Spiral Wave Nucleation (Interfaces, Pulses and Waves in Nonlinear Dissipative Systems : RIMS Project 2000 "Reaction-diffusion systems : theory and applications")

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E. Meron\ddagger

Department

_of

Energyand EnvironmentalPhysics, BIDR,and PhysicsDepartment,

$Ben$-Gurion University, Sede Boker Campus _84990, _Israel

$\ln$bistablesystems,the stabilityof frontstmcturesofteninfluencesthedynamics of extended patterns. We

show how thecombinedeffect ofaninstability to curvaturemodulationsandproximitytoapitchfork front

bifurcation leadsto spontaneousnucleationofspiralwavesalong thefront. This effectis demonstratedby

directsimulationsofaFitzHugh-Nagumo (FHN)model and bysimulationsof order parameterequations

for thefront velocityand curvature. Spontaneous spiral-wave nucleationoftenresultsinastate of

spatio-temporal disorder involvingrepeated events of spiral wavenucleation,domainsplittingand spiralwave

annihilation.

I. INTRODUCTION

Spatio-temporal disorderis aubiquitous phenomenon in ex-tended nonequilibrium systems yetthemechanisms

respon-sible for it are only partially explored. A generic

mecha-nism for theonset of disorder inperiodicpattemsconsists

ofphase instabilities followed by the formation of phase sin-gularities whichappear asdislocation defectsorvortices [1].

This mechanism has been observedinnumerical simulations ofthe complex Ginzburg-Landau equation$[2, 3]$, andin ex-periments, e.g. electro-convectionin liquid crystals [4]. A

considerableefforthasbeendevoted toelucidatingthenature

ofthetransition from the regular phase regimetotheregime wherevorticesordefects spontaneouslyappear[5-9]. Other mechanisms involving instabilitiesofperiodicpatterns,

lead-ingtospiral breakup, have been reported recently[10-13]. In this article we present a mechanism for the onset of spatio-temporal disorder associatedwith

_front

structures.This type of disorder is illustrated by a numerical solution of a bistable$\mathrm{F}\mathrm{i}\iota \mathrm{z}\mathrm{H}\mathrm{u}\mathrm{g}\mathrm{h}$Nagumo (FHN) typereaction-diffusion

system(Fig. 1).An almost flatfront,connecting thetwostable

uniformstationarystates,begins traveling throughthe system.

The frontrepresentsan “up”state(black)invadinga“down”

state(white).Initial nonuniform perturbations of the front

po-sition grow intowiggles which nucleatepairsofspiral

waves

(see the Appendix fora differentview of the middle frame in Fig. 1). _{The solution then evolves into}_adisordered state with repeatedeventsofdomain splitting and spiral-wave

nu-cleation and annihilation. Similar phenomena have been ob-servedin experimentsonthe$\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{r}0\mathrm{c}\mathrm{y}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{d}\mathrm{e}-\mathrm{i}_{\mathrm{o}\mathrm{d}\mathrm{u}1}\mathrm{a}\mathrm{t}\mathrm{e}-\mathrm{S}\mathrm{f}\mathrm{i}\mathrm{t}\mathrm{e}$(FIS) reaction$[14, 15]$.

1I. THENIB FRONTBIFURCATION AND SPONTANEOUS

FRONT REVERSALS

The keytounderstanding the disorder associated with front structuresisapitchfork bifurcation in the velocity ofa propa-gatingfront. The front bifurcationisrepresented by the

equa-tion

$C^{3}-(\lambda_{C}-\lambda)c=0$, ₍₁₎

where$C$is the velocity ofaflat front and $\lambda$ is acontrol pa-rameter. Thecorresponding bifurcation diagram is shownin

Fig. 2. Astationary front$(C=0)$becomes unstable belowa

criticalparametervalue, $\lambda=\lambda_{c}$. At thatpointtwonew

sta-ble front solutions appear, representing an up stateinvading

adownstate $(C>0)$ andadownstateinvadingan upstate

$(C<0)$. Hereafter,werefer to these front solutionsasto“UD front”and “DUfront”,respectively.

This type of front bifurcation has been derived for period-ically forced oscillatory systems [16] andfor bistable FHN typemodels[17-19]. Experimental observations of front bi-furcations, or supporting evidence for their existence, have

been reported in Refs. $[20, 21]$ for liquid crystals, and in

Refs. $[22, 23]$ for chemical reactions. The stationary and

counter-propagating frontsaresometimes referredtoasIsing and Blochfronts, respectively, and the bifurcation itselfas a

nonequilibrium Ising-Bloch(NIB)bifurcation$[16, 18]$.

Physical realizations of front bifurcations usually involve perturbations that unfold the pitchfork form of Eq. (1) into[24]

$C^{3}-(\lambda_{c}-\lambda)c+\nu_{1}+\nu_{2}C2=0$ . ₍₂₎

An asymmetry between the

up

anddown statesisanexample of suchaperturbation. In the followingweconfineourselves

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FIG. 1: The evolution of anunstable front solution connecting theup state (black) to the down state (white) in a bistable FHN reaction-diffusionsystem. Theframes, from left toright, represent the pattern solutionatsuccessivemoments in time.

Perturbations on the flat traveling frontgrow and nucleate spiralpairs along the front line (seethe Appendix foradifferent view of the middleframe). The resulting disorderedstateischaracterized by domain splitting and spiral-wave nucleation and annihilation.

FIG. 2: The pitchfork front bifurcation. At$\lambda=\lambda_{c}$ the

sta-tionary$(C=0)$front solution becomes unstabletoa pairof

counterpropagating traveling fronts. spanned by$C,$$\lambda$ and

$\nu_{1}$ is shown in Fig.3. Thesignificance of smallvariations of$\nu_{1}$ in the vicinity of the pitchfork bifur-cation point,$\lambda=\lambda_{c},$ _{$\nu_{1}=0$}, isnowevident: perturbations may induce transitions between the upper andlower sheets

andreversethedirection of front propagation. Notice that far-ther from the bifurcation point thevariations of$\nu_{1}$ mustbe largerin ordertoinduce front reversal.

Thedynamics ofasingleflat frontmayalso be changed by slow dynamicalprocessessuchas anapproachtoa boundary, an interaction with another front, _or development of

curva-ture. Figure4 shows atypical graph of the normal velocity ofa front, $C_{n}$, versus its curvature, $\kappa$, for fixed $\lambda$ nearthe front bifurcation. The figure

was

obtained using asingular perturbation analysis ofanFHNmodel, assuming slow

cur-vature dynamics with respect to the time scale of front

re-versal [25]. The hysteretic shape, similar tothe graph of$C$

versus

$\nu_{1}$for fixed$\lambda$(seeFig.3),suggeststhat front reversals canbeinduced by smallcurvatureperturbations. Since

curva-ture isan intrinsic dynamicalvariable,reversals of propaga-tiondirectionoccurspontaneouslyasthecurvatureofafront changes.Spontaneous front reversalsincatalyticreactions on

FIG.4:Normal front velocity$C_{n}vs$curvature$\kappa$in the vicin-ity ofthefrontbifurcation. The development of small negative

curvaturemayinducea transitionfromaUDfront$(C_{n}>0)$

toaDUfront$(C_{n}<0)$. $.\mathrm{A}$ _,.

platinum surfaces have indeed been observed forparameters

in thevicinitya front bifurcation[22]. Experimental obser-vations of front reversals induced by boundaries have been reported in[23].

Imaginenow a flat UD front thatis unstabletotransverse

perturbations (i.e. an instability to curvatureperturbations). As the frontpropagates,alternatingsegmentsalong the front acquire negative and positivecurvatures. For parametersthat place thesystemnearthe the frontbifurcation,whereFig. 4

applies, segmentswithnegativecurvaturewill eventually re-versepropagation direction. Such local front reversals involve thecreationoftransition

zones

between the counterpropagat-ing UD and DU fronts. Thesezonesform thecoresofrotating spiral

waves.

Thescenariosketched above providesa heuristic

explana-tionfor the spontaneous nucleation of spiral

waves

in Fig. 1

which precedes theonsetof spatio-temporal disorder. Similar

argumentshold for otherintrinsicperturbations suchasfront

interactions. IndeedFig. 1 includes manyevents where lo-calfront reversalsareinducedby theinteractionsbetween

ap-proaching domains. An example of suchaneventis shownin

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FIG.3:Top: The surface$C^{3}-(\lambda_{c}-\lambda)c+\nu_{1}=0$inthespacespanned by$C,$$\lambda,$$\nu_{1}$(thecuspcatastrophe).First triad: Asection

at$\nu_{1}=0$showing thepitchforkbifurcation(center),andsectionsat$\nu_{1}<0$(left)and$\nu_{1}>0$(right)showing unfoldings of the

pitchfork. Second triad: Sections of the surfaceatconstant$\lambda$ showing the hysteresispoint(center)with single valued(left)and multivalued(right)relationsawayfromthe hysteresispoint.

Fig. 5. Similarprocesseshave beenobserved in experiments

(Fig.6)ontheFISreaction$[14, 15]$.

II..I.

THEDYNAMICS OF FRONT REVERSALS

Algebraic relations like that displayed in Fig. 4areuseful for predictingtheonsetofspiral-wave nucleation: nucleation

eventsbecome feasible whenasingle valued relation becomes

multivalued(orhysteretic). Such relations donot, however, contain information about the nucleationprocess itself. To study the nucleation process a time-dependent approach for thefront dynamicsis needed. Near thebifurcation,the asymp-toticfront dynamicsaregoverned by bothatranslational

de-greeoffreedom andan order parameter associated with the bifurcation: the frontvelocity$C$

.

InRefs._{$[26, 27]$} wederived

asymptoticdynamicequationsfor asingleunperturbed

one-dimensional frontusingFHN type models. Theequationsfor

the frontposition,$X$_,andvelocity are

$\dot{X}=C$_, ₍₃₎

$\dot{c}=(\alpha_{C^{-}}\alpha)c-\beta c^{3}$

.

These uncoupledequationsdescribe theconvergenceto

con-stantspeedmotion Ising(zerospeed)andBloch fronts. Order

parameterequations for front propagation inonespace

dimen-sionnear aNIB bifurcation have also been derived in Ref.[28] tostudy the effects of external fixed heterogeneities.

The effect ofcurvature is to couple the two equations in

a way that allows for front reversal. Theequations for the dynamicsofasingle two-dimensional front with smooth cur-vature,$\kappa$,are[29]

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FIG. 5: A closer lookatfrontinteractions in the numerical solution of the FHN model from Fig. 1. The repulsiveinteraction

betweenapproaching frontscausesthemtoreversedirection. The reversalis followed by domain splitting.

FIG. 6: Patterns in the$\mathrm{F}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{c}\mathrm{y}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{d}\mathrm{e}- \mathrm{I}\mathrm{o}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{e}-\mathrm{s}_{\mathrm{u}1}\mathrm{f}\mathrm{i}\mathrm{t}\mathrm{e}$reaction show interactionsleadingto front reversals followed by domain splittings. Theseimagesarefromexperiments performed in the Center for Nonlinear DynamicsattheUniversity of Texasat

Austin.

and

$\frac{\partial C}{\partial t}=(\alpha_{c}-\alpha)c-\beta c^{3}+\gamma\kappa+\gamma 0+\frac{\partial^{2}C}{\partial s^{2}}$

$- \frac{\partial C}{\partial s}\int_{0}^{S}\kappa C_{n}d_{S’}$, (5)

where$C_{n}$,the normal front velocity,is given by

$c_{n}=C-D\kappa$, (6)

and$s$is the arclength coordinate along the front. In deriving

theseequationsanasymmetry between theupand downstates

has been introduced. Equation (4), for thecurvatureofthe

front,follows from purelygeometricconsiderations$[30, 31]$.

Equation(5),forthespeedof thefront,is validnearthe front bifurcation and the boundary of instabilityto transverse

per-turbations$[25, 29]$. Theintegralterminbothequations

repre-sents“advection” of changesin$C$and$\kappa$from thestretching

of the arclengthover time. Note thatawayfrom the front$\mathrm{b}\mathrm{i}$,

furcation where thetimescaleassociated with frontreversal,

$(\alpha_{C}-\alpha)-1$, isshort,$C$isnolongeraslow variable andcan

beeliminated adiabatically. Foracircularfront,equation(5) then reduces toanalgebraic relation between the normal front velocity anditscurvaturesuch

as

theonein Fig.4.

We have computed numerical solutions of Eqs.(4)and(5) starting with an almost flat UD frontas aninitial condition. Figure 7 pertains to parameter values in the Bloch regime

wh$e\mathrm{r}\mathrm{e}$the UD and DUfronts$\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{x}\prime \mathrm{i}\mathrm{S}\mathrm{t}$ and areboth stable to

transverseperturbations. The initial frontconvergestoa flat

UDfrontpropagatingat constantspeed. Crossing the

trans-verse

instability boundary

causes

perturbationson thefront

togrow.Nearthe NIBbifurcation,thegrowingcurvature trig-gers the “nucleation” of a frontsegmentwithopposite velocity

asshownin Fig.8.The frontstructuresin theC-splane,

sep-aratingsegmentswithpositive and negativevelocities,pertain

tospiralwavesin the physical two-dimensional plane.

IV. CONCLUSION

We have presented a new mechanism for spontaneous spiral-wave nucleationin bistable media that leadsto spatio-temporal disorder. Unlike most other mechanisms which

in-volvedestabilization of periodic patterns (see however [32,

33]), this mechanism involves destabilization of

_fronts

and

mayinducespatio-temporal disorder fromasingle frontstate.

Thedynamicalequations,(4)and(5),for thefrontspeed and curvature,describe theasymptoticbehavior of frontsnearthe front bifurcation and thetransverseinstabilities. These equa-tionscapture the process ofspiral-wave nucleation and can

be used to analyze the transition from the stable curvature

dynamics shown inFig. 7to dynamicsinvolving nucleation eventsshownin Fig. 8. Itwould beinteresting tofind ifan

intermediate parameterrange existswhere the curvature

fluc-tuatesbut

no

nucleationeventsoccur.

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FIG. 7: A solutiontoequations (4)and(5)when thetwo-dimensional frontis stableto transverseperturbations. The frames

in the toprowshow the timeevolution of$C(s)$ (thin line)and $\kappa(s)$ (thickline). The solutionconvergestoa constant speed

flat$(\kappa=0)$traveling front. The frames in the bottomrowdisplay the corresponding dynamics of the front in the physical

two-dimensionalplane.

APPENDIX

The model used to demonstrate the nucleation of

spiral-vortices in Fig. 1 is a doubly-diffusive version of the $\mathrm{F}\mathrm{i}\mathrm{t}\mathrm{Z}\mathrm{H}\mathrm{u}\mathrm{g}\mathrm{h}$-Nagumo Equations

$u_{t}=\epsilon^{-1}(u-u^{3}-v)+\delta^{-}1u_{x}x$ ,

$v_{t}=u-a_{1}v-a0+v_{xx}$.

Theparameters$a_{0}$and$a_{1}$ arechosen such that theequations

havetwostable uniform solutions(bistable)and$\epsilon$and $\delta$ con-trol th$e$typeand stability of front solutions between those two

stablestates.

Figure9showsacloseup of the middle frameinFig. 1.The

transverseinstability oftheoriginal front solution has already

caused the formationof spiral-vortex pairs along the front. Thespiral-vorticesareidentified with the crossing points of

thezerocontourlines of the$u$ and$v$ fields. At thesepoints

thenormal front velocityiszero. On either side of the

spiral-vortex thefrontpropagatesinopposite directions.

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\dagger _{Electronic address:} _{aric@lanl. gov}

\ddagger _{Electronic address:} $\mathrm{e}\mathrm{h}\mathrm{u}\mathrm{d}\Theta \mathrm{b}\mathrm{g}\mathrm{u}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{l}.$b9u.ac.il

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