andrei goryachev, raymond kapral and hugues chate- synchronization defect lines

8/3/2019 Andrei Goryachev, Raymond Kapral and Hugues Chate- Synchronization Defect Lines

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International Journal of Bifurcation and Chaos, Vol. 10, No. 7 (2000) 15371564c World Scientific Publishing Company

SYNCHRONIZATION DEFECT LINES

ANDREI GORYACHEV and RAYMOND KAPRALChemical Physics Theory Group,

Department of Chemistry,University of Toronto,

Toronto, ON M5S 3H6, Canada

HUGUES CHATECEA Service de Physique de lEtat Condense,

Centre dEtudes de Saclay, 91191 Gif-sur-Yvette, France

Received October 22, 1999

Spatially distributed reactiondiffusion media where the local dynamics exhibits complex os-cillatory or chaotic dynamics are investigated. Spiral waves in such complex-oscillatory orexcitable media contain synchronization defect lines which separate domains of different oscil-lation phases and across which the phase changes by multiples of 2. Such synchronizationdefect lines arise from the need to reconcile the rotation period of a one-armed spiral wavewith the oscillation period of the local dynamics. We analyze synchronization defect lines andshow how to classify them and enumerate their types. In certain parameter regions the spa-tially distributed system exhibits line defect turbulence arising from the nucleation, growthand destruction of defect lines. The transitions to line defect turbulence may be characterizedby power law behavior of order parameters and may be described as nonequilibrium phasetransitions.

1. Introduction

Reactiondiffusion media with local excitable oroscillatory elements support a variety of chemicalwaves which have been investigated extensively be-cause of their relevance for a number of biologicaland chemical processes [Kapral & Showalter, 1996;Mikhailov, 1994; Walgraef, 1997]. An importantfacet of nonlinear wave patterns, especially in oscil-

latory media, is that they organize the synchroniza-tion of local oscillations.

Many naturally occurring oscillations havecomplicated temporal patterns. Well-known exam-ples are mixed mode oscillations, period-doubled os-cillations or even chaotic oscillations. In this paperwe describe chemical waves and pattern formation,as well as issues related to the synchronization ofoscillations, in spatially distributed media wherethe local dynamics has a complicated periodic or

chaotic structure. Our focus is on the existence,structure, stability and dynamics of spiral waves,which are one of the most commonly observed pat-terns in the laboratory and in nature.

1.1. Simple spiral waves

Pulse-like waves of excitation propagate in excitablemedia when the stable stationary state experiences

a perturbation that exceeds a threshold. If the frontof such a wave is broken the free ends will curl toform spiral waves which occupy the entire mediumand persist as self-sustained patterns. In oscilla-tory media spiral waves arise from the diffusion-mediated synchronization of concentration oscilla-tions in neighboring spatial p oints. A single spiralwave in an oscillatory medium is shown in Fig. 1.

The spiral wave in Fig. 1 is an example of a sim-ple spiral wave. The concentration profile at time t

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1538 A. Goryachev et al.

Fig. 1. A spiral wave in a disk-shaped oscillatory mediumwith no-flux boundary conditions and dynamics describedby the Rossler reactiondiffusion equation in the period-1regime. The cz(r, t) concentration field at a single time in-stant is displayed.

of such a spiral wave can be obtained from a rota-

tion of the concentration field at time t0 throughsome angle. The wave propagation can be con-sidered as a rotation of a stationary concentrationprofile around the spiral core. The local concentra-tion varies periodically with time at arbitrary pointsof an oscillatory or excitable medium supporting aspiral wave. Although the mechanisms leading tothe periodicity are different, in either case the lo-cal dynamics, represented as an orbit in concentra-tion phase space, has the form of a simple closedloop which can be projected onto a two-dimensionalphase plane without self-intersection. We shall call

this period-1 dynamics to distinguish it from morecomplex periodic patterns discussed b elow. In sucha circumstance, it is always possible to introduce aphase variable which uniquely parameterizes theorbit. The phase field (r, t) for a medium witha spiral wave contains a point topological phasedefect [Mermin, 1979] in the spiral core such thatthe integral of the phase field gradient taken alongany contour encircling the defect yields a nonzerovalue,

1

2(r, t) dl = nt . (1)

The signed integer nt is the topological charge of thedefect. The net topological charge of a medium withperiodic b oundary conditions is invariant; there-fore, the defects are born or annihilated in pairswith opposite charge. In media with no-flux bound-aries, the birth and destruction of defects mayalso occur on the boundaries; however, the sumof the topological charges of the medium and the

boundary is constant. Although nt can take on anyinteger value, we shall be concerned with commonlyobserved cases where only defects with nt = 1corresponding to one-armed spiral waves are stable[Hagan, 1982].

1.2. Complex oscillationsTogether with simple period-1 dynamics, almost allknown oscillatory reactions also display complex-periodic, aperiodic or chaotic behavior [Scott,1991]. A complex-periodic oscillation is character-ized by many peaks per full period if viewed as atime series of a single concentration variable and,in a suitably chosen concentration phase space, itcorresponds to an orbit which loops several timesbefore it closes onto itself. Figure 2 shows exam-ples of complex oscillations commonly seen in the

laboratory and as the solutions of model reactionequations. Period-doubled oscillations have periodTn 2

nT0, where T0 is the period of the orbit whichspawned the period-doubling cascade, and are char-acterized by 2n loops in a phase space representa-tion of the orbit [cf. Figs. 2(a) and 2(c)]. One period

a

c

b

Fig. 2. Examples of complex-periodic oscillations: (a) aperiod-4 oscillation in a system with a period-doubling cas-cade; (b) 13 mixed-mode oscillation; (c) a period-4 oscillationon a torus. Left panels show phase portraits of orbits in athree-dimensional phase space while the right panels showthe time series of a single dynamical variable.


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Synchronization Defect Lines 1539

of mixed mode nm oscillations (MMO) consists ofn consecutive large-amplitude loops followed by msmall-amplitude loops [cf. 2(b)]. We shall be con-cerned with situations where the local dynamics ina spatially-extended system exhibits such temporalpatterns.

Complex-periodic dynamics is found also inexcitable media. In some excitable systems re-

excitation is possible before the previous relaxationcycle is complete but the response produced bythe excitation is weaker. Since the execution ofa smaller relaxation loop generally takes a shortertime, by the time the next excitation pulse arrivesthe medium has recovered completely and a nor-mal response is produced. Then the entire patternrepeats. As a result, the system shows excitable be-havior with alternating amplitude and the effectiveperiod of the local dynamics doubles. Such a phe-nomenon is observed, for example, in cardiac tissue[Chialvo et al., 1990] and is termed supernormal ex-citability. To illustrate this phenomenon consider amodel excitable system of the form [Barkley, 1991;Giaquinta et al., 1994]

u

t=

1

u(1 u)

u

v + b

a f(v)

+ D2u ,

v

t= u v ,

f(v) = exp

(v v0)2

2

,

(2)

where v is a nondiffusive variable and both u andv are functions of time and space. The excitabledynamics of system (2) consists of two fast and twoslow stages. If displaced from the stable fixed point(u = 0, v = 0) to the right of the unstable branchof the nullcline u = 0, it quickly reaches upper sta-ble branch u = 1. It follows this branch until vreaches sufficiently large values and then jumps tothe lower stable branch u = 0, along which it slowlyrelaxes to the stationary state. To add a supernor-mal excitability to (2) the unstable branch of thenullcline u = 0 is modified by the function f(v)

so that for suitably chosen parameters , and v0it nearly touches the nullcline v = 0 (see Fig. 3).As a result, if another excitation is applied to thesystem (2) before it reaches the stationary state, itmay execute a second, smaller excitable loop beforeit finally returns to the stable state.

The spatiotemporal dynamics of system (2) wasstudied in a one-dimensional array forced by an ex-ternal pacemaker with varying period [Giaquinta

Fig. 3. Phase plane (u, v) plot of a 12 type trajectory cal-culated at a p oint in a super-excitable medium. Two stablebranches (u = 0, u = 1) and one unstable branch of the null-cline u = 0 are shown as solid lines and the nullcline v = 0by a dashed line.

et al., 1994]. Nontrivial behavior consisting ofmixed-mode wave forms 1m was observed for cer-

tain values of the forcing. We will show below thatinstead of an external pacemaker one can use a spi-ral wave as a self-sustained source of excitation inthe medium to induce such dynamics.

Complex or chaotic oscillations are quite com-mon and expected to be generic for systems withmore than two local scalar fields. Little is knownabout the spatial patterns and the spatiotemporaldynamics of media with such local oscillations. Inthe following, we show that in most circumstances

the basic spatial patterns still exist and, in particu-lar, spiral waves are easily observed; however, newfeatures appear giving rise to qualitative changes inthe overall synchronization of the medium.

To describe these systems a number of newterms need to be defined and some well-establishednotions, such as oscillation phase, require gener-alization. In Sec. 2 we describe the basic phe-nomenon, synchronization defect lines, which havebeen observed in experiments and numerical sim-ulations. Next, a phase formalism is developed todescribe these defect lines and demonstrate the ne-

cessity for their formation in complex-periodic me-dia. The study of synchronization defect lines iscontinued in Sec. 4 which considers a period-4 os-cillatory medium in some detail. Section 5 devel-ops an approach to the classification of defect linesand presents a complete solution to this problem forperiod-doubled media with period-2n local dynam-ics. The role of defect lines in the transition fromcomplex-periodic to chaotic dynamics is considered


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in Sec. 6. Finally, our conclusions are summarizedin Sec. 7.

2. Synchronization Defect Lines

Consider a system described by the reactiondiffusion equation

c(r, t)

t= R(c(r, t)) + D2c(r, t) , (3)

where c(r, t) is a vector of time-dependent concen-trations at point r in a two-dimensional physicalspace, D is the diffusion coefficient and the localkinetics is specified by the vector-valued functionR(c(r, t)). As a specific example of a reactiondiffusion system with complex local dynamics wetake R(c) to be given by the Rossler model [Rossler,1976] with Rx = cy cz , Ry = cx + Acy, Rz =

cxcz Ccz + B. The behavior of this system wasinvestigated for C in the interval [2., 6.] with theother parameters fixed at A = B = 0.2. We ob-served that the spatially-distributed medium un-dergoes period-doubling bifurcations. Simulationsdesigned to characterize the period-doubling tran-sitions were performed on a disk-shaped domain(R = 256, D = 1.6) with a single spiral wave inthe center [Goryachev & Kapral, 1996a; Goryachevet al., 1998]. The spiral wave concentration profilein the period-1 regime was taken as the initial condi-tion. The parameter C was gradually incremented

and a long equilibration time was allowed for the dy-namics to reach the new system state at each newvalue ofC. This Rossler reactiondiffusion mediumwith a spiral wave undergoes a first period-doublingbifurcation at C = C1 3.03. Within the period-2domain, the spiral wave acquires a global structurethat differs from that in a simple periodic medium.Figure 4 shows the cz(r, t0) concentration field at asingle time instant t0.

The most prominent feature of this spiral wavewith period-2 local dynamics is the presence of aline, labeled in Fig. 4, which connects the spi-

ral wave core to the boundary. Along this line thepattern is displaced by one wavelength. While thewave propagates, the dislocation line remains sta-tionary and its position is traced by the free endsof the propagating high-amplitude wave segments.The dislocation line is nearly straight with a gen-tle bend where it approaches the spiral wave core.The concentration time series at any two nearbypoints on opposite sides of this line are shifted in

Fig. 4. Spiral wave in the Rossler medium with no-fluxboundary conditions and period-2 local dynamics at C =3.84. Concentration field cz(r, t) is shown as gray shades.The solid line depicts the curve. Dynamics on the smallarc segment that transversally cuts the curve will be de-scribed below.

time relative to each other by one period of rota-

tion of the spiral. The passage of a high-amplitudewave maximum through one observation point issynchronized with the passage of a low-amplitudemaximum through the other and vice versa. Wecall this line a synchronization defect line. Two ar-bitrary states of the spiral wave evolution cannotbe transformed into each other by simple rotationof the entire concentration field. After one turn ofthe spiral the high and low maxima interchange andit is only after two spiral revolutions that the con-centration field is restored to its initial value.

Synchronization defect lines were studied

also in the WillamowskiRossler [1980] reactiondiffusion system where period-doubling in spiralwave dynamics was observed [Goryachev & Kapral,1996a]. In this system the line connecting thespiral core to the medium boundary had a spiralgeometry and slowly rotated with a frequency muchsmaller than that of the oscillation frequency. All ofthe other characteristic features of synchronizationdefect lines described above were the same as thoseseen in this reactiondiffusion model.

Synchronization defects are not confined tosystems where the complex dynamics arises from

period-doubling and were found also in the modelsystem with super-normal excitability describedearlier [cf. Eq. (2)] [Goryachev & Kapral, 1999].The parameters and , which control the sizeof the bell-shaped hump on the nullcline u = 0,were varied to change the preference for forma-tion of low-amplitude waves over high-amplitudewaves. For small and the medium formednormal, high-amplitude spiral waves with 10 local


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Fig. 5. Spiral wave with period-3 dynamics. Color-codingof the v field is indicated in Fig. 3.

dynamics, while for large and the relaxationof excitation always took a short path and low-amplitude 01 waves with a short wavelength wereobserved. Complex-excitable dynamics was foundin the parameter region between these domains. Inthis region the medium supports mainly aperiodic,stable spiral waves. Pure period-3 dynamics wasfound in a subdomain of this region.

Figure 5 shows a spiral wave in this period-3parameter region. The entire concentration fieldslowly rotates around the spiral core with a con-

stant angular velocity . The period-3 dynamicsis manifested in a coordinate frame rotating withangular velocity and centered at the spiral core.Indeed, it takes three rotations of the spiral for theconcentration field to return to itself. In Fig. 5, oneclearly sees two dislocation lines emanating fromthe spiral wave core and subtending an angle of ap-proximately 1800.

2.1. Experimental observationsof defect lines

Synchronization defect lines have been observed ex-

perimentally in a spatially heterogeneous BZ reac-tion system consisting of thin (0.7 mm) layer of aMn-catalyzed BZ reaction mixture with a mono-layer of the surfactant-derivative of the ruthenium-bipiridil complex on its top [Yoneyama et al., 1995].Spiral waves were created by mechanical disruptionof chemical waves spontaneously emerging at thedish boundaries and the dynamics of the waves onthe surface of catalyst and in the bulk of the solution

was monitored using fluorescence microscopy. Scrollwaves, whose propagation on the surface and in thebulk showed no differences, were observed in a ni-trogen atmosphere above the solution.

When the nitrogen atmosphere was replaced byair, the spiral on the surface developed collapsedwave segments with subthreshold amplitude whichappear on fluorescent images as completely blank

spaces. These collapsed segments, which are cre-ated at the tip of the spiral, propagate to the bound-ary as the wave unwinds from the tip. The scrollwave rotating in the underlying solution had uni-form amplitude and, thus, propagation failure oc-curred only on the catalytic surface. The frag-mented surface spiral wave preserved its integrityand free ends of normal fragments exhibited notendency to curl with the formation of new spi-ral waves. In the initial stages of the evolutioncollapsed fragments were formed randomly and noorder could be seen in the succession of the twotypes of wave front. However, with time the dy-namics locked into the 11 regime with strict alter-nation of high- and low-amplitude waves so that thesuprathreshold response on the surface occurred onevery second passage of the wave front in the under-lying solution. As a result the wave on the surfacedeveloped into the wavelength-doubled form shownin Fig. 6.

One immediately recognizes a period-doubledspiral waveform with characteristic alternation ofhigh and low maxima and clearly visible synchro-

nization defect line like that shown in Fig. 4. Theconfiguration in Fig. 6 is stable and was observed inthe experiment for more than 30 min until the con-centrations of consumable chemicals in the mediumfell below some critical value and it was no longerpossible to detect fluorescence of the wave.

Very recently, a set of controlled experimentson the BZ reaction under different conditions wascarried out by Park and Lee [1999] with the aimof observing synchronization defect lines and de-termining some of their properties. The wave pat-terns were seen in a continuously fed unstirred re-

actor composed of a 0.65 mm thick polyacrylamidegel membrane sandwiched between two well stirredreservoirs of reagents. Patterns can be controlledby varying the compositions and flow rates of thereservoirs and can be observed for arbitrarily longtimes since the reagents are continuously refreshed.

For certain composition values and flow ratesthey observed a complex pattern of spiral waveswhich, upon analysis, was shown to comprise


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Fig. 6. Spiral wave with synchronization defect line on thesurface of the ruthenium catalyzed BZ reagent. Panel (b)shows continuation of the defect line seen in panel (a). Re-produced with permission from [Yoneyama et al., 1995].

period-2 and period-3 spirals along with domains

with more complicated dynamics separating thesespiral wave regions. Both the period-2 and period-3 spirals exhibited synchronization defect lines, insome cases more than one defect line. For exam-ple, the period-2 spirals had either one or three de-fect lines. The defect lines themselves had spiralgeometry and slowly rotated with some angular ve-locity, similar to those observed in simulations onthe WillamowskiRossler reactiondiffusion system[Goryachev & Kapral, 1996a].

We shall comment further on additional fea-tures seen in these experiments below since the

observations of Park and Lee also provide experi-mental evidence for line-defect turbulence which isdiscussed in Sec. 6.

2.2. Structure of synchronizationdefect lines

The structure of the period-2 spiral wave inFig. 4 may be analyzed by introducing a polar

Fig. 7. The change of the local dynamics across the curve.Upper panel shows 1cz() along the arc at radius r0 = 130indicated in Fig. 4 for three values of C: (a) C = 3.04,(b) C = 3.10, (c) C = 3.84. Bottom: two cz(t) concentrationtime series calculated at points r1, 3 = (r0 = 130, 0.05)on opposite sides of the curve.

coordinate frame (r, ) with the origin at the coreof the wave. To quantitatively describe the sharpchanges in the local dynamics across the synchro-nization defect line, one needs to find a local orderparameter able to distinguish between the variousloops of the local dynamics. In the present case, itis convenient to take advantage of the nature of thetime series cz(r, t) and define the local order pa-rameter as 1cz(r), the difference between two suc-cessive maxima of cz(r, t). Figure 7, upper panel,shows 1cz() for three values ofCas function of thepolar angle along the arc segment that crosses the

curve. To a very good approximation, the numer-ical data show that 1cz(r, , C) varies like

cz(r, , C)

A(r)

C C1 tanh[r( )(C C

1 )] ,

(4)

where is the angular position of the curve and is a numerical factor. The overall


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Fig. 8. Loop exchange in local orbits as the curve iscrossed. Right column displays the cz(, t) time series cor-responding to the orbits in (cx, cy, cz) phase space shown inthe left column. To highlight the exchange the first half ofan oscillation period for each orbit is colored differently fromthe second half.

amplitude radial profile A(r) is similar to that ofa spiral wave in a simple oscillatory medium. The

C C1 factor reflects the supercritical nature ofthe period-doubling bifurcation. The curve isthus an exponentially localized object whose widthremains approximately constant with r and varieslike 1/(C C1 ). As C approaches C

1 from above,

the width increases until it becomes comparableto 2 and the curve ceases to be a well-definedobject.

To demonstrate that the curve corresponds toa one-dimensional synchronization defect, two cz(t)concentration time series calculated at nearby spa-tial points on either side of the curve are shownin the bottom panel of Fig. 7. The two oscillationsare shifted relative to each other by half a period.The nature of this shift can be understood from theobservation of the local orbit loop exchange whichoccurs on the scale of the curve width. Figure 8

shows local orbits calculated at five different pointson the arc shown in Fig. 4 with r0 = 130 and mono-tonically increasing .

As one traverses the arc, the larger, outerloop of the local orbit constantly shrinks while thesmaller, inner loop grows. At = , both loopsmerge and then pass each other exchanging theirpositions in phase space. (Compare top and bot-

tom panels of Fig. 8.) The behavior of the 1cz()order parameter along the arc provides only lim-ited information on the loop exchange. As onesees from Fig. 8, at the exchange p oint = not only are the cz maximum values of two loopsequal (1cz() = 0), but the entire loops coincidein phase space and the local oscillation is effectivelyperiod-1.

Similar transformations of local orbits are ob-served in the excitable system shown in Fig. 5 asthe defect lines are crossed. Figure 9 shows thev(r, t) time series at four neighboring locations inthe medium along a path intersecting one of thedefect lines. Panel (a) is a plot of the normal 12

dynamics seen on one side of the defect line. Everythird minimum is lower than the two preceding min-ima and corresponds to the larger relaxation loopin the phase space plot. As one approaches thedefect line, the large-amplitude loop shrinks whileboth small-amplitude loops grow [Fig. 9(b)]. Then,one small-amplitude loop begins to grow faster thanthe other and at some point becomes larger thanthe still shrinking large-amplitude loop [Fig. 9(c)].

Fig. 9. Change in character of local dynamics across one ofthe defect lines seen in Fig. 5. See text for details.


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This loop exchange process continues until the newlarge loop attains a size equal to that of a large-amplitude loop and the other two loops shrink tothe size of the small-amplitude loop. The net resultof this transformation is that the local time serieson opposite sides of defect line appear to be shiftedin time relative to each other by one third of thefull period.

3. Phase Description ofSynchronization Defects

The phenomena described in the previous sectionappear in systems with diverse reaction kineticswhere complex oscillatory or excitable dynamicsarises through completely different mechanisms.Nevertheless the structures of the spiral waves inthese systems are similar, indicating the existenceof a general organizing principle independent of the

particular reaction mechanism. We now explainwhy defect lines are created to reconcile the rotationperiod of the spiral wave and the complex period ofthe local dynamics. The arguments presented hereapply to a general complex-periodic system and arebased on the observed local and global dynamics ofthe medium and not on the details of the local reac-tion mechanism. Therefore, the discussion appliesto oscillatory as well as driven and autonomous ex-citable systems with spiral waves.

To approach this problem one needs to considerhow local complex-periodic or chaotic dynamics is

synchronized by diffusion into a global coherent pat-tern in the form of a spiral wave. The notion ofphase plays a key role. In the p eriod-1 regime thephase is an explicit, single-valued function of theconcentrations = (c) and uniquely parameter-izes a local orbit. After the first period-doublingthis parameterization is no longer unique since theorbit does not close on itself when changes by2. The definition of phase for complex-periodicand chaotic oscillations is a difficult problem[Rosenblum et al., 1996] and several possible defini-tions for the phase in such systems have been pro-

posed. [Osipov et al., 1997] For all such extendeddefinitions the phase is no longer an observable sinceit cannot be calculated as a function of the instanta-neous concentrations. In this section we generalizethe definition of phase and show that although theabsolute value of phase remains an arbitrary quan-tity, the phase difference between two oscillations ofthe same periodicity can be defined in a meaningfuland robust way.

3.1. Phase of complex-periodicoscillations

Consider the orbit of a complex-periodic oscillationin a three-dimensional concentration phase space.We assume that the phase (t) of an arbitraryperiod-n oscillation is a scalar function of timewhich increases by 2n for each full period of the

oscillation, Tn. In many cases can be defined interms of an angle variable in a suitably chosencoordinate frame. The phase variable [0, 2n)then takes the form

= + 2m , (5)

where m is an integer with values from 0 to n 1.For an n-periodic orbit, n of its points lie in a semi-plane = 0 for any 0 [0, 2) (cf. Fig. 10)and the value ofm serves to distinguish points withidentical . While is a single-valued function

of the original dynamical variables, the phase ofa complex-periodic oscillation is not an observablesince a knowledge of the entire orbit is required inorder to calculate m. At a p eriod-doubling bifurca-tion, the parameterization given by the phase vari-able changes discontinuously as m takes on twiceas many possible values. This feature of the defi-nition does not result in artificial discontinuities ofany physical quantity since only the nonobservablepart 2m undergoes a jump. Figure 10 illustratesthe definition of phase for a period-2 orbit. Be-fore the trajectory closes on itself the radius-vectorexecutes two full turns and the orbit is naturallyparameterized by [0, 4).

Fig. 10. Period-2 orbit in the (cx, cy, cz) concentrationphase space. Two points A and A lying in the semiplane = 0 have phases A = 0 and A = 0 + 2 which differby 2.


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The phase (t) of an n-periodic oscillation maybe written as

(t) = (0) +

t0

()d , (6)

where

() = A0 +

k=1

Ak cos

2k

Tn + Bk sin

2k

Tn

is a Tn-periodic function of time with average A0 =2n/Tn. Performing the integration in (6) we canrewrite (t) as

(t) = (0) + A0t + C+ (t) ,

C =Tn2

k=1

Bkk

,(7)

where (0) and C are constants depending on theinitial conditions and the definition of (), and(t) is a Tn-periodic function with zero average.

Consider two oscillations with the same pe-riod Tn described by the same orbit in phase spacebut shifted in time such that (t) = (t). SinceA0 = A

0 = 2n/Tn, for the phase difference (t) =

(t) (t) we get

(t) = (0)(0)+Tn2

k=1

Bk B

k

k+(t)(t) .

(8)

Averaging (8) over Tn gives

=1

Tn

Tn0

[(t) (t)]dt

= (0) (0) +Tn2

k=1

Bk Bk

k, (9)

because both integrals of (t) and (t) vanish sep-arately. Since both oscillations differ only by a timeshift t [0, Tn) it is obvious that

(t) = (t t)and thus

(t) = (0) + A0(t t) + C+ (t t) .

Performing the average of (t) again and equatingthe result to (9) we finally obtain

= (0) (0) +Tn2

k=1

Bk B

k

k= A0t .

(10)

The quantity

= A0t = 2nt/Tn ,

unlike (t), is independent of time and correspondsto the time delay of one oscillation with respect tothe other expressed in fractions of the full-periodphase increment 2n. For harmonic oscillations ofthe form A cos(t + ) with = 2/T, coin-cides with the well-known notion of phase . Notethat does not depend on the precise definitionof the phase (t) as long as it is assumed that (t)changes by 2n in one full period of the oscilla-tion. Moreover, in Eqs. (6)(10) the phase (t) isnot assumed to be a monotonic function of timeand (t) can change its sign. We only require that(t) has period Tn and average value 2n/Tn. Thisallows one to define even in those cases wherea (, , z) frame cannot be chosen so that growsmonotonically with time and uniquely parameter-

izes the orbit. The quantity also possesses theimportant property of additivity. Consider three os-cillations described by the same orbit in phase spaceand same period Tn but nevertheless nonidentical1(t) = 2(t) = 3(t). From (10) one finds thatthe pairwise differences 12, 23, 13 satisfy therelationship

13 = 12 + 23 . (11)

From the above discussion it follows that one canuse the time averaged quantity as the phasedifference for complex-periodic oscillations and this

definition of will be used below.

3.2. Phase description ofthe medium

In a spatially-distributed system the shape of the lo-cal orbit generally depends on r and, therefore, twooscillations at points r1 and r2 cannot be matchedby simple time translation of one oscillation by t.Nevertheless it is possible to define (r1, r2) if oneassumes that the local oscillations in the mediumhave the same period Tn everywhere apart from,

perhaps, some sets of points with zero measure.Consider two points in the medium r and r anda path connecting them so that for all r thelocal oscillations have period Tn. Let us partitionthe curve into short segments by a set of pointsri , i = 0, 1 . . . l such that r0 = r

, rl = r. The

length of the segments can be chosen sufficientlysmall so that due to the continuity of the mediumthe local dynamics at the adjacent points ri and


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ri+1 is described by the same orbit and the phasedifference (ri, ri+1) is defined in the sense of theprevious section. Then (r, r) can be defined asa sum of (ri, ri+1) along . Applying (10) and(11) one finds

(r, r) =l1

i=0

(ri, ri+1)

=l1i=0

i+10

i0 +

Tn2

k=1

Bi+1k Bik

k

= l0 00 +

Tn2

k=1

Blk B0k

k. (12)

Defined as a sum of infinitesimal phase shiftsA0t(ri, ri+1), the quantity (r

, r) can be for-mally equated to A0t(r

, r) with t(r, r) being

the time translation which one should apply to theoscillation at point r, together with the simultane-ous deformation of the orbit, to make it correspondwith oscillation at point r. From (12) it followsthat (r, r) does not depend on the choice of and the partition by ri since the contributions ofall intermediate points ri cancel. Taking the con-tinuous limit in (12) gives the integral form of thedefinition

(r1, r2) =

r2r1

(r) dl , (13)

where dl is an infinitesimal tangent to . The in-dependence of(r1, r2) on the path of integrationendows it with some of the properties of a potential.On the other hand, due to the periodic nature ofthe phase, a nonzero circulation of along closedloops is allowed as long as it is equal to an integermultiple of the full-period increment 2n.

Consider the phase variable field (r, t) at atime t and a contour integral of the phase gradi-ent (r, t) taken along a closed loop which sur-rounds the core of the spiral wave. For an arbitrary

point r0 , this loop is just a path in the mediumwhich starts at r0, described by the local n-periodicoscillation phase (r, t), and returns to the samepoint with same oscillation phase (r, t). Thus, theintegral may take only values 2nk equal to multi-ples of the full-period phase increment. From (5)the phase field is given by

(r, t) = (r, t) + 2m(r, t) ,

and we find(r, t) dl =

(r, t) dl

+ 2

m(r, t) dl . (14)

For a one-armed spiral with topological charge nt =

1, at any given time t,(r, t) dl = 2 ,

regardless of the periodicity of the local dynam-ics. This follows from the fact that, by defini-tion of , the (r, t) field of a one-armed spiralwave in a medium with any periodicity is equiva-lent to that of a one-armed spiral wave in period-1medium. Given that the full-period phase incre-ment is 2 only in the case of period-1 dynam-

ics with m(r, t) 0, for period-doubled dynamicsthe integration of m(r, t) along must yield anonzero contribution to balance (14). Since m(r, t)is an integer function, its value changes discontinu-ously with time and space so that m(r, t) can bedifferent from zero only at a finite number of pointswhich should possess a special, nongeneric type oflocal dynamics. Consider the period-2 medium de-scribed in Sec. 2. There it was shown that all themedium points with such nongeneric dynamics con-stitute a continuous curve connecting the core ofthe spiral wave and the boundary, so that any con-

tour encircling the core must intersect at leastat one point. The local oscillations on opposite sidesof the curve were found to be shifted by half aperiod, which for a period-2 oscillation is equal toa 2 phase translation.

It can be shown explicitly that on the curvethe phase (r, t) undergoes a jump by 2. Ifone monitors the local dynamics while traversingthe medium along the arc segment (r = r0, [ , + ]) intersecting the curve (seeFig. 4) one finds a progression of period-2 orbitswhose loops rapidly converge to each other in phase

space. For all < the two loops can bedistinguished from each other and are parameter-ized by [0, 4). At = the loops coa-lesce and the parameterization is given by period-1 phase [0, 2). At the point = +, +0 the oscillation is again period-2 butthe loops are exchanged. Maintaining continuity ofthe parameterization (e.g. m = 0 on the smallerloop and m = 1 on the larger) one finds that


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( , t) = ( + , t) + 2 for small > 0and any t. Obviously, as 0 the discontinuity inthe concentrations can be avoided only if at = both and + 2 parameterize the same point onthe orbit. This implies a period-1 oscillation whichis indeed found at .

A simple example of the spatial distribution ofthe oscillation phase is presented in Fig. 11 which

schematically shows a period-2 medium with spi-ral wave and single curve. A circular contour is parameterized by the angle [0, 2) and(0, ) is plotted in the third dimension abovethe corresponding point on the contour. As onefollows clockwise, the phase difference (0, )grows monotonically up to = where it experi-ences sudden 2 jump and then increases continu-ously until the full-period phase increment of 4 isreached at the point of the departure 0.

Similar considerations apply to the system withsuper-normal excitability shown in Fig. 5. For themedium with period-3 local dynamics the continu-ity argument (14) requires that the total phase in-crement along is equal to 6. Taking into con-sideration that the crossing of one of the defectlines leads to a 2 phase shift, this requirementis easily satisfied by the presence of the two de-fect lines seen in Fig. 5. Indeed, the total phaseincrement = 2 resulting from the integra-tion of(r, t0) along , excluding its intersectionswith defect lines, plus the two 2 phase jumps atthe intersection points gives the expected 6 phase

increment.

Fig. 11. Advance of the oscillation phase along contour inthe period-2 medium. The local angle variable field (r, t0)of a spiral wave is schematically shown by gray shades.

4. Synchronization Defects in Mediawith Higher Periodicity: Period-4

The continuity argument presented in the previoussection requires that the sum of the phase jumps as-sociated with defect lines emanating from the spiralwave core equals the full-period phase increment ofthe local dynamics 2nk minus 2 from the spiral

wave itself. It does not specify which types of syn-chronization defect are possible in a medium withgiven periodicity or how they are organized.

As n increases one expects the diversity of de-fects will grow since the total phase shift asso-ciated with defect lines increases. Experimentalinvestigation of systems with high periodicity is dif-ficult since the size of the parameter domain whereperiod-n dynamics is found often rapidly decreaseswith n. Given the high dimensionality of parame-ter spaces for real systems, this makes experimentalobservation of complex-periodic regimes with largen difficult. Moreover, noise, which is inevitablypresent in every experiment, obliterates the finefeatures of the dynamics with large n and effec-tively sets an upper limit on the observable peri-odicity. Nevertheless, higher periodic dynamics hasbeen observed in chemical systems; for instance, theperiod-3 spirals in the Park and Lee experiment[1999] and the early stages of the period-doublingcascade in the well-stirred BZ reaction [Roux &Swinney, 1981]. Therefore, it is important to con-sider dynamics with n > 2. As an introduction tothis topic in this section we consider systems withperiod-4 dynamics arising from period-doubling ofthe period-2 dynamics described earlier. In the nextsection we generalize these considerations to period-doubled dynamics of any order and to systems withgeneral period-n dynamics.

As the parameter C is increased, theRossler medium supporting a spiral wave undergoesthe second period-doubling at C2 = 4.075. Thetransition to period-4 dynamics results in the dou-bling of the spiral wavelength, while the wave pre-serves its one-armed geometry. Therefore, the in-stantaneous concentration profile of the spiral wavenormal to its front exhibits four different maxima

corresponding to four different loops of the local or-bit. It is convenient to number these loops accord-ing to their position in phase space starting from theinnermost loop. Then, up to cyclic permutations,the wave maxima follow each other in the order 4 1 3 2. This is the order in which the loops arevisited during one full period of the dynamics. Inperiod-2 media the identification of synchronization


12/28


(a)

(d)(c)

(b)

Fig. 12. Synchronization defect lines in Rossler media with period-4 dynamics for C = 4.30: (a) instantaneous cz(r) con-

centration profile of a single spiral wave is shown b oth by elevation and color; (b) synchronization defects in the mediumshown in panel (a) are visualized using the procedure described in text; (c) result of evolution from different initial conditions;(d) minimal defect configuration with one 1 line connecting spiral wave cores in a domain with periodic boundary conditions.

defect lines based on the instantaneous concentra-tion profiles was possible due to the significantdifference between the loops of the local orbit inthe concentration phase space. For n 4, wheresuch differences are more subtle, different tech-

niques are needed to identify synchronization de-fects. To achieve this the cz concentration max-ima were computed at every point in the mediumduring one spiral rotation and color-coded. Fig-ure 12 shows the result of this procedure for two


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(a) (b)

Fig. 13. Structure of synchronization defect lines in the period-4 regime: (a) 1 defect, (b) 2. Upper row of panels showslocal cz(t) time series calculated on the opposite sides of corresponding defect lines. Lower row of panels displays correspondingloop exchanges (see text). Lines of the same color correspond to cz() maxima calculated at the same time instant.

disk-shaped media and a system with two spiralwaves in a domain with periodic boundary condi-tions at C = 4.30. Defect lines can be seen as ra-dial discontinuities in color. (The spiral-shaped dis-continuity is the inevitable artifact of the method:points lying on opposite sides of this line are col-lected at times separated by one quarter of the fulloscillation period.) After one spiral rotation thecolored bands exchange according to the permu-tation

12343421

defined by the order of loop succes-

sion. Naturally, it takes four spiral rotations forthe pattern to return to itself. The homogeneity of

colors away from the boundary seen in Fig. 12(b)demonstrates that the local dynamics strictly fol-lows a period-4 pattern whose characteristics areindependent of location in the medium. Conversely,significant fluctuations of color near the boundary(disk-shaped media) and where the two spiral wavescollide (periodic boundaries) show that the local dy-namics is irregular in these regions. This shock linephenomenon is discussed in detail in Sec. 6.

Visual inspection of Fig. 12 shows the existenceof two types of defect line which we denote as 1and 2. As in the case of the prototypical curve,a certain value of the phase shift, expressed in frac-tions of the full-period phase increment 8, is asso-ciated with crossings of the 1 and 2 curves. Fig-ure 13 (upper panels) shows the local cz(r, t) timeseries calculated on opposite sides of the 1 and 2curves. Similar to the curve, crossing of 2 leadsto a half-period shift which amounts to a 4 phasetranslation. The specific nature of the half-periodtime shift is such that it is not possible to determinewhich oscillation is advanced, since a translation byTn/2 forward is equivalent to a shift by Tn/2 back-ward. This is not the case for the crossing of the 1line where the oscillation acquires a quarter-period+2 or minus quarter-period phase shift 2 6,depending on the direction of crossing.

As in the period-2 medium, the phase discon-tinuity on the 1 and 2 curves results from loop


14/28


exchanges. It is convenient to denote the loop ex-changes by permutations which specify for everyloop with number of the oscillation pattern on oneside of the defect, the loop with number m whichis executed by the local dynamics on the other sideof the defect at the same time instant. Thus, forthe crossings of the 1 curve in opposite directions,different loop exchange permutations are assigned:12343421

for the +2 shift and

12344312

for 2. A simple

representation of the complex loop exchange pro-cess can be achieved by the following procedure.Consider an arc segment which intersects a line de-fect and is parameterized by the angle . The cal-culation ofcz concentration maxima during one fulloscillation period T4 at every point on the segmentgives four curves which intersect at those values of where the corresponding loops exchange. The re-sult of this procedure applied to both 1 and 2curves is shown in Fig. 13 (lower panels). One seesthat in the middle of the 2 curve there exists apoint at which the local orbit is period-2 and, thus,every two points with phases and + 4 becomeequivalent. The phase shift by 2 on the 1 curveformally takes place at the point with period-1 dy-namics. This requires the intersection of all fourcurves in Fig. 13 at a single point which is struc-turally unstable. Therefore in the simulations onefinds that the exchange

12343421

is distributed over

few neighboring mesh nodes where the dynamicsappears to be represented by noisy (finite thick-ness) period-1 orbits.

The asymptotic configuration of defect lines inperiod-4 media depends both on boundary and ini-tial conditions. If the medium shown in Fig. 4 istaken as the initial condition and the parameterC is increased past the transition to the period-4dynamics, the curve undergoes a transformationinto the 1 curve. Although such a configurationwith a single 1 curve connecting the wave coreand the boundary is the minimal line defect con-figuration which satisfies continuity condition (14),it is certainly not the only possible one. If the ini-tial conditions are taken sufficiently far from the

asymptotic state, in the course of evolution themedium can develop a number of defect lines of bothtypes which either connect the spiral wave core tothe boundary or form loops with both ends on theboundary [see Fig. 12(c)]. Once formed, these con-figurations freeze due to the pinning of the free ends

of the curves to the boundary. In simulationson domains with periodic boundary conditions, allspontaneously formed curves take the form ofclosed loops and lines joining the wave cores. Inthe absence of boundaries these metastable config-urations anneal to the minimal configuration whereall closed loops are eliminated and every pair of op-positely charged defects is connected by a common1 curve.

5. Classification of SynchronizationDefect Lines

In the previous sections we have shown that thephase singularities associated with line defects arerelated to the exchange of loops in the local orbits.This fact suggests that there is a relation betweenthe types and properties of defect lines that exist incomplex-periodic media and the topological organi-

zation of local orbits. In this section we show howto enumerate the possible types of synchronizationdefect and determine their properties for a period-nmedium. For 2n-periodic oscillations which consti-tute a period-doubling cascade this question can beanswered in full, while for the general case we givean algorithm for its solution.

One can represent the loop exchanges pertain-ing to a particular type of defect line symbolicallyby a permutation which, for loop mi, i = 1, . . . n,of a period-n orbit specifies another loop mj, j =1, . . . n, j = i into which it transforms under theexchange. The exchange permutations contain in-formation on the value and sign of the associatedphase shift. To extract this information one needsto include the state of the local oscillation into thesymbolic description. Since the phase jump on the curves is always an integer multiple of 2, to char-acterize the jump one needs only a coarse-graineddescription with 2 being the minimum detectablequantum of the phase change. At t = t0 let thephase point of the period-n orbit be somewhere onthe m1th loop,

1 at t = t0 + Tn/n on the m2th loop,

and so on (where ml [1, n], l [1, n], m = l)until at t = t0 + Tn the phase point returns tothe m1th loop and the pattern (m1, m2, . . . , mn)repeats. The symbolic string s = (m1, . . . , mn)constructed in this way captures the most signif-icant gross features of the oscillation pattern it

1As in the case of the introduction of phase, the partition of the local orbit into loops is arbitrary but the same at differentpoints of the medium; e.g. by the semiplane = 0.


15/28


describes. In this representation the oscillationshifted by 2 forward relative to s is given by thestring (m2, . . . , mn, m1) while the oscillation shiftedby 2 backward reads (mn, m1, . . . , mn1). Obvi-ously, any phase translation by 2k is representedby one of the n cyclic permutations of the symbolicstring s. To find how the crossing of particular linedefect affects the phase of the local oscillation oneneeds to act with the corresponding exchange per-mutation on a trial symbolic string and compare theresult to the initial state. Consider as an examplethe 2 curve described by the exchange permuta-tion

12342143

. Acting with it on the trial state (4132)

one finds 1234

2143

(4132) = (3241) ,

which corresponds to 4 phase shift of the initialstate. This result does not depend on the choice

of the trial state. Action of the exchange permu-tation on any of the four different period-4 stringsresults in another string translated by 4 relativeto the initial string. In the general case of period-n dynamics, the exchange permutations correspondto operators which map the set of all possible sym-bolic strings onto itself. Below we will show thatthese operators form a group.

To classify all synchronization line defects ex-isting in the period-n medium and to find the as-sociated phase shift values, one needs to identifyall possible types of loop exchange allowed by the

topology of the orbit and select those that cor-respond to nontrivial phase translations. To ap-proach this problem one needs a means to char-acterize a period-n orbit topologically. Considera period-n orbit, Pn, consisting of n loops in theconcentration phase space (cx, cy, cz). Using thecylindrical coordinate system introduced earlier,we may project Pn on the (, ) plane preserv-ing its original orientation and 3D character by ex-plicitly indicating whether self-intersections corre-spond to over or under crossings. This proceduremaps Pn onto a closed braid [Birman, 1974] Bn.Figure 14 illustrates the construction of the braidrepresentation for the P4 orbit. The direction ofthe flow is indicated by the arrows and the loopsare numbered according to the convention intro-duced in Sec. 6. It is convenient to subdivide theclosed braid Bn into the open braid Bn (separated

(a)

(b)

Fig. 14. Projection of the period-4 orbit of theWillamowskiRossler system on (a) the (cx, cy) plane and(b) the corresponding closed braid B4.

by dashed lines in Fig. 14) and its closure wherethreads run essentially parallel to each other.

A general open braid2 bn is a topological ob-ject which consists of n oriented threads forming atangle as they run from one end to the other. Eachcrossing of the threads i and i + 1 is assigned anelementary braid i if thread i overcrosses threadi + 1 and 1i if it undercrosses thread i + 1 [seeFig. 15(a)]. Any braid bn is described by a braidword that lists all elementary braids i as they areencountered from the top of bn to its bottom (con-

vention for direction). For example, B4 as shown inFig. 14 is described by the braid word 32132.If i and j are on the threads with nonoverlap-ping numbers their order can be exchanged and thecorresponding commutation relation reads

ij = ji , |i j| 2 .

2We reserve the notation Bn for braids corresponding to period-n orbits, while bn denotes any braid with n threads.


16/28


Fig. 15. Conventional designations and basic braid moves:(a) definition of elementary braids i and

1i ; (b) type 2

Reidemeister move; (c) type 3 Reidemeister move, and(d) first Markov move ( represents arbitrary braid).

Without violation of its topology bn can be trans-formed into braids described by different braidwords according to the set of elementary moves[Birman, 1974].

The type 2 Reidemeister move,

i1i =

1i i = 1 ,

allows one to simplify braid words by stating thatthe two threads that appear to cross but, in fact,are not entangled can be deformed into unit braid

consisting of parallel threads [see Fig. 15(b)]. The type 3 Reidemeister move,

ii+1i = i+1ii+1 ,

illustrated in Fig. 15(c) completes the set of ele-mentary operations.

By closing a braid bn bn one makes it an ob-ject of more general topological nature. Indeed, anylink or knot can be represented as a closed braid.For closed braids two additional moves can be ap-plied to change the braid word.

Type 1 Reidemeister (or second Markov) movewhich does not preserve the number of loops isnot allowed in our studies of periodic orbits whoseperiodicity given by the number of loops is an es-sential invariant feature.

The first Markov move

i = i ,

where belongs to the set of open braids, isshown in Fig. 15(d) and allows fragments of thebraid to propagate around the closure of bn sothat they can change their position from the topof bn to the bottom and vice-versa.

5.1. Loop exchanges as braid moves:

period-2 example

We now examine possible loop exchanges for theperiod-2n period-doubled orbits and show how theseexchanges influence the corresponding patterns ofoscillation (unless specifically stated, the index ndesignates the power of 2 and not the full period-icity 2n). A period-2 orbit is represented by thesimplest nontrivial braid B2 = 1 shown in Fig. 16.

Consider the first Markov move M(1)1 which prop-

agates the elementary braid 1 by 2 around thebraid B2 in a direction opposite to the flow. The su-

perscript in parentheses denotes the power n whilethe subscript numbers the moves consecutively. Al-though the braid B2 remains unchanged as a resultof this move, one finds that the two loops have inter-changed their positions. To indicate this, the loopsof B2 are drawn in different line styles in Fig. 16.The left panel shows B2 before the application of

M(1)1 and the right panel after. The interchange

of loops induced by M(1)1 can be described by the

exchange operator A(1)1 represented by the permu-

tation 1221. In the framework of the coarse-grained

symbolic description, a period-2 oscillation has onlytwo nonidentical states s1 = (12) and s2 = (21).

Action of A(1)1 on one of them gives the other and

vice-versa

A(1)1 s1 =

12

21

(12) = (21) = s2 ,

A(1)1 s2 =

21

12

(21) = (12) = s1 .

One sees that the action of A(1)1 is equivalent to

the translation of the oscillation pattern by half aperiod, which for the period-2 oscillation amountsto a 2 phase shift. To undo the result of the

application of M(1)1 one needs to perform the re-

verse move (M(1)1 )

1 which propagates 1 along the

flow or repeat M(1)1 once more. Thus, in terms

of the exchange permutations the inverse opera-

tor (A(1)1 )

1 induced by (M(1)1 )

1 is equivalent to


17/28


Fig. 16. Propagation of 1 around the B2 braid results inthe exchange of loops.

A(1)1 and produces the same action on the symbolic

strings. From the properties discussed above, the

operator A(1)1 , which symbolically represents the

transformation of the period-2 oscillation on the curve, generates a group

(A(1)

1)1 = A

(1)

1, (A

(1)

1)2 = (A

(1)

1)2 = 1 . (15)

In this way the exchange operators and algebraic re-lations between them can be established for period-doubled orbits with higher n. A detailed discussionof the period-4 case is presented in the Appendixwhere we consider all allowed braid moves and as-sociated exchange operators for the P4 orbit andshow that the 1 and 2 lines discussed above arethe only types of synchronization defect in a period-4 medium.

5.2. General case

A generalization of the formalism for the period-2and period-4 (see Appendix) cases to a period-2n or-bit P2n may be derived from the observation of thestructural organization of the corresponding closedbraids B2n. This does not require the enumerationof all possible braid moves whose number quicklygrows with n. Indeed, to find all types of synchro-nization defect allowed in the period-2n medium oneonly needs to identify those moves that result innontrivial translations of the oscillation pattern.

A braid B2n+1 can be obtained from B2n bydoubling each thread of B2n and adding a singlecrossing on the top of the braid to preserve the sim-ple connectivity of the construction. The braid B2narising as a result of n successive iterations of thisprocedure can be subdivided into n nonoverlapping

structurally similar blocks of braids (n)m , m = 1, n.

This principle of structural organization is illus-trated in Fig. 17 for B8 and its three braid blocks

Fig. 17. Braid B8 constructed for the period-8 orbit.

are shown in a series of boxes with decreasing size.The analysis shows that these blocks can be movedas individual entities along B2n without interfer-

ence, and produce an exchange of the loops alongwhich they move. The essential parts of these trans-

formations can be represented by a set M(n)m of 2

moves of structural blocks (n)m so that M

(n)1 cor-

responds to the largest block and results in an ex-

change involving all the 2n loops, M(n)2 corresponds

to the movement of the next smaller braid blockand results in the exchange of 2n1 loops, and soon. The transformations of the time trajectoriesresulting from exchanges of loops can be again de-

scribed by the action of permutation operators A(n)m

on symbolic strings sj . The fact that the crossingblocks move independently implies that operators

A(n)m commute. The geometry of B2n also defines

the basic algebraic property of the A(n)m . The gen-

eralization of (15) gives

(A(n)i )

2i = 1, A(n)i A

(n)j = A

(n)j A

(n)i , (16)

where i, j [1, n]. Those compositions of exchangeoperators which are induced by moves returningB2n to itself (see Appendix for examples of moveschanging the configuration of the braid) yield trans-

lation operators T2k where k [1, 2n1]. To iden-tify these operators one does not have to enumer-ate all possible compositions, whose number growsexponentially with n. Indeed, since translation by2k is equivalent to k times a translation by 2 and,thus T2k = (T2)

k, it is only necessary to find thegenerator T+2 of the subgroup of translators T2k .For the period-2n orbits this problem can be solvedexactly for any n.


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As mentioned earlier, the configuration of B2ndetermines the succession rule which states the or-der in which the loops are visited during one periodof oscillation. For example, consider again B4 (seeFig. 14). Following the direction of the flow one canfind the order in which loops succeed each other af-ter they pass through the tangle of B4. Writing theresult as a permutation 12343421 one sees that it corre-sponds to the permutation representation of T+2,since phase translation by 2 is essentially equiv-alent to the advance of the oscillation by T2n/2

n

prescribed by the succession rule. This result obvi-ously holds for any n. Thus, if the entire braid B2nis propagated around its closure by 2, the resultis formally equivalent to the composition of all el-

ementary moves M(n)m applied one by one and the

net loop exchange is given by the succession rule.Therefore, the sought-for operator T+2 is induced

by the composition M(n)1 M

(n)2 M

(n)n and any

translation operator T2k is given by

T2k =

n

m=1

A(n)m

k, (17)

where k = [2n1, 2n1]. Indeed, from the resultsof the previous section for n = 2 it follows that

T4 = (A(2)2 )

2 = (A(2)1 A

(2)2 )

2 = (T2)2 .

In principle, every nontrivial translation operatorT2k corresponds to a different type of synchro-nization line defect and in the period-2n oscillatorymedium one should be able to find 2n1 differenttypes of curve. Thus for period-8 system thefollowing types of line defects with correspondingexchange operators are possible:

(3)1 (2) A

(3)1 A

(3)2 A

(3)3 ,

(3)2 (4) (A

(3)2 )

2 (A(3)3 )

2 ,

(3)3 (6) A

(3)1 (A

(3)2 )

3 (A(3)3 )

3 ,

(3)

4 (8) (A

(3)

3 )4

.

A half-period translation of period-8 oscillation

which occurs at the crossing of the (3)4 curve is

induced by the propagation of the simple crossing

(3)3 four times around the closure of B8. By anal-

ogy with the cases n = 1, 2 one can predict that

in the middle of the (3)4 curve there exists a point

where the oscillation is period-4. The analysis of the

exchange operator (A(3)2 )

2(A(3)3 )

2 leads to the con-

clusion that on (3)2 one should find a point with ef-

fective period-2 dynamics. The operator A(3)1 which

stands for the exchange of the period-2 bands ofperiod-8 orbit appears in the above list twice: for

the (3)1 and

(3)3 curves. Therefore, period-1 dy-

namics should be observed on both lines. Since it is

unlikely to observe the exchange of all eight loopsin the same medium point due to its structural in-stability, the actually observed dynamics should begiven by thick, noisy period-1 orbits. In the nextsection we show that pure period-2n local dynam-ics with n 3 is not observed in media with spiralwaves. Nevertheless, p eriod-8 local dynamics canbe observed in media without spiral waves (see be-low). Indeed, all four possible curve types withpredicted characteristic features have been found insuch media.

For the more general case of period-n local dy-

namics where n is not a power of 2 resulting froma period-doubling cascade, the following algorithmcan be employed. First, the local orbit Pn shouldbe mapped onto a closed braid Bn. Then throughthe analysis of the topology of Bn the blocks ofelementary braids which can be moved indepen-dently should be identified. Their 2 propagationaround the closure of Bn will give a set of essen-tial braid moves with corresponding exchange oper-ators. Those compositions of the essential moveswhich lead to the identity transformation of Bnshould result in some nontrivial phase translations.

These phase shifts, in turn, should correspond topossible types of curve.

6. Line Defect Turbulence

Thus far we have considered the description andanalysis of line defects in systems with period-2 and period-4 dynamics in the period-doublingcascade. Now, again using the Rossler reactiondiffusion equation as our model system, we exam-ine its behavior as the parameter C is increased

causing the system to enter chaotic regimes. Wedescribe the nature of these regimes for spatial con-figurations with and without spiral waves and showhow their structure is related to the dynamics ofsynchronization line defects.

In the computations presented below, we con-sider both systems with a single spiral in mediumwith no-flux boundary conditions, as well as sys-tems with periodic boundary conditions supporting


19/28


several spiral waves. For the Rossler reactiondiffusion system, configurations with many spi-ral waves form spatially blocked structures. Theblocked structures form irregular cellular patternswith cells centered on the spiral wave cores. Thepolygonal boundaries of the cells are delimited byshock lines where the spiral waves from neighboringcells collide. Figure 12(d) shows a simple example

of such a blocked cellular pattern with two spiralwaves.

6.1. Transition to chaos in thepresence of spiral waves

Figure 18 shows the bifurcation diagram for theRossler medium supporting a solitary spiral wave,measured at a point r in the bulk region, i.e. awayfrom both the spiral core and boundary regions. Toshow the first two period-doubling bifurcations, the

local order parameter mcz(r) was redefined as themaximum difference b etween cz concentration max-ima calculated at point r in one full period T4 ofthe period-4 oscillation. In the p eriod-2 regime thisquantity coincides with 1cz(r) introduced earlier,while in the period-4 domain it gives the differencebetween the fourth (the highest) and first (the low-est) cz maxima. The dependence ofmcz(C) for theRossler ordinary differential equations is also shownfor comparison in Fig. 18. One sees that the period-doubling bifurcations in the bulk of the medium areshifted to the right along the parameter C axis with

respect to bifurcations observed in the ODE. In-deed, the local dynamics in the medium controlledby a spiral wave resembles the dynamics of the ODEat some C which is always less than the value of Cin the PDE. The value of C = C C depends onthe wavelength of the wave pattern in the mediumand vanishes as . If more than one spiralwave is present, similar considerations apply to theshock lines between spiral domains where incomingwaves collide. The normal wave pattern is brokenon these shock lines and the local dynamics is alsoshifted to higher C values compared to the dynam-

ics in the bulk. In fact, at C values correspondingto the period-4 dynamics in the bulk, the dynam-ics on the shock lines already exhibits chaos whichmanifests itself in the fluctuations of color seen inFig. 12. Chaos first occurs on the shock lines, forC values approximately equal to those for the onsetof chaos in the Rossler ODE, in agreement with thefact that the phase gradient is zero along the shocklines.

Fig. 18. Domain of period-doubling oscillations in theRossler medium supporting a spiral wave. The upper redcurve is for the Rossler ordinary differential equation.

The entire period-doubling cascade to chaos isnot observed when the bifurcation parameter C isincreased further. One finds instead a complex sce-nario which involves turbulent stages characterizedby spontaneous formation, erratic motion and pro-liferation of synchronization defect lines. To char-acterize line-defect turbulence quantitatively oneneeds a means to both visualize the dynamics of the lines and measure their density. The color-codingtechnique introduced earlier allows only visualiza-tion. To solve this problem scalar fields were defined[Goryacher et al., 1999] in the following way: dur-ing four consecutive rotations of the spiral wave thevalues of the cz(r) concentration maxima Ai(r), i =1, 4 were collected at every point in the medium andsorted so that A1(r) A2(r) A3(r) A4(r). The

scalar fields are defined as 1(r) = A4(r) A1(r)and 2(r) = A4(r) A3(r). By construction, 1(r)and 2(r) take on fixed, nonzero values at pointsin the medium with period-4 dynamics and vanishat points where the loop exchanges occur. Indeed,1(r) decreases to zero on the 1 curves while 2(r)vanishes on both the 1 and 2 curves. The 1(r)and 2(r) fields allow one both to determine thelengths of the curves and to visualize them. Fig-ure 19, panel (c), shows the 2(r) field at C = 4.3 fora medium with periodic boundary conditions sup-porting a spiral pair. The corresponding (r, t0)

and cz(r, t0) fields are shown in panels (a) and (b),respectively. The cores of spiral waves, seen as blackdisks, are connected by a common 1 curve whichappears as a wide, nearly straight black line. At thisvalue ofC the dynamics in the bulk of the medium(spiral cores, 1 curve, and shock lines excluded)is given by a period-4 pattern [see Fig. 20(a)]and single 1 curve connecting the cores consti-tutes the minimal configuration which satisfies the


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Fig. 19. Representation of defect lines by the 2(r) fieldfor the medium with two spiral waves at (c) C = 4.30 and(d) C = 4.32. Panels (a) and (b) show, respectively, the(r, t0) and cz(r, t0) fields calculated for the medium inpanel (c).

continuity condition (14). The shock lines, wherethe spiral waves collide, are seen in panels (c) and(d) as one vertical and two horizontal strips wherethe gray shades are inhomogeneous. This inhomo-geneity in the 2(r) field indicates that the localdynamics on the shock lines is different from thatin the bulk. Indeed, the calculations show that the

local dynamics on the shock lines at C = 4.3 isgiven by two-banded chaos [see Fig. 20(b)]. Thechaotic dynamics on the shock lines gives rise tospatially coherent fluctuations of Ai(r) fields seenin Fig. 19(c) as darkly shaded breathing spots.Sufficiently large fluctuations (like that indicated bythe arrow in the figure) may result in the creationof bubbles domains delineated by circular 2curves [cf. Fig. 19(d)]. For C C2 = 4.306, thebubbles are formed with a size smaller than a cer-tain critical value and collapse shortly after birth.As C increases beyond C2, the bubble nuclei begin

to proliferate, forming large domains whose growthis controlled by collisions with spiral cores or otherdomains (see next subsection for details about thedynamics of lines).

The transition to line-defect turbulence changesthe character of the local dynamics observed in thebulk of the medium. As the 2 lines propagatethrough the medium the associated loop exchangesthe result in an effective band-merging in the orbits

Fig. 20. Time series of the cz concentration maxima: (a andb) C = 4.30, (c and d) C = 4.42 and (e and f) C = 4.7. Leftpanels show the local dynamics at a point in the bulk, whilethe right panels show the dynamics on the shock lines. Timeis in units of thousands of spiral revolutions. In panels (c)and (e), respectively, the crossing of the cz maxima reflect thepassages of an 2 or 1 line through the observation pointin the medium.

of local trajectories so that they take the form oftwo-banded chaotic trajectories [cf. Fig. 20(c)]. AsC increases past C 4.5 the local dynamics failsto exhibit a period-4 pattern in the intervals sep-

arating line-defect passages and shows thick four-banded orbits whose bands grow in width with in-creasing C. The width of chaotic bands varies frompoint to point and while in some locations one ob-serves their merging, in others there might be a dis-tinct gap between them. The local dynamics on theshock lines also progresses towards less-structuredchaos. At C = 4.42 [cf. Fig. 20(d)] it already ex-hibits one-banded chaos.

As the parameter C approaches the second crit-ical value C1 4.557 the shock lines begin tospontaneously nucleate bubbles of 1 lines. At

C > C1 the newly-born domains delineated by 1lines begin to proliferate and fill the medium witha perpetually changing turbulent pattern. The de-fect line density can be used as an order parameterto characterize transitions to turbulence quantita-tively. From the i(r), (i = 1, 2), fields the lengthsi of the i lines may be computed. The i linedensity is then i(t) = i(t)/L where L is the lin-ear dimension of the square array. Although the


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Fig. 21. Transition to (left) 2- and (right) 1-line turbulence. (Green) System with two spirals shown in Fig. 19; (red)system of the same size with four spirals; (blue) system with two spirals but twice the linear dimension, L = 512.

instantaneous values ofi(t) fluctuate considerably,the balance between defect line growth and destruc-tion results in a statistically stationary average den-sity i of i lines. The time-averaged i line densi-ties i as a function of C are plotted in Fig. 21. Inall simulations the average intensity of noise on theshock lines was estimated from the calculations atC < Ci and subtracted. Likewise, for constructionof the diagram in Fig. 21(a) the contribution of thestationary 1 line (see Fig. 19) was estimated andsubtracted. The behavior of both averaged densi-ties beyond the corresponding critical points showsa power law dependence,

i(C) (C Ci)i , (18)

with parameters (C2 = 4.306, 2 = 0.25 0.01)and (C1 = 4.557, 1 = 0.49 0.02) respectively.Such behavior indicates that both transitions toturbulence can be considered as nonequilibriumphase transitions [Goryachev et al., 1999].

The bands of the local orbits continue to growin width and merge at C 4.7. Globally this tran-sition can be identified as spontaneous nucleation of2 bubbles homogeneously in the medium. Thesechaotic 2 lines are loci of medium points where

two chaotic bands of the local orbit shrink and athick period-2 orbit is formed. With increase in Cthe width of these lines decreases and their motionin the medium becomes totally erratic. For C > 4.8the 2 lines cease to exist as well-defined entities.Qualitatively the same scenario applies to the dis-appearance of 1 lines. At C 5.0 one starts tofind bubbles of 1 lines not only in the shock zonesbut also in the bulk. This indicates the transition

of the local dynamics to one-banded chaos homoge-neously in the medium. For C 5.1 no defect linescan be identified and further increase in C does notresult in any qualitative changes.

It is interesting to follow the fate of the spiralwave in chaotic regime. Even deep in the chaoticregime, where the local dynamics is represented byone-banded chaotic orbits, spatially coherent spiraldynamics can be still seen in the medium [Kleveczet al., 1991; Brunnet et al., 1994; Goryachev &Kapral, 1996a, 1996b]. Figure 22, left panel, showsthe local angle variable field (r, t0) for C = 5.9.A clearly visible image of the spiral wave with dis-

torted equal-angle lines = 0 signifies that chaoticoscillations remain synchronized [Rosenblum et al.,1996] in the sense of period-1 phase = . Asthe execution of any loop of the Rossler chaotic at-tractor takes approximately the same time and themotion occurs with nearly constant angular veloc-ity d/dt 2/T0, where T0 is the period of spiralrevolution, the chaotic spreading of the local trajec-tories does not lead to break-up of synchroniza-tion. To demonstrate that the medium is indeedin a turbulent regime in which the amplitude of thelocal oscillations is disordered, the maximum cz am-

plitude color-coding procedure described in Sec. 4was applied. One sees that the randomly varyingcz maxima constitute a turbulent pattern with veryshort correlation length.

In the experiments of Park and Lee [1999] onthe BZ reaction, configurations containing severalperiod-2 and period-3 spiral waves separated by re-gions with complex dynamics were seen. Parameterdomains were also investigated where the system


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Fig. 22. Spiral wave in the deep chaotic regime at C = 5.9.

exhibited turbulent dynamics. Within the turbu-lent regimes bubble-shaped lines and irregular line dynamics were observed, akin to the line-defectturbulence phenomena described above.

6.2. Dynamics of line defects

By following the dynamics of synchronization linedefects in the turbulent regimes described above,one can make a number of observations important

for understanding the factors governing growth, mo-tion and destruction of lines. Figure 23 showssix consecutive frames of the 2(r) field for a longinterval of the evolution of 2-line turbulence atC = 4.38. When a supercritical nucleus of the 2-line domain is born on the shock line, it grows to-ward the spiral wave cores positioned on oppositesides of the shock line. As mentioned earlier, everyspiral core is confined to a cell formed by the shocklines and, thus, within one such cell there existsonly one center of attraction for the proliferatingdefect lines. Therefore, depending on whether the

nucleus was formed on one shock line or the inter-section of two shock lines, a domain may experienceattraction to two or more spiral cores.

The dynamics of 1 lines in the regime of 1-line turbulence exhibits qualitatively the same fea-tures as 2-line dynamics. In the process of growththe domain is likely to collide with other expand-ing domains. The contact of 1 and 2 lines re-sults in the formation of one 1 line. Where the

lines coalesce, the phase shift across the 1 linechanges sign. Indeed, the composition of the loopexchanges corresponding to 1 and 2 lines resultsin a loop exchange which can be attributed to an-other 1 line with opposite sign of the phase shift:2 + 4 = 2. The contact of two lines of thesame type occurs according to the scenario whichis sketched in Fig. 24(a). When two defect linescollide, they reconnect with change of their topol-ogy and net length. At the instant of collision an

unstable state is formed in which the lines seem tointersect. The break-up of this state always occursin the direction given by the bisectrices of the twoacute angles. As a result the two initial defect linesbreak and two new lines are formed as shown inFig. 24(a). Two bends with high curvature quicklystraighten and the newly-formed lines depart fromeach other. Note, that this process leads to the re-duction of the net length of lines. The majorstages of the reconnection process are sketched inthe bottom of panels (c)(e) in Fig. 23. Panel (d)captures the intermediate crossover state (indicated

by the arrow) which is difficult to detect due to itsshort lifetime. This elementary event which takesplace locally when two lines come in contact cangive rise to a variety of topologically different globalconfigurations. For example, the contact of two do-mains may result in the formation of one large do-main, or a large domain with a small one in itsinterior. The case where one domain forms insidethe other is of particular interest [see Fig. 24(b)]. In


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Fig. 23. The evolution of the medium with 2-line turbulence at C = 4.38. All panels are equally separated in time by 24spiral revolutions and show the 2(r) field coded by gray shades.

Fig. 24. Interaction of defect lines in the turbulent regime: (a) reconnection of lines following contact, (b) formation ofbubbles disconnected from the shock line (indicated by two vertical solid lines), and (c) reconnection of lines in the wave core(indicated by black disk). The arrows between the pictures show the sequence of events, while arrows on the pictures designatethe direction of line-defect motion.


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the process of their recombination two new domainsdisconnected from the shock line are formed. Theyquickly acquire a circular shape and begin to drifttoward the wave cores simultaneously shrinking insize. In most cases the latter process predominatesand the domains collapse before they can reach thecores.

Configurations with more than three lines

emanating from the same spiral wave core appearto be unstable, at least for the Rossler medium wehave studied. This is probably due to the fact thatthe size of the core is insufficient to accommodatemore than three lines. Nevertheless, such con-figurations play the role of intermediate states inthe process of line reconnection in the wave core.Figure 24(c) schematically shows the events thattake place in the panels (c) and (d) of Fig. 23 inthe neighborhood of the lower spiral core. When an2-line loop reaches the core, an unstable configura-tion with one 1 and four 2 lines is created. As inthe case of line contact, the unstable configurationbreaks so that the two lines that form the sharpestangle produce a loop which disconnects from thecore. Thus, reconnection of defect lines can also oc-cur through the unstable line configuration at thespiral wave core.

From these observations it follows that the evo-lution of the size and shape of a domain is controlledby the balance of two competing factors: propaga-tion of defect lines along the phase gradient directedto the spiral wave cores which results in line growth,

and the tendency of diffusion to eliminate curvatureand reduce the length of defect lines. To investi-gate the interplay of these two factors a series ofsimulations was carried out on a rectangular sys-tem of size Lx Ly without spiral waves but with ashock line where the nucleation of lines may takeplace. Periodic boundary conditions were used inthe x direction and constant concentrations corre-sponding to those in the spiral core were imposedat y = Ly/2. To break the spatial translationalsymmetry of the system the fixed concentrationson the core boundaries were taken to have the

form c0 + (r), where c0 is the concentration in thespiral core and (r) is a spatial white noise con-tribution. All other parameters were unchanged.In this geometry wavetrains of plane waves emit-ted by the constant concentration boundaries col-lide in the center of the system to form a shock line.The domains randomly formed on this shock linehave very simple finger-like shapes, normal to theshock line, and consist of two straight segments with

Fig. 25. Dependence of the domain propagation velocity,v, versus the curvature, 1/R, of the tip for C = 4.30. Atypical finger-shaped 2 domain is shown in the right panel.

approximately semi-circular caps (see Fig. 25). Thedomain growth velocity v along y varies with theradius R of the arc of the growing tip as

v = vp /R (19)

where the quantities vp = 0.126, the velocity of a

planar 2 line along y, and = 0.658 were calcu-lated from the simulation data presented in Fig. 25.Since the width of small domains is approximatelyequal to 2R, one can estimate the critical size thatmust be exceeded for domain expansion. FromEq. (19) the critical radius is Rc = 5.228, which isin good agreement with Rc directly measured fromthe observation of domains whose shape does notchange with time. Also from (19) it follows thatthe velocity v is independent of the distance tothe core boundary and is solely defined by theproperties of the 2-line domain the curvature

1/R of its growing tip. The investigation of thec(r, t) field has shown the absence of concentrationgradients along the y axis of the medium. On thebasis of these results it was concluded that the de-fect lines propagate along the local phase gradient(r, t). In media with spiral waves, where thephase gradient is always directed to the wave core,this results in the motion seen as the propagationof defect lines to the wave core.

6.3. Line-defect turbulence insystems without spiral waves

It is interesting to contrast the phenomena de-scribed above for the Rossler system with spiralwaves with those observed in the same mediumwithout spiral waves. We studied this case byconsidering initial conditions where spatial inho-mogenieties were sufficiently weak that spiral waveswere not formed in the course of the dynam-ics. We observed that the onset of line-defect


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nucleation occurs at the same critical values Ci aswhen spiral waves are present. In the absence oflarge-scale phase gradients, the entire medium be-haves like the shock regions separating spiral wavecells, defect lines do not grow and the increase ofi(C) arises essentially from the enhanced nucle-ation rate. This leads to a different form of theonset of line turbulence characterized by different

critical exponents (1 = 1.22, 2 = 0.53) [Gory-achev et al., 1999]. These values are difficult toestimate because of fluctuations in the i fields notassociated with fully developed lines. The factthat 1 and

2 are significantly different from 1

and 2 supports the conclusion that the characterof the transitions is different in the spiral and spiral-free systems.

7. Conclusions

In this paper we presented a study of synchro-nization defect lines in two-dimensional complex-periodic and chaotic media. As for simple periodicmedia, stable spiral waves in such systems are one-armed and, therefore, characterized by a topologicalcharge nt = 1. In one turn of such a spiral wavethe local complex-periodic dynamics executes onlya fraction of the full oscillation period and the con-centration field does not return to its initial state.For example, in the period-2 medium it takes tworotations of the spiral for the concentration field tobe restored. This conflict between the spiral rota-

tion period and the period of the local oscillationis reconciled by the presence of a synchronizationdefect line connecting the wave core to the bound-ary or to another spiral wave core with oppositetopological charge. The average of the continuitycondition (14) over the full period Tn of the localn-periodic dynamics gives

1

2n

(r) dl = k , (20)

a complex-periodic analog of (1) for a topologicalphase defect in a period-1 medium. As follows from

(14), the full contour increment of 2nk consistsof two different parts. While 2 is contributed bythe integration of (r) along , the remaining2(nk 1) is gained by adding up phase jumps onthe intersections with synchronization line defects.Thus, k in (20) does not characterize the topologi-cal phase defect whose charge remains 1. Insteadthe value of k is defined by the particular configu-ration of the line defects emanating from the spiral

wave core. Consider, for example, the period-4 casedescribed in Sec. 4. Condition (20) selects possibleconfigurations of curves emanating from the core.If the topological charge of the spiral wave core ispositive (nt = +1) then the minimal configurationwhich satisfies (20) has one 1 curve with phaseshift 2 and corresponds to k = 0. The other con-figuration for k = 0 is given by three 1 curves with

one positive and two negative signs (2+22).For k = 1 one also finds two admissible configura-tions. One is given by 1 (+2) and 2 (4) curvesand the other consists of two 2 curves and one 1curve with negative sign.

Synchronization defects also play an importantrole in the transition to chaos. In this transition,media supporting spiral waves exhibit li

andrei goryachev, raymond kapral and hugues chate- synchronization defect lines

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