Systems Phase

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Phase Fronts and Synchronization Patterns in Forced Oscillatory Systems

EHUDMERON*

The Jacob BlausteinInstitutefor^{DesertResearch}and the PhysicsDepartment, Ben-Gurion University, Sede BokerCampus84990,Israel

(Received15 April1999)

This is a reviewofrecentstudiesofextendedoscillatorysystemsthat aresubjected to peri- odictemporal forcing.Theperiodic forcingbreaks the continuous time translationsymmetry and leaves a discretesetof stable uniformphase states.Themultiplicityofphase statesallows forfront structuresthat shiftthe oscillationphase by[nwhere n 1, 2,...,hereafter[n-

fronts. The main concern here is with front instabilities and theirimplicationsonpattern formation.Mosttheoretical studieshavefocusedonthe2" 1resonance where thesystem oscillatesathalfthedrivingfrequency. Allfront solutions in this case are -fronts.Athigh forcing strengths only stationary fronts exist. Upon decreasing the forcing strength the stationaryfronts losestability to pairsofcounter-propagating fronts.The coexistence of counter-propagating frontsallows fortravelingdomainsandspiralwaves.Inthe4:1reso- nancestationary -frontscoexist with/2-fronts.^Athigh forcingstrengthsthestationary -fronts are stable and standing two-phase waves, consisting of successive oscillatory domainswhosephasesdifferby

,

prevail.Upondecreasingtheforcingstrengththe station- ary -frontslosestabilityanddecomposeintopairs of propagating]2-fronts.The instabil- ity designates a transitionfrom standing two-phase wavesto traveling four-phase waves.

Analogous decompositioninstabilities have been found numerically inhigher 2n" 1 reso- nances.The availabletheoryisusedto accountfor a fewexperimentalobservationsmade on thephotosensitive Belousov-Zhabotinsky reactionsubjected to periodic illumination.

Observationsnotaccounted forbythetheoryarepointedout.

Keywords." Forced oscillations,Resonances,Phasefronts, Instabilities, Patterns

I. INTRODUCTION

A

conspicuous property of systemsdrivenfar from equilibrium is the possible appearance of persis- tent oscillations

[1-5].

Theonsetofoscillationshas extensively been studied in the contextofchemical

E-mail:[email protected].

217

reactions. In spatially extended reactions it often involvesspatialphasevariationsthat leadtotravel- ing wave phenomena. Biological rhythms provide another manifestation of persistent oscillatory dynamics. They occur in unicellular and multi- cellular organisms and cover a wide range of

periodicities, tens of milliseconds to years. The oscillating systems encountered in nature are not isolated and quite often the interaction with the environment takes theformofaperiodic forcingin time. One example is the entrainment of cardiac cells in the atrioventricular node to signals gener- ated at the sinoatrial node. Circadian rhythms entrainedbythe 24h day-night periodicity provide anotherexample.

Most theoretical studies of periodically forced oscillatory systemshavefocusedonfrequency lock- ing phenomena and the onset ofchaos in single oscillator models

(or

^circle maps) [4,6-10]. Fre- quency locking refers to the property of a forced system to oscillate at a frequency co which is a rational fraction of the forcing frequency _{cot in} some range of the latter. These ranges of reso- nant behavior getwider as the forcing strengthis increased, and ^are commonly refer to as Arnold tongues. The fractional frequencies a forced sys- tem can realizefollow the Farey rule: between the tonguescol: co n mand_cof CO k thereexiststhe

(n + k):(m +

l) tongue.

Another property of forced systemsisthe coexis- tenceof multiplestablephase states, corresponding to uniform oscillations with different fixedphases.

The multiplicity of phase states becomes particu- larly significant in coupled oscillator arrays or in oscillatory media, for different oscillator groups or spatial domains may oscillate with different phases, forming spatial patterns [11]. In the 2:1 tongue there are two stable phase states whose phases differ by rr. They allow for two-phase patterns involving alternatingdomains of the two phase states. The boundaries between these alter- nating domains, hereafter re-fronts, have been studied recently by Coullet et al. [12]. ^{They have} foundafrontbifurcationreminiscent of the Ising- Blochtransition in ferromagnetswithweak aniso- tropy. The bifurcation, now referred to as the nonequilibrium Ising-Bloch

(NIB)

bifurcation, renders a stationary (Ising) front unstable as the forcing strength is decreased, and gives rise to a pair of counter propagating

(Bloch)

fronts

(see

Fig.3).Thebifurcationdesignatesatransitionfrom

standing two-phase patternstotravelingtwo-phase patterns[13].

In the 3:1 tongue there ^are three stable phase states givingrise to traveling three-phase patterns [14]. The boundaries between any pair of phase states form

2rc/3-fronts

(shift the phase of oscilla- tion by

2rr/3).

The four stable phases in the 4:1 tongue allow for either two-phase or four-phase patterns depending on the forcing strength. At strong forcing stationary w-fronts ^are stable and standingtwo-phase patterns prevail. Asthe forcing strengthisdecreased stationary w-frontslose stabil- ity and decompose into pairs of propagating

rc/2-

fronts

(see

Fig.

8).

Thew-frontinstability designates a transition from standing two-phase patterns to traveling four-phase patterns

[15].

Recent experi- ments onthe photosensitive Belousov-Zhabotinsky reactionsubjectedtoperiodicilluminationdemon- stratethe existence of two andthree-phase patterns [19]. Some ofthe observed patterns are shown in Fig. 1.

In this paper I will review the mathematical analysis ofthe front instabilities described above, discuss the implications they bear on pattern for- mation, and use them to interpret some of the experimentalobservations. The mathematical^ana- lyses to be reviewed rely oncontinuum models of oscillatory systems. Such models apply not only to continuous media, such as chemical reactions, but alsoto discrete systems like coupled oscillator arrayswhen thecouplingisstrong enough.Wewill assume aninstability ofauniformstationary state to uniform oscillations. Near the instability

(the

Hopfbifurcation)the system’s dynamicsis govern byauniversalequationfor theenvelope

(or

ampli-

tude)

of the oscillations.Webeginwith a discussion ofthisequation.

II. ENVELOPE EQUATION APPROACH Let C(x,

t)

representthe setof dynamicalvariables ofa given system, and let

C-C0

^{be a stationary}

uniform state of the system. We assume that the state

Co

^{loses stability, as a} control parameter R

1:1 4:3 (Front) 2:1 (Labyrinth) 3:1

11 12

FIGURE Experimentalphase diagramfor the rutheniumcatalyzed Belousov-Zhabotinsky reaction periodicallyforced with pulsesof spatially uniformlightfroma videoprojector. The diagram shows frequencylockedregimes observedas afunction of the forcingfrequency,fp (CVr^{in the}text).Patternsareshown in pairs, oneabovetheother, attimesseparated byAt 1/fpexcept for the 1:1 resonance where At=1/2fp. Striped boxes on the horizontal axis mark forcing frequency ranges with the same frequency-lockingratio. Reprinted bypermission fromNature (Petrov etal., 388, 655-657) copyright (1997) Macmillan Maga- zinesLtd.

exceedsacriticalvalueR.,to uniform oscillations at frequency _a0(Hopfbifurcation at zero wavenum-

bet).

Beyondthe bifurcation andcloseto

R.

^{the set}

C(x,

t)

^canbe representedas anasymptotic expan- sion inpowers ofA

v/(R ^R.)/R, <<

^1:

C(x, t) Co + AC + AZCz + AC3

^+...,

(1)

where

C C0A(x, t)

^expicvot

+

^c.c.

(2)

Here

C0

^{is asetof}constants and "c.c." standsfor the complex conjugate. The amplitude

A(x, t):=

fI(X,T),

^{where X=}

x/-x

and

T=Xt,

depends weaklyonspace andtime. Theslow temporalvaria- tionsofAstemfromthesmallgrowthratesof per- turbations nearthe bifurcationpoint, and theweak spatialdependenceis aconsequence ofthenarrow band of growing wavenumbers.

Assume

now that the system is periodically forced atfrequency

cr

na0wheren isan integer.

TheequationforAadmits then theuniversalform

At (# + iu)A + (1 + ict)Axc -(1 + i/3) AlZA ^{+ %A}

^*’-1

where the subscripts andxdenote partialderiva- tives with respect to time and space, and all the parameters are real. The proximity to the Hopf bifurcation implies #<<^1. The amplitude equa- tion

(3)

can be derived for specific models using standard methods [20]. The general form of the equationcanbededuced from symmetryconsidera- tions. Inparticular, theforcingtermA^*n-1 follows from the discrete time translation symmetry, ^t--+

-+- 27r/cf (the

amplitude equation should beinvari- ant under thetransformationA^--,Aexpi2rc/n)[21 ].

Inpractical situations theperiodic forcing often containsharmonicsof the mainfrequencycol.Thus, forcing a system at cofco0, for example, may contain forcing components at 2co0,3co0,..., and the corresponding terms,

72A*,%A*2,...

^{in the}

amplitude equation should beconsidered aswell.

III. THE 2:1 RESONANCE A. TheNIB Bifurcation

An oscillating system that is forced at approxi- mately twice its natural frequency is described by the amplitude equation

(3)

^with n 2. This equa- tionhas two stablephasestates(uniform solutions) whose phases arg(A) differ by rr. In addition, the equation supports front solutions connecting the twophasestatesasthespatial^coordinategoes from

-oc to

+.

The front solutions ofEq.

(3)

have been studiedbyCoulletetal.[12].Consider first the gradient version of this equation, obtainedby set- ting v c

fl

^0[22]

At- #A + Axx IA ^2A

^+’y2A*.

⁽⁴⁾

The term "gradient" refers to the existence of a

Lyapunov

(or

freeenergy)functionalfor

(4)

which is minimizedby the dynamics [23]. The two stable phase states of Eq.

(4)

^are A

+A0

^where

A0 v/# +

^{"y2. One type} of front solution connecting these states exists for all (positive) _"Y2values. It is given by

A(x)-I(x; cr)-o-Ao

^tanh

(2 ^A0x )

where c +1 is the front polarityassociated with the reflection symmetry x--+-x of the

Eqs. (3)

and (4). The front solution

(5)

has a zero phase, arg(I) 0, and is referred to as Ising front inanal- ogy toIsingwalls inferromagnets.

At ")/2

#/3

^{the Ising front} loses stability and a pair of new fronts solutions appear

(for

^a given polarity)

A

(x) B+ (x; or) erA0 tanh(kx) -+- it/sech(kx), (6)

where r/-

v/#- 3"T2

^{and k-}

2x/f.

^{The phases,}

arg(B+/-), associated with these front solutions are not zero but rather rotating clockwise and anti- clockwise by rc as x increases from -oc to

+oc.

Itis in this sense that thesefrontsolutions resemble the Bloch domain walls inferromagnets. They are consequentlyreferredtoas Blochfronts.

Because of the gradient nature of

(4)

and the symmetry of the two phase states, all fronts solu- tions are stationary. The remarkable finding of Coullet etal. [12] wasthat any of the nongradient termsinEq.

(3)

withn 2whosecoefficientsareu, c

and/3

makes the two Bloch fronts propagating in oppositedirections whileleavingthe Isingfront stationary.The coexistence ofcounter-propagating Bloch fronts is a nongradient effect that does not exist in equilibrium systems. The Bloch front velocities aregivenby

C q-o-

3rot/A0

2k(3# "y2) (-u + fl# + (c fl)’Y2). (7)

Figure 2shows ^abifurcationdiagramfor theNIB bifurcation basedon Eq.

(7).

Equation

(7)

^can be derivedby writingafront solutionofEq.

(3)

with n 2 intheform

A(x, t) Ao(x- ct) +

^eR,

assuming the coefficients u, c,

and/3

are of order e

<<

^{1. Here,}

Ao(z)

is eitherthe Isingfront solution

0.02

(

Ira(A)

U

o:

y

-0.02

R

FIGURE 2 The NIB bifurcation for front solutions of Eq. (3) with n=2. For -y>%=/,/3 there is a single stable Isingfront with zero speed (solid line). For-y<% the Ising front is unstable (dashed line) and a pair of stable counter- propagating Bloch fronts appears (solid lines). Parameters:

# 1.0,u 0.01,c fl ^{0.0.(Theparameter}3’ is^")/2inthetext.)

0 2 4 6 8 I0 12 0 2 4 6 8 i0 12

X X

300

200

IOO

FIGURE 3 The NIB bifurcation in the 2:1 resonance:

space-time plots of arg (A) showing an unstable stationary Ising frontevolvingintoleft (a) andright(b)traveling Bloch frontsbeyondtheNIBbifurcation.

of(4), I(z; or),^orthe Bloch frontsolutions,B(z; or), and R represents higher order corrections. Using thisform in

(3)

and applying solvabilityconditions atordereleadtotheresultc 0 for

A0

^I,andthe

expression

(7)

for A0=B+/-. Another view of the NIB bifurcation is shown in Fig. ^{3 which} shows spacetimeplotsofunstable Ising frontsthatevolve into counter-propagating Bloch fronts. As willbe discussed inthenextsection theNIBfront bifurca- tionhassignificant effectsonpattern formation.

B. Implications onPatternFormation

Beforeembarking onpatternformation aspects of the amplitude equation

(3)

(withⁿ 2) let us con- siderthe moregenericcasewhere the twouniform phase states are not symmetric (under

A---+-A).

The symmetry can be broken by adding to the forcing a component at the

system’s

oscillation frequency, _wfw. The amplitude equation will nowread

At (# + ^iu)A + (1 + ^io)Ax.,

-(1 + ifl) AI2A + "72A* +

"71.

(9)

The effect of the 1:1 forcing term, "yl, is to grant velocitiestoallfrontsolutions, even in thegradient case. The expression

(7)

forthe Bloch front veloc- ities nowreads

c-

+or ^37rr/A0 2k(3# 3/2)

3crd0"l

(--/]

^-}-

fl#

^nc

((Y fl)’)’2) (o)

assuming_"1 is small.

Imaginenow apair of Isingfronts with different polarities

.

^{Such a pair forms a domain of one}

phasestate inabackgroundofthe other. Depend- ing^onthesign

of’71

this domain eitherexpandsor shrinks. In both cases a uniform phase state will eventually prevail.Inthesymmetric case,")/1 0,the attractive interactions betweenIsing fronts should lead inprinciple to a uniform state, but since the interactions are exponentially ^{small most} often patterns appear as frozen standing-wave patterns.

The situationchanges beyond theNIBbifurcation

(’2 < #/3)

where counter-propagatingBloch fronts coexist. For now, a combination of two distinct Blochfrontswith differentpolarities forma travel- ing domain. Due to the different propagation speedsofthetwoBloch frontsthetraveling^domain mayeitherexpand ^orshrink. Inthe former case a uniform state will be reached, but the shrinking domain may reach an equilibrium shape that tra- vels invariably, because of the repulsive interac- tions between Bloch fronts. Asymptotic traveling domain solutions are shown in Fig. 4. Thus, the coexistence of the counter-propagating Bloch frontsallowsfor asymptotic traveling patterns[13].

The interaction between a pair of Bloch fronts andthe formationofastable travelingdomaincan

A o.o

--Io0

A o.o

e(A) Irrt(A)

0 20 40 60

X

FIGURE4 Traveling domain solutions to Eq. (9) ^{near the} NIBbifurcation.(a) Astable"up"domain,"yl<0.(b) Astable

"down" domain -y>^0.

be studied by writinga traveling domain solution as [13]

A(x, t) B+[x- x,(t); +1] + B_[x- Xr(t); -1]

Ao + ^{R(x, t),} (11)

wherethe variables,

xr

^{and x, are the positions of}

the leading (right) and trailing (left) Bloch fronts, B/ aregivenby(6),andRisasmall correction term.

The two polarities

(or +1)

^are necessary to con- struct a domain bounded by the fronts. The two typesof Blochfronts,^B_andB/,make the domain traveling.Weassumethatthe systemisnearlysym- metric

(]-y] <<

1), that it is close to theNIB bifur- cationand thatthedomain ismuch wider than the width ofthe fronts. Following themethods of[24]

the following equation for the domain width, L

X --X1,has been derived:

kAo- 3"71 12A0e

^-2kL

^{+ 6A0r/2e -/L.}

The first term ontheright hand side describesthe effect of the broken symmetry between the two

L

20’

15- 10-

0.05 0.I0 0.15 0.20 0.25 0.30 0.35

7

FIGURE5 The distance, L, between thefront and back of a travelingdomain solutionfor"Y2 nearthe NIB bifurcation.

The solid and dashed lines represent the stable and unstable branches solutions from Eq. (13). ^{The crosses are data from} direct numerical solution ofEq. (9). Parameters: _#=1, u=

0.01,_"l -0.001,c=/3=0.(Theparameter is")/2inthetext).

Blochfronts;the initial domainexpands

(1 >

0)^or shrinks

(T <

0) in time when the leading front is faster or slower than the trailing one. The second term describes an attractive front interaction gen- eratedbythe realpartsof the Blochfrontsolutions.

The lastterm, generatedby the imaginary parts of the Bloch front solutions,describes alonger range repulsive interaction. The repulsive interaction strengthens_{as "Y2}isdecreased belowtheNIBbifur- cation point, _"Y2 #/3, and becomes dominant at sufficientlysmall"y2values.

Solutions describing domains traveling at con- stantspeeds areobtained by

setting/,

0 in

(12).

The solutions to theresulting quadraticequation in z exp(-kL)are

L

-k-’

in

(/]2_+_ ^V/f]4

^_+_

^4A0"Yl ) ⁺

^2k-1

^ln2A0.

(13)

Consider the case wherewide domains shrink, ^or

’l<0.

^{At the NIB} bifurcation point,

#-32--0, travelingdomain solutions do notyet appear unless_{71 -0.}They appearin asaddle-node bifurcationonly for _"72

< %(’71) _< #/3

^where

%(’1)

solves

(#- 3,7p)

²

4v/#

^{/ 7p} 7

I.

Graphs of these solutions in the

L-2

^{plane are shown in Fig. 5.}

Theupperand lowerbranches representstableand

unstablesolutions.Also showninFig. 5 areresults fromdirect numerical solutionsof

Eq. (9)

showing the stable traveling domain branch. The shape of thestabletravelingdomainis shownin Fig.

4(a).

IntwospacedimensionstheNIBbifurcationhas anotherinteresting implication: itallows forspiral wave solutions. Figure 6 shows the time evolution

a

b

d

FIGURE 6 Simulation of the two-dimensional version of Eq. (3)withn 2showingthedevelopmentofarotatingspiral wavebeyondthe NIB bifurcation(2<#/3). The left column is

IAI

and the right column arg(A)in the x-y plane. (a) An unstable Ising frontconsistingof twosegments perturbedsoas toinitiate convergence to different Bloch fronts.(b),(c)and(d) Thesubsequentevolutiontowardarotatingspiralwave.

ofaw-frontfor_"?’2

< #/3.

The initial front consists oftwo segments that converge to different Bloch fronts.Thisleads to^atwist motionthat evolves into arotating spiralwave

[29].

IV. THE4" 1 RESONANCE A. TheDecomposition Instability

The dynamics within the 4:1 resonance tongueis governed (close ^to the Hopf bifurcation and for small detuning)by

Eq. (3)

^with n 4. The param- eter_#canbe scaledoutby rescalingtimespaceand amplitudeas t-#r,x

V/#/2z

^andB

A/x/- fi

Bt (1 + iuo)B + 1/2(1 + ic)Bxx

(1 -+- i/3)]BI2B

-Fy4

B*3, ₍₁₄₎

whereu0

u/#.

Consider first thegradient version obtainedby setting_u0 c 0:

B, B^q--

1/2Bxx IBI2B ^-+-

⁷⁴

^{B*3. (15)}

Equation

(15)

has four stable phase states for 0

<

%

<

^shownbysolid circles inFig. 7:B+/-I ^+k

(A)

-1.0

-I.0 0.0 1.0

fig(A)

FIGURE 7 Uniform states and front solutions ofEq. (15) in thecomplex B plane.The dots represent the 4spatiallyuni- formphasestates.The solid linesarethe r-front solutions and the dashed linesarethe7r/2-fronts.The thin lines in the circle are phase portraits offront solutions at successive time steps showingthecollapseofar-front intoapairofr/2-fronts.

and B+/-i=+iA, where A

l/x/1-

^{y4. Front solu-}

tions connecting pairs of these states divide into two groups,rr-fronts and

rc/2-fronts.

^{The re-fronts,}

shown inFig. 7as solidlines,aregiven by B-1-,+I

B+

^tanhx,

B-i-,+i

B+

^tanhx.

⁽¹⁶⁾

The

rc/2-fronts

^{are shown in Fig. 7 by the dashed}

curves. For the particular parameter value

/4--1/2

theyhavethe simpleforms

B+l-++i--- ^[1 +i-(1 -i)tanh x], B-i-+ I1 + (1 + ^i)tanh x], B+i-+-I

^-B-i-,+I,

B-l-+-i -B+I--++i.

(17)

Additionalfront solutions follow from the invari- ance of Eq.

(15)

^under reflection, ^x--+-x. For example,the symmetric counterpartsof

B+;_++(x)

and B+l__i(x are B+l_++i(x)=B+i_++l(-X ^and /_._+(x)

+

^__(-x).

Considernow the nongradient system

(14).

The maineffectof thenongradientterms is tomake the

r/2-fronts

^{traveling. The} nongradient terms have noeffect onthe --fronts which remain stationary.

To see this assume a traveling solution

B(x-ct)

of Eq.

(14)

and project this equation on the translational mode

B’.

For r-fronts the resulting condition

c(B;2)-O Bo(z)-Atanhz,

implies c-0

(the

brackets denoteintegration ^over the whole line). For

rc/2-fronts

^with

4-1/2

^the.fol-

lowing expressionisfound:

(0

3

^{), (19)}

where A-

v/-/2. ^A

perturbation analysis around

4-

^{shows that theexpression}

⁽¹⁹⁾

^{for the speed}

remains valid for small deviations of 4 from^There-fronts

⁽¹⁶⁾

^{aresimilar totheIsingfront in}

.

the 2:1 resonance and like theIsingfronttheylose stability as the forcing strength, 74, is decreased.

Stability analysis of the re-fronts indicates that the instability ^occurs at

4-1/2.

The nature of the instability,however,isquitedifferent.Itis adegen- erate instability leading to asymptotic solutions that ^arenot smooth continuations of the unstable stationaryre-fronts,unlike theNIB pitchforkbifur- cation. Figure 8 shows aspace-timeplot ofarg(A) analogous to Fig. 3. The initial unstable re-front decomposesintoapairof

rc/2-fronts

^{travelingtothe}

0 2 4 6 8 10 12 0 2 4 6 g _{10 12}

X X

300

200

100

FIGURE 8 The decomposition instability in the 4:1 reso- nance: Space-time plots ofarg(A) (solutions ^ofEq. (3) ^with

n 4)showing thedecomposition ofan unstablerr-front into apair ofrc/2-frontstravelingtothe left(a)or totheright(b).

The pairs ofrc/2-fronts enclose grey colored domains whose oscillationphases areshiftedbyre/2 ^withrespect tothe black and white domains. Parameters in Eq. (3): #=1.0, u=0.02,

")/4 0.3.

right orto the left dependingon initialconditions.

Along with the re-front decomposition an inter- mediatephasestate(thegreydomain)appears.This behavior is found arbitrarilyclose tothe instability point, and in this sense the new solutions are not smoothcontinuations ofthe re-front solution. The instability has been analyzed by Elphick et al.

[15,25] using

Eq. (14)

^near

4-1/2. ^A

^{briefdescrip-}

tionofthisanalysis follows.

Again,^weconsider first thegradientversion

(15).

Introducingthe new variables

(20)

Equation

(15)

is written as

2

U3

^d

U2 V2

u, u _{+ -i}

-5 -i u,

2

V3

^d

V2

v, v + vxx _{--5 -i} ^u

²

v’

(21a) (21b)

where

At the instability point,

%-1/2,

^{the two equations}

decoupleand admit solutions of the form U-

0-Bo(x- xl),

V-

0-2B0(x- x2), (22)

where

Bo(x) X/-/2

^tanhx,

0-1,2-+l,

^{and x, and}

x2 are arbitrary constants. An intuitive under- standingof thisfamily ofsolutions canbeobtained by expressing these solutions back interms of the complex amplitude B. For 0-1----0-2-- for exam- ple, thesolution

(22)

^isequivalentto

B(x;

x1,

x2) a-i-++l (x x1) + B+I_++i(x x2) .

When

[X

²

X11

^{---> OOthisformapproachesapair of}

isolated

rc/2-fronts:

B B_i_++l

(X-

Xl

),

^{X Xl,}

and

B

B+++i(x- x2),

^x,- X2.

When_x2-xl-0 it reducesto the w-front

B_;_+;.

Defining a meanposition,

,

^{and an orderparam-}

eter, )4,by

theone-parameter family of solutions,

{B(x; , X)

) C

R}, where/)(x; , X) B(x;

x,

x2),

represents

rc/2-front

pairs with distances, 2X, ranging from zerotoinfinity.

For

I%- 1/21- ^{Id[ <<}

1, the weak coupling between the two Eqs.

(21a)

and (21b) ^induces slow drift along the solutionfamilyB(x;xl,

x2).

A pair solu- tion is now written as

(23)

whereuandvarecorrectionsof orderd.Equations of motion for _xl and _{x2 or}for and follow by inserting these forms in Eqs.

(21a)

and (21b) and applying solvabilityconditions atorder d:

(24)

V-

--

^d

^J(z)

^dz,

⁽²⁵⁾

where

J(x) 6( a- a-3)

^q-

(1 3a-2)G(a),

G(a) (1- a-2) ln(l ⁺ aa)

with a tanh

2X.

^Notethat

^{Eqs. (24)}

^and

⁽²⁵⁾

^are

valid to all orders inX and^{to linear order}around

%-1/2.

^{Figure 9 shows the potential}

^V(X9

^ford>0

(54>1/2)

and d<0. There is only one extremum point,;V 0, ofV. Ford

>

^{0 it isaminimum}and X convergestozero.Pairsof

rr/2-fronts

^{witharbitrary}

initialseparation, x2-x, attract oneanother and

V o.00

-2 0 2

FIGURE9 Thepotential V(X). (a) For d>0 the extremum at X=0 ^{is a minimum} and X converges ^to 0 (a w-front).

(b)For d<0the extremum is amaximum and X diverges^to

-+-oc(isolatedpair ofrc/2-fronts).

eventually collapse to a single re-front (xl x2 or X

0).

Inpractice,thecollapse processisnoticeable only for relatively small separations. For d<0 the extremum point, X 0, is a maximum and diverges to

-+-oc.

Are-frontdecomposesinto apair of

re/2-fronts

^{which repelone anotheras shown in} Fig. 8 for the nongradient system

(3).

In the gra- dient case both re and

re/2-fronts

^{are stationary} (inthe absenceofinteractions). ^Sincethe potential

V(X)

becomes practicallyflat at finite values, the pairof

7r/2-fronts

^do not seem to departfrom one another at long times. Figure 7 shows the decom- positionprocessofar-front in thecomplexBplane.

Starting with the B_l-+l re-front, represented by the thick solid phase portrait, the time evolution (thin^solidphase portraits)istoward thefixedpoint

B+i

and the dashed phase portraits representing

the pair of

re/2-fronts B+I-++i

^and

B+i--l.

^Because

of the parity symmetry -X, an appropriate perturbation of the initial

B-1-+1

^{re-front could}

haveled thedynamicstoward thepair

B+l--i

^and

B-i-+-l.

^{Notice that for d=0, 0,}

2

^{0, and}

we recover the two-parameter family ofpair solu- tions

B(x; , ^X)

^{witharbitrary and} X.

The derivation ofEqs.

(24)

and

(25)

^can easily be extended to the nongradient case assuming u0, a and

/3

^are small. The X equation ^remains unchanged.The (equationtakesthe form

02

IyFu(x.

^q--

oFa()(.)

^q--

flFfl()), (26)

o

whereF,

F

^and

F

^areoddfunctions ofX and^do not vanishwhend= 0

[25].

When

Ixl-

^theright

handside of

(26)

converges to

-(u0-/3),

^{the speed}

of a

re/2-front

^{solution ofEq.}

(14).

The X ^{0 solu-} tion (representing ^a

re-front)

remains stationary

(4- 0)

ⁱⁿ the nongradient case as well. At _’74

(d-0)

itlosesstability and decomposesinto apair of

re/2-fronts

^{whichapproachthe}asymptotic speed Thedegeneracy ofsolutions at

"74-

^{is liftedby}

adding higher order terms to the amplitude equa- tion

(14).

These terms are smaller by a factor of

#

<<

than the terms appearing in

(14)

and their effect is noticeable only in a /z-neighborhood of

"74-1/2. ^Apart

^{from this small}parameter range the overall behaviordoesnotchange [25].

B. Implications onPatternFormation

Like in the 2:1 resonance the re-front instability designates atransitionfrom frozen standing^waves totraveling^waves. The coexistence ofre-fronts and re/2-fronts, however, allows inprinciple both two- phaseandfour-phase patterns.The re-front decom- position instability contains information not only about the transition from standing to traveling waves, butalsoabout the parameter regimes where two-phaseandfour-phasepatternsareexpected to be seen.For_’74

> 1/2

the interactionbetweenapair of propagating

re/2-fronts

is attractive as indicated

by the single minimum of the potential

V(X) (see

Fig. 9). The minimum at X⁼⁰ corresponds to a 7r-front and according to Eq.

(26)

the 7r-front is stationary. As a result, traveling four-phase pat- ternsconsisting of

7r/2-fronts

convergetostationary two-phase patterns consisting of 7r-fronts. When

")/4

<

^the interaction between

r/2-fronts

^becomes repulsive. Stationary r-fronts, corresponding to the maximum of

V()

at =0, are unstable and decompose intopairs oftraveling

r/2-fronts.

^Asa result stationarytwo-phase patternsdestabilize and evolve into four-phase traveling patterns.

Figure

10(a)

shows ^astably rotatingfour-phase spiral wave for ")/4

< 1/2"

^Figure

^10(b-d)

^{show the}

collapse ofthisspiral wave into a stationary two- phase patternas’4 is increasedpast

.1/2.

^Thecollapse

begins atthe spiral ^corewherethe

7r/2-front

^inter-

actions are the strongest.

As

pairs of

7r/2-fronts

attract and collapse into r-fronts, the core splits into two vertices that propagate away from each other leavingbehindatwo-phase pattern.

V. HIGHER2n:1 RESONANCES

Numerical studies of amplitude equations for higher resonances suggest the existence of --front decompositioninstabilities in2n resonanceswith n

>

^1.Thefollowing generalization hasbeenconjec- tured:withinthe 2n tongue

(n >

1),upon decreas- ing"Y2n,a7r-frontmaylosestabilityby decomposing inton propagating

r/n-fronts.

^{Considerfor exam-} plethe equation

Bt 1/2 Bxx + (1 +

^iuo)g

+ #41B 2B

-+- #61BI4B ⁺

^"y4

^B*3

^q-,y6

^{B*5. (27)}

The normal form equation up to fifth order con- tainsmanymoretermswhose coefficients were set to zero for simplicity. Figure 11 shows the decom- position in the complex B plane of a 7r-front withinthe6 tongue

(76 - ⁰⁾

^{into three}

rr/3-fronts.

Figure 12 shows a space-timeplot of the decom- position instability within the 6:1 tongue. The initial unstable re-front decomposes into three

d

FIGURE 10 Numerical solution ofa two-dimensional ver- sionofEq. (14)showingthecollapseofarotating four-phase spiral-wave into a stationary two-phasepattern. The left col- umn is [A[ ^{and the} right column arg(A) in the x-y plane.

(a) The initial four-phase spiral ^wave (computed with")/4<

1/2)"

(b) The spiral core, a 4-point vertex, splits into two 3-point vertices connected by ^a r-front. (c) A two-phase pattern developsasthe 3-pointvertices further separate. (d)The final stationary two-phase pattern. Parameters: ")/4 0.6, u0:0.1, oz--/3 0,x [0,64],y [0, 64].

7r/3-fronts, traveling to the left or to the right depending on initial conditions. Along with this process two intermediate phase states appear between the original white and black phases. A similardecomposition instability has been observed withinthe 8:1 tongue.

,’7 0.0

-1.0

@

-1.0 0.0 1.0

R(A)

FIGURE 11 Phase portraits of front solutions atsuccessive time steps (thin ^solid lines) showing the decomposition of a re-front(thick^solidline)^{into three}rc/3-fronts (dashedlines)ⁱⁿ the 6:1 resonance tongue. Parameters in Eq. (27): %=0.9,

/*4 1.0,/*6 1.0. All other parametersare zero.

0 2 4 6 8 10 12 0 2 4 6 8 10 12

X X

300

200

1oo

FIGURE 12 Decomposition of a --front into three rc/3- fronts in the 6:1 resonance band. The figures show space- timeplotsofarg(B) usingnumerical solutionsofEq. (27)with parameters _% 0.9, /*4 1.0, /*6 1.0, u 0.1. All other parametersare zero.

VI. THEORY VS.EXPERIMENT

The main difficulty in confronting the available theory with the experiments on the forced Belousov-Zhabotinskyreaction is that the experi- ments were carried out far from the Hopf bifur- cation while the theory is valid only close to the bifurcation.

At

high forcing strengths, within the 2:1 tongue, standing two-phase patterns were observed. This observation is consistent with the behavior in the Ising regime that the theory pre- dicts. Atsufficiently lower forcing strengthsatran- sition to traveling waves is observed [16], ^as the theory predicts ^too.

However,

no indications for Blochphase frontsandphased lockeddomainshave sofar beenfound.This behavior may be attributed to the relaxational nature of the oscillations far from the Hopf bifurcation, and consequently to large phasegradientsthat develop. These gradients mayprevent convergenceto the two uniformphase states atlowforcing strengths.

Direct studiesof theNIBbifurcation havebeen carried outonliquidcrystals subjected to rotating magnetic fields [17,18]. ^This system, like forced oscillatory media, ^canbe modeled by

Eq. (3)

^with

n=2 [17]. Experimental observations of decom- position instabilties of re-fronts have not been reported so far. The lowest resonance to display suchaninstability,4 1,wasbeyondthescopeof the experiments ^onthe forced Belousov-Zhabotinsky reactionreportedin

[19].

Someof theexperimentalobservationsshown in Fig. ^can nevertheless be accounted forusingthe available theory. The absence of multiple stable phase states within the 1:1 resonance does not allow for domainpatterns as inhigherresonances (althoughnonuniformphase dynamics arising from phaseinstabilitiesoftheuniform statemay

occur).

This is consistent with the uniform oscillations shown in the first pair offrames on the left. The two phase states that coexist within the 2:1 reso- nance allow for two-phase patterns as shown in the third and fourth pairs of frames

(from left).

These patterns ^are standingwaves suggesting that these observations weretaken athighillumination

intensities within the Ising regime. The two other patterns shown in Fig.

(second

and fifthpairs of

frames)

correspondtothe 3:2and 3 resonances.

In both cases three uniform phase states coexist.

Three phase patterns have been observed within the 3:1 resonance only. Theyconsistof successive domains with shiftsof

2-/3

^intheoscillationphases.

The boundariesof thesedomains

(27r/3-fronts)

^were found to drift very slowly. In other experiments withinthe 3 tongue traveling threephase patterns were observed [16]. Numerical studies ofEq.

(3)

with n- 3 have indeed foundthreephase traveling waves(spiral

waves)

[14].

Some other experimental observations are not yet understood.Thebubble patternswithinthe 3:2 resonance(secondpairofframesinFig.

1)

^{are not} simple three-phase patterns although the power spectrum at any spatial point shows well defined peaks atmultiples of

cf/3.

^{The bubbles randomly} appearanddisappearandthemechanismthatgov- ernsthis behavior is not known. The transition to labyrinthine patternswithinthe 2 resonanceasthe forcingfrequency,cf, is increased is also notfully understood. Simulations of a forced Brusselator model within the 2:1 resonance reproduced the transition tolabyrinthine patterns[33,34].The tran- sitionhas been attributed toatransverseinstability ofanIsing front.Transversefront instabilities lead- ing to labyrinthine patterns through fingering and tip spliting have been found earlier in bistable (unforced) reaction-diffusion models [26-28]. A neccessarycondition is repulsive frontinteractions that rule out merging ofgrowing fingers, but the origin of such interactions in the context ofEq.

(3)

is not clear. The decoration that appears on the experimental

2-/3-fronts

^{within the}3:1 resonance suggestthe existence ofatransverseinstability but notheoretical accounthas yetbeenoffered.

There are also theoretical predictions that have not been tested yetin experimentslikethe --front decomposition instabilityinthe4:1 resonance and itsimplicationsonpatternformation. Inthe vicin- ity of^theNIBbifurcationcomplex spatio-temporal behaviorinvolving spontaneous nucleationofspi- ralwavesmayarise[27,29]. Atheoretical accountof

this behavior has been given in the context of activator-inhibitor systems [30-32]. Complex spatio-temporal dynamicsnearthetransitionfrom standing to traveling waves within the 2:1 reso- nance have also been found experimentally in the periodically illuminated Belousov-Zhabotinsky reaction[16], butnoattempthas yetbeen madeto interpret these observations.

Vll. CONCLUSION

Ihavepresentedhereashortreviewof recent stud- ies of extended dissipative oscillatory media sub- jectedtotemporal periodic forcing.Mostattention hasbeengiventothe analysis ofre-front instabilities within even 2n:l resonances and to the implica- tions theybear on pattern formation. Despite the theoretical progress described in this review many aspects ofperiodically forced oscillatory systems are still not understood. The discussion of experi- mental findings in the previous section points toward^afewofthem. Inparticular, theeffect ofthe distance to theHopfbifurcationhas to bestudied.

Additional open questions include: (i) dynamics withinoverlappingresonanceswhichbecomes rele- vant athigh forcingstrengths, (ii)forcingatmulti- ple frequencies, e.g. periodic forcingthat contains harmonics and (iii) coupled front-phasedynamics (involving instabilities of uniform phase states) which may shed light on the onset of spatio- temporalchaotic behaviors.

I did not discuss here periodic forcing ofnon- oscillatory extendedmediasuchasexcitable media [35]^andgranular systems[36-39]. Nor^didIdiscuss conservative ornearlyconservativesystemsthat are periodically forced[40-44].All these systemsshare commonfeatures but alsodiffer inmany aspects.

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