalmedium [11].

Though the analysis of pacemakers and wave sinks has beenperformed for a model system described by the CGLE, theresults may remain qualitatively correct for other oscillatorymedia. For example, in experiments with the light-sensitiveBelousov–Zhabotinsky reaction, the oscillation frequency caneasily be controlled by changing the local light intensity.In this way, heterogeneities with different shapes, sizes andstrengths can be created. Experiments in this direction havebeen done by several groups [17,18]). By varying theseparameters and the reaction conditions, it would be possibleto create different pacemakers and maybe also observe theonset of desynchronization in this system. Also, experimentsof the CO oxidation reaction on platinum have recently beenperformed [12]. By pointing a laser beam onto the surface,the temperature is locally increased and the desorption of COis enhanced. The temperature shift of the “heterogeneity” iscontrolled by the laser power. In this way, the kinetics ofthe system is changed locally by external means, as assumedhere. The experiments in the oscillatory regime of the COoxidation show outgoing and ingoing target patterns andlocalized outgoing waves. The outgoing target patterns can beeasily explained by the aforementioned model, since there, thefrequency is locally increased and the dispersion is positive.

To summarize, we have presented new results on targetwave formation through heterogeneous pacemakers in twodimensions. We emphasize the existence of an effective criticalradius for the core, the impact of wave sinks on the dynamics,and the possibility of circular (or more complex) line defectsproduced by strong local frequency shifts.

Acknowledgments

M.S. wants to thank Vanessa Casagrande and ChristianNeissner for useful discussions.

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