PHYSICAL REVIEW E 68, 056307 ~2003!

Reactive dynamics of inertial particles in nonhyperbolic chaotic flows

Adilson E. Motter,1,* Ying-Cheng Lai,2 and Celso Grebogi3

1Max Planck Institute for the Physics of Complex Systems, No¨thnitzer Strasse 38, 01187 Dresden, Germany2Departments of Mathematics, Electrical Engineering, and Physics, Arizona State University, Tempe, Arizona 85287, USA

3Instituto de Fı´sica, Universidade de Sa˜o Paulo, Caixa Postal 66318, 05315-970 Sa˜o Paulo, Brazil~Received 23 April 2003; published 20 November 2003!

Anomalous kinetics of infective~e.g., autocatalytic! reactions in open, nonhyperbolic chaotic flows areimportant for many applications in biological, chemical, and environmental sciences. We present a scalingtheory for the singular enhancement of the production caused by the universal, underlying fractal patterns. Thekey dynamical invariant quantities are theeffective fractal dimensionand effective escape rate, which areprimarily determined by the hyperbolic components of the underlying dynamical invariant sets. The theory isgeneral as it includes all previously studied hyperbolic reactive dynamics as a special case. We introduce aclass of dissipative embedding maps for numerical verification.

DOI: 10.1103/PhysRevE.68.056307 PACS number~s!: 47.70.Fw, 05.45.2a, 47.52.1j, 47.53.1n

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Many chemical and biological processes in fluids acharacterized by a filamental distribution of active particalong fractal invariant sets of the advection chaotic dynaics. These fractal structures act as dynamical catalysts foreaction, which is relevant for a variety of environmenprocesses in open flows, such as ozone depletion in thmosphere@1# and population dynamics of plankton in thoceans@2#.

The study of active processes in open chaotic flowsattracted a great deal of interest from the dynamical syscommunity @3,4#. Most of the studies have been performfor time-dependent two-dimensional incompressible flowin the limit of weak diffusion. The flow is nonturbulent, buthe particle dynamics is considered to be chaotic~Lagrangianchaos! and the active particles to interact with one anothwithout modifying the flow. The advection dynamics of suparticles can be cast in the context of chaotic scatterwhere incoming tracers spend some time in a mixing~scat-tering! region before being scattered along the unstamanifold of the chaotic saddle. As a result, the productsthe reaction concentrate along a fattened-up copy of thestable manifold, giving rise to the observed fractal patter

Although filamental patterns have been observed inture, a clear relation between the observed value of the ftal dimension and the underlying advection dynamicsbeen lacking. For example, in the ‘‘flow past a cylindesystem previously considered@3#, the dimension of the unstable manifold is known to be 2 but the relevant dimensgoverning infective and collisional reactions is about 1This lack of relation, while not reducing the importancethe previous phenomenological characterization of filamedistributions of active particles, has led to some skepticabout the merit of the dynamical system approach toproblem. In general, the advection dynamics can be chaterized as either hyperbolic or nonhyperbolic. In hyperbochaotic scattering, all the periodic orbits are unstablethere are no Kolmogorov-Arnold-Moser~KAM ! tori in thephase space, while the nonhyperbolic counterpart is

*Electronic address: motter@mpipks-dresden.mpg.de

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quently characterized as having both chaotic and marginstable periodic orbits. Fundamental assumptions in sworks are that~1! the active particles are massless pointlitracers and~2! the advection dynamics of these particleshyperbolic@5#. However, in realistic situations, the Lagranian dynamics is typically nonhyperbolic and the active pticles have finite size and inertia. Indeed, fully hyperbosystems are quite rare and represent very idealized situaas the advection dynamics of tracers in fluids is usually cstrained to have a nonhyperbolic character because of noboundary conditions at the surface of obstacles. Obstaare at the same time the origin of Lagrangian chaos andorigin of nonhyperbolicity. Even away from obstacles aboundaries, chaotic motions of tracers typically coexist wregular motions. In addition, the individual active particlare often too large to be regarded as noninertial, as iscase for many species of zooplankton in the sea. Therefnonhyperbolic and inertial effects are prevalent in natureexpected to play an important role in most environmenprocesses. A question of physical importance is then: Whappens to the reactive dynamics when assumptions~1! and~2! are dropped?

In this article, we present a scaling theory for the reactdynamics of inertial particles in nonhyperbolic chaotic flowThe key concepts in our framework are theeffective fractaldimensionand effective escape rate, which are respectivelydefined as

De f f~«!52d ln N~«!

d ln «, ~1!

ke f f~«!52d ln R~n!

dn, ~2!

whereN(«) is the number of«-squares needed to cover threlevant fractal set, andR(n) is the fraction of particles thatakes more thann5n(«) steps to escape from the mixinregion ~see below!. As a representative application of theconcepts, we show, for autocatalytic reactions of the fo

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MOTTER, LAI, AND GREBOGI PHYSICAL REVIEW E68, 056307 ~2003!

A1B→2B, that the area covered byB particles in the steadystate obeys the following scaling law:

AB;F s

eme f f /(22De f f)21G 22De f f

, ~3!

where me f f[(ke f f1k)t, t is the time interval betweensuccessive reactions~time lag!, s is the reaction range, ank is the contraction rate due to dissipation. For nonhypbolic flows in two dimensions,De f f,2 and ke f f.0 arenontrivial functions of the scale«, which can often be re-garded as constants over a wide interval, even thoughD5 lim«→0De f f(«)52 and k5 lim«→0ke f f(«)50. We find,surprisingly, thatDe f f and ke f f are significantly differentfrom D andk, respectively, not only for noninertial but alsfor inertial particles, even though the advection dynamicsthe latter is hyperbolic, meaning thatscarsof the nonhyper-bolic conservative dynamics are observable in the hyperbdynamics of slightly dissipative systems (k!1). The previ-ous relations for noninertial particles in hyperbolic fluids apear as a particular case of our results.

The nature of the chaotic scattering arising in the contof particle advection in incompressible fluids may chanfundamentally as the mass and size of the particles arecreased from zero. Physically, this happens because odetachment of the particle motion from the local fluid mtion. For spherical particles of finite size, the particle velocv[dx/dt is typically different from the~time-dependent!fluid velocity u5u(x,t) and, in first order, is governed by thequation@6#

dv

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dt52a~v2u!. ~4!

The parameters area53r f /(r f12rp) and a5 23 a/St,

wherer f and rp are the densities of fluid and particle, rspectively, and St is the Stokes number, which goes to zin the limit of pointlike particles@7#. For neutrally buoyantparticles, the mass ratio parameter isa51, while for aerosolsand bubbles we havea,1 anda.1, respectively. The inertiaparametera determines the contraction rate or dissipationthephase space~x,v!, which for incompressible flows can bshown to be22a. In the limit a→`, the dynamics is pro-jected on a surface defined byv5u, which corresponds to theadvection dynamics of point particles. Theconfiguration-spaceprojection of the particle motion is strongly influenceby a @8#. For small inertia ~large a), “•v'a21(a21)“•@(v•“)u#5a21(a21)(s22v2), wheres andv areproportional to the strain rate and vorticity of the fluid@9#,respectively. The behaviors of bubbles and aerosols arequalitatively different. For instance, along a closed orbaerosols are pushed outward, while bubbles are pusheward. We first consider bubbles, whose configuration-spdynamics is dissipative when the vorticity overcomesstrain rate.

Dynamically, the inertial effects are effectively those dto dissipation, so that the transition to finite inertia is equivlent to a transition from open Hamiltonian to dissipative d

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namics. It has been recently shown@10# that, while hyper-bolic dynamics is robust, nonhyperbolic chaotic scattertypically undergoes a metamorphosis in the presence ofbitrarily small amount of dissipation. For nonhyperbolscattering in open Hamiltonian systems, particles can spelong time in the neighborhood of KAM tori, resulting in aalgebraic decay for the survival probability of particles in tscattering region. As a consequence, the fractal dimensiothe invariant manifolds is the phase-space dimension@11#.This should be contrasted with the hyperbolic case, whdecay is exponential and fractal dimension is typicasmaller. The dissipation, however, may convert marginastable periodic orbits of the KAM islands into attractors. Tsurvival probability then becomes exponential, the dimesion of the chaotic saddle becomes fractional, and the ovedynamics of the scattering process becomes hyperbolic.

To understand the meanings ofDe f f and ke f f for fractalsets arising in the transition from Hamiltonian nonhyperboto weakly dissipative chaotic scattering, we consider a Ctor set, which is constructed in the interval@0,1# according tothe rule that in thenth time step, a fractionDn5g/(b1n)1d is removed from the middle of each one of theN52n21 remaining subintervals, whereb, g, andd are con-stants. The conservative case corresponds tod50, which ischaracterized by an algebraic decay withn of the total lengthremaining, given byR(n);n2g for n@b, and by a unityfractal dimension for the invariant set,D51. The removedfraction Dn decreases at each time step and, as a resusystematic change of scales is induced, resulting in a nself-similar invariant set that becomes denser as we gosmaller scales. The relevant consequence is that thecounting dimension converges slowly to 1, leading toscale-dependent effective fractal dimensionDe f f'12g/ ln «21 for small «. Similarly, the effective escape ratbehaves aske f f'g/n'g ln 2/ln«21, wheren5n(«) is de-fined as the number of iterations needed to make the lenof each remaining subinterval smaller than«.

The limiting dynamics changes drastically and acquiproperties of hyperbolic dynamics when a small amoof dissipation is allowed, which is modeled by 0,d!g/b.In particular, the total length of the remaining intervadecays exponentially,R(n);(12d)n for n@g/d2b,and the dimension of the invariant set becomes smaller t1, namely,D5 ln 2/ln@2/(12d)#. At finite time, however,the transition from the conservative to the dissipative cis much smoother. For ln«21@b, the effective fractaldimension and the effective escape rate areDe f f(«)' ln 2/ln@2/(12d)#2g8/ ln «21 and ke f f(«)' ln(12d)21

1g8ln@2/(12d)#/ ln «21, respectively, whereg8[g/~12d!.The key feature is that unrealistically small scales arequired to resolve the limiting values of the fractal dimensiand the escape rate, rendering them physically irrelevant.instance, to obtainDe f f.0.95, scales«,10220 may be re-quired. Thus, the physically important characteristics offractal set are the effective dimension and escape rate.

We now present a physical theory for the scaling law~3!,valid for autocatalytic reactions in two-dimensional timperiodic flows. Consider the area covered byB particles inthe open part of the flow~i.e., region where particles even

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REACTIVE DYNAMICS OF INERTIAL PARTICLES IN . . . PHYSICAL REVIEW E68, 056307 ~2003!

tually escape to infinity! and, to be specific, that the time lat is integer multiple of the flow’s period. After a sufficientllong time from the onset of the reaction, the reagentB isdistributed along stripes of approximately uniform widtmimicking the unstable manifold. The average widthe of thestripes changes aperiodically over time until the steady sis reached, when it undertakes the periodicityt of the reac-tion. We assume that the reaction is sufficiently close tosteady state so thatDe f f(e) and ke f f(e) can be consideredconstant over time. This condition is not very restrictive bcause for many systemsDe f f and ke f f are essentially constant over several decades~see below!. Therefore, for scaleslarger thane, the area covered byB particles can be regardeas a fractal characterized by dimensionDe f f(e) and escaperateke f f(e).

Let e (n21)(t) and e (n)(0) denote the average widths othe stripes right before and right after thenth reaction@3#,respectively. Between successive reactions, the stripes sdue to escape and dissipation as follows:e (n)(t)5e (n)(0)e2he f ft, where he f f5(ke f f1k)/(22De f f) playsthe role of aneffective ~contracting! Lyapunov exponent,while k accounts for the nonconservative contribution. Whthe reaction occurs, the widening due to the reaction is pportional to the reaction range:e (n11)(0)2e (n)(t)}s. Thearea covered byB particles right before the (n11)th reac-tion A B

(n) satisfiesA B(n)}@e (n)(t)#22De f f. These relations can

then be combined to yield the following recursive relatifor the area:A B

(n11)5e2me f f@(A B(n))1/(22De f f)1cs#22De f f,

whereme f f5(ke f f1k)t andc is a constant geometric factoFrom the conditionA B

(n11)5A B(n) , our main scaling~3! fol-

lows for the areaAB in the steady state@12#. This scalingholds for both noninertial and inertial particles, regardlesswhether the flow is hyperbolic or nonhyperbolic. The hypbolic case with inertial particles is studied in Ref.@13#. Thescaling~3! represents a further step toward generality sincis also valid for nonhyperbolic flows.

To make possible a numerical verification of the scallaw ~3!, it is necessary at present to use discrete-time mTo construct a class of maps that captures all essentialtures of continuous-time chaotic flows, we note the folloing: ~1! the fluid dynamics, determined bydx/dt5u, is em-beddedin the particle’s advection equation and is recovein the limit a→`; ~2! the phase-space contraction is detmined by a ~irrespective ofa!; ~3! for small inertia, theconfiguration-space contraction is proportional toa21(a21). For an area-preserving mapxn115M (xn), represent-ing the dynamics of a time-periodic incompressible fluidpossible choice for the correspondingembedding maprepre-senting the inertial particle dynamics isxn122M (xn11)5e2a@axn112M (xn)#, where the factors involvinga andaare naturally imposed by the particle dynamics@14#. This canbe written as

xn115M ~xn!1dn , ~5!

dn115e2a@axn112M ~xn!#, ~6!

wherex andd can be interpreted as the configuration-spacoordinates and the detachment from the fluid velocity,

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spectively, so that~x, d! represents the phase-space coornates. This class of embedding maps can be a paradigaddress many problems in inertial advection dynamics acaptures the essential properties of Eq.~4!. In particular, it isuniformly dissipative, with phase-space contraction requal toe22a; the noninertial dynamicsxn115M (xn) is re-covered in the limita→`; and the configuration-space contraction rate is proportional toe2a(a21) for e2a(a21)!1, in agreement with the distinct behavior expectedaerosols and bubbles. Therefore, for finitea, a rich higherdimensional dynamics witha-dependentx-space projectionis expected. Next we consider such a dynamics for botha.1anda,1.

To simulate the flow, we consider a two-dimensional arpreserving map that has a pronounced nonhyperbolic chater @11#: (x,y)→@l(x2w2/4),l21(y1w2)#, where w[x1y/4 andl.1 is the bifurcation parameter. The dynamicsnonhyperbolic forl&6.5. Forl54, for example, there is amajor KAM island in thexy space, as shown in Fig. 1~a!.Also, from Fig. 1~a!, one can see tangencies betweenstable and unstable manifolds in the neighborhood ofKAM island, which is a signature of nonhyperbolicity. It iwell established that, within the nonhyperbolic region, tdimension of the invariant manifolds isD52 and the escaperate isk50 @10,11#. When this map is embedded in Eqs.~5!and ~6!, for a.1, the xy projection of the resulting four-dimensional map is dissipative in the mixing region~KAMislands and their neighborhoods!. In this regime, the dissipation stabilizes marginally stable periodic orbits in the KAislands of the conservative map, converting the KAM islanand neighborhood into the corresponding basin of attracof the newly created attractors, as shown in Fig. 1~b!. Thebasin itself extends around the mixing region, mimicking tstable manifold of the conservative dynamics. As a result,tangencies between the invariant manifolds apparently dispear, suggesting that the advection dynamics of bubbleticles is hyperbolic. For a,1, on the other hand, theconfiguration-space projection expands in the mixing regand almost all the orbits eventually escape to infinity. Hoever, for small inertia anda close to 1, particles in the regions corresponding to KAM islands of the conservative dnamics and neighborhood arealmost trappedin the sensethat the time it takes to escape is much larger in these regthan outside them. These regions are neglected in our ansis of the open part of the flow, as shown in Fig. 1~c!, be-cause filamental structures cannot be resolved inside the

Numerical simulation of the autocatalytic reaction is peformed by dividing the mixing region with a grid where thsize of the cells represents the reaction ranges. Particles areplaced in the center of the cells. When a reaction takes pin a cell occupied by aB particle, all the cells adjacent to iare infectedwith B particles@15#. We assume thatA is thebackground material and that the reaction takes place simtaneously for all the particles at time intervalst @16#. If westart with a small seed ofB particles near the stable manifoldafter a transient time a steady state is reached whereB par-ticles are accumulated along a fattened-up copy of thestable manifold, as shown in Fig. 1~d! for massless point

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MOTTER, LAI, AND GREBOGI PHYSICAL REVIEW E68, 056307 ~2003!

(c) (b)

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FIG. 1. Forl54, ~a! KAM island ~light gray!, and stable~gray! and unstable~black! manifolds fora5`. ~b! Fixed point attractor~blackdot!, basin of attraction~gray!, and unstable manifold~black!, for a51.05 anda51. ~c! Stable~gray! and unstable~black! manifolds fora50.95 anda51, outside the region covered by the almost trapped orbits~light gray!. Particles are launched with initial velocity matchinthe fluid velocity (d050). ~d!–~f! Corresponding area covered byB particles in the ‘‘open’’ part of the flow right before the reaction, fort55ands5531023.

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particles, in Fig. 1~e! for bubbles, and in Fig. 1~f! for aero-sols. In the computation, particlesA andB are set to have thesame mass ratio and inertia parameters. To compute thfective fractal dimensionDe f f of the unstable manifold, weuse the uncertainty algorithm@17# applied to the first-orderapproximation of the inverse map. The effective dimensturns out to be constant over many orders of magnitudevariations in« and it is approximately the same for bononinertial and slightly inertial bubble particles (De f f51.73 for «.10215), while it is somewhat smaller foslightly inertial aerosol particles (De f f51.68 for «.10215), as shown in Fig. 2~a!. Strong evidence of the scaing law ~3! is presented in Fig. 2~b! for two different valuesof the time lagt, where the scaling exponent is consistewith De f f51.73 for noninertial and bubble particles, an

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with De f f51.68 for aerosol particles. We see that evthough the areaAB changes with the inertial properties of thparticles, thescalingof AB remains essentially the same fobubbles, as expected from our Cantor-set model.

It is instructive to compare this result with the reactiofree dynamics. The dimension of nonhyerbolic invariant scan be argued to be integer by mean of a zoom-in techniqwhere a fast numerical convergence is achieved by focuon the densest parts of the fractal@11#. The reaction, how-ever, has aglobal character, as it takes place along the ustable manifold around the entire mixing region. This makthe convergence of the relevant effective dimensiontremely slow, and that is why the effective dimension is aparently constant.

In summary, we have shown that the dynamical syst

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FIG. 2. Forl54: ~a! Effective dimension of the unstable manifold as computed from the uncertainty method, wheref («) is the fractionof «-uncertain points in the linex50, 0,y,0.1, of the time-reversed dynamics.~b! Scaling of the relative areaAB covered byB particles@in the region shown in Figs. 1~d!–~f!, right before the reaction# as a function of the reaction ranges for two choices of the time lagt. Inboth plots, stars correspond to noninertial particles (a5`), circles to bubble particles witha51.05 anda51, and plus signs to aerosoparticles witha50.95 anda51. The aerosol data in~a! are shifted vertically downward for clarity.

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REACTIVE DYNAMICS OF INERTIAL PARTICLES IN . . . PHYSICAL REVIEW E68, 056307 ~2003!

approach to the reactive dynamics in imperfectly mixflows also applies to realistic situations where nonhyperband inertial effects are relevant. The rate equations of retive processes are primarily governed by finite-time dynaics and as such change smoothly in thenoninertial→inertialtransition, which is in sharp contrast with the metamorphoundergone by the long-term and asymptotic dynamicsreaction-free particles. We have focused on autocatalyticactions, but our results are expected to hold wheneverreaction front mimics the underlying unstable manifold athe activity takes place along the boundary of a fattenedfractal. Examples of this kind of process include infectireactions~e.g., combustion@18#! and collisional reactions ingeneral ~e.g., A1B→2C, where an unlimited amount o

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materialA is present and materialB is continuously injectedin the vicinity of the stable manifold@3#!. Finally, we observethat our analysis does not rely on the existence of a wdefined fractal set in the advection dynamics. The resremain valid as long as effective values for the fractal dimsion and escape rate can be properly defined and are appmately constant over the relevant interval of observatiThis is important for environmental processes, whose unlying dynamics is only partially understood@13#.

A.E.M. and Y.C.L. were supported by AFOSR througGrant No. F49620-03-1-0290. C.G. was supported by Fapand CNPq. A.E.M. thanks Tamas Te´l for illuminating discus-sions.

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@1# S. Edouard, B. Legras, F. Lefevre, and R. Eymard, Nat~London! 384, 444 ~1996!.

@2# E.R. Abraham, Nature~London! 391, 577 ~1998!.@3# Z. Toroczkai, G. Ka´rolyi, A. Pentek, T. Tel, and C. Grebogi,

Phys. Rev. Lett.80, 500 ~1998!.@4# See, for example,Active chaotic flow, focus issue of Chaos12,

~2002!, and references therein.@5# Inertial effects on reactive particles have been previously c

sidered for turbulent flows@R. Reigada, F. Sague´s, and J.M.Sancho, Phys. Rev. E64, 026307~2001!# and for cellular flows@T. Nishikawa, Z. Toroczkai, C. Grebogi, and T. Te´l, ibid. 65,026216~2002!; Z.H. Liu, Y.-C. Lai, and J.M. Lopez, Chaos12,417 ~2002!#.

@6# M.R. Maxey and J.J. Riley, Phys. Fluids26, 883 ~1983!.@7# Equation~4! is an approximation of the Maxey-Riley equatio

obtained by neglecting the Faxe´n corrections, the BassetBoussinesq history force, and the term@~v2u!•“#u @A. Babi-ano, J.H.E. Cartwright, O. Piro, and A. Provenzale, Phys. RLett. 84, 5764~2000!#.

@8# I.J. Benczik, Z. Toroczkai, and T. Te´l, Phys. Rev. Lett.89,164501~2002!.

@9# M.R. Maxey, Phys. Fluids30, 1915~1987!; E. Balkovsky, G.Falkovich, and A. Fouxon, Phys. Rev. Lett.86, 2790~2001!.

@10# A.E. Motter and Y.-C. Lai, Phys. Rev. E65, 015205~2002!.@11# Y.-T. Lau, J.M. Finn, and E. Ott, Phys. Rev. Lett.66, 978

~1991!.@12# A similar scaling law can be derived in the continuous-tim

limit t→0.@13# T. Tel, T. Nishikawa, A. E. Motter, C. Grebogi, and Z. Toroc

zkai ~unpublished!.

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@14# For a51, this equation reduces to the first-order approximtion of the bailout embedding map introduced by@J.H.E. Cart-wright, M.O. Magnasco, and O. Piro, Phys. Rev. E65, 045203~2002!# to study the advection dynamics of neutrally buoyaparticles.

@15# For numerical convenience, the autocatalytic reaction islowed by coalescence, so that when more than oneB particle isfound in the same cell, they are all replaced by a singleBparticle in the center of the cell. Without coalescence, the nuber of B particles diverges because of the shrinking duedissipation, but the areaAB still reaches a steady state. Sincwe focus on theareacovered byB particles, our results do nodepend on the coalescence.

@16# The time lagt is assumed to be on the order of or smaller ththe relevant time scale of the advection dynamicst f

51/ke f f(s). For t@t f , the reaction undergoes an emptyintransition and the Cantori structures near the KAM tori canbe neglected. In this regime, almost all the particles alongunstable manifold escape between successive reactions,the hypothesis that reagentB is distributed along approxi-mately uniform stripes breaks down outside the Cantori. Hoever, since Cantori are obstacles to the transport of particfor small enoughs, a high concentration of particles can stbe found inside the hierarchy of Cantori structures. For detwe refer to A. P. S. de Moura and C. Grebogi~unpublished!.

@17# C. Grebogi, S.W. McDonald, E. Ott, and J.A. Yorke, PhyLett. 99A, 415 ~1983!.

@18# I.Z. Kiss, J.H. Merkin, S.K. Scott, P.L. Simon, S. Kalliadasand Z. Neufeld, Physica D176, 67 ~2003!.

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