Resonant Chemical Oscillations: PatternFormation in Reaction-Diffusion Systems

Anna L. Lin

Department of Physics, Center for nonlinear and complex systems, Duke University, Durham, NC27708-0305

Abstract. Using the Belousov-Zhabotinsky (BZ) chemical system we explore the resonant re-sponse of spatially-extended oscillatory and excitable media to periodic perturbation. Resonance inexcitable media is particularly relevant to biological systems, where excitable dynamics (thresholdresponse to stimulus and refractoriness) are common. Methods to quantify spatio-temporal patternswill be discussed and the resonant patterns in excitable and oscillatory media will be compared. Ex-perimental observations are compared to the results from numerical simulations of the Brusselatorand FitzHugh-Nagumo models and from a forced complex Ginzburg-Landau amplitude equation.

INTRODUCTION

A simple system such as a driven damped pendulum and complex systems such asall living systems, from individual organisms to an entire ecosystem, are examples ofdriven dissipative systems that exist far from thermodynamic equilibrium. A common,perhaps universal feature of sustained non-equilibrium systems is their tendency to formpatterns. Investigations of pattern formation and the transitions between them, as thenon-equilibrium analogue of thermodynamic phase transitions, have been pursued todevelop far from equilibrium physics.

Pattern formation has been experimentally investigated in chemical, fluid, granular,and liquid crystal systems. Amplitude equations provide one theoretical framework tounderstand the origins of non-equilibrium pattern formation [1]. The equations are basedon system symmetries and independent of the microscopic differences among systems.

In our experiments we study resonant pattern formation in a spatially-extended oscil-latory chemical system periodically forced with light. We use numerical simulations toprovide insight and guide our experiments, and discuss our experimental results in thecontext of theory when possible.

RESONANT PATTERN FORMATION IN A CHEMICAL SYSTEM

When an oscillatory nonlinear system is driven by a periodic external stimulus, the sys-tem can lock at rational multiples p : q of the driving frequency. The frequency rangeof this resonant locking at a given p : q depends on the amplitude of the stimulus; thefrequency width of locking increases from zero as the stimulus amplitude increases fromzero, generating an “Arnol’d tongue” in a graph of stimulus amplitude vs stimulus fre-

quency. Physical systems that exhibit frequency locking include electronic circuits [2, 3],Josephson junctions [4], chemical reactions [5–8], fields of fireflies [9, 10], and forcedcardiac systems [11–13]. Most studies of frequency locking have concerned either mapsor systems of a few coupled ordinary differential equations (ODEs). The Arnol’d tonguestructure of the sine circle map has been extensively studied, and the theory of period-ically driven ODE systems has been well developed [14], but there has been very littleanalysis of frequency locking phenomena in partial differential equations (PDEs), ex-cept for a few studies of the parametrically excited Mathieu equation with diffusion anddamping [15–17] and the parametrically excited complex Ginzburg-Landau equation[18, 19]. Our interest here is in the effect of periodic forcing on pattern forming sys-tems such as convecting fluids, liquid crystals, granular media, and reaction-diffusionsystems. Such systems are often subject to periodic forcing (e.g., circadian forcing ofbiological systems), but the effect of forcing on the bifurcations to patterns has not beenexamined in experiments or analyzed in PDE models of these systems.

We have conducted an exploratory study of the effect of periodic forcing on a reaction-diffusion system, the Belousov-Zhabotinsky (BZ) reaction, which is examined in aregime in which the homogeneous reaction (i.e., the well-stirred system, described byODEs) is oscillatory. Spiral waves form in this reaction-diffusion system in the absenceof any external forcing. We can perturb the system using light, since the chemicalkinetics are photo-sensitive. As we shall describe, illumination of the spiral patterns withperiodically modulated light is found to lead to a change in the pattern, and the particularpattern that emerges depends on the locking ratio p : q and on the amplitude and thefrequency of the forcing. We were surprised to find also that bifurcations in the patternscan occur within a single resonant tongue. We have conducted numerical simulationson a PDE model and have found behavior qualitatively similar to that observed in theexperiments.

We now briefly discuss Arnol’d tongues and frequency locking in maps, ODEs, andPDEs. Next we introduce the forced complex Ginzburg-Landau equation, an amplitudeequation describing a generic oscillating field, and then present our laboratory experi-ments and our numerical simulations of two model reaction-diffusion systems, the Brus-selator and the FitzHugh-Nagumo models.

The sine-circle map

Locking has been very well studied for the sine-circle map [20–25]:

θn

1 θn

Ω K2π

sin 2πθn mod 1 (1)

where Ω is (in the present context) the ratio of the natural unperturbed oscillationfrequency to the number of forcing cycles, and K is the strength of the nonlinearity. Thesine circle map can be obtained from consideration of two coupled oscillators. Fig. 1shows the Arnol’d tongue structure for the sine circle map. Within a tongue labeled

FIGURE 1. Phase diagram for the the sine circle map showing the Arnol’d tongues of locking (darkregions) associated with labeled rational winding numbers. Above the critical line K 1 the tonguesoverlap while below the critical line the tongues are uniquely defined. Taken from [26].

p : q, the system is locked, that is, the winding number (where θn is not mod 1),

W limn ∞

θn θ0

n(2)

is given by a fixed value, p : q (loosely, W is the number of full cycles that the systemcompletes per period q). Between the tongues, W is irrational (the behavior is quasi-periodic). At K 0, the irrational winding numbers have the full measure of the unitinterval (although the rational numbers are dense), but as K 1, the Arnol’d lockingtongues grow to the full measure of the unit interval; for larger K, the tongues overlapand the behavior is very complicated.

Simple periodically forced nonlinear ODEs can yield an Arnol’d tongue structurethat is in the same universality class as that for the sine circle map [25]. Laboratoryexperiments of two coupled oscillators have also yielded a similar Arnol’d tonguestructure [4, 27, 28]. We are aware of only one other experimental observation of thestructure of Arnold tongues in a spatially extended system; the observation of lockedflow regimes near 2:1 resonance in externally forced Rayleigh-Bernard convection [28].

The forced complex Ginzburg-Landau equation

We are interested in the generality of our experimental results and so introduce here ageneric model of a forced oscillating field with diffusion, the complex Ginzburg-Landau(CGL) equation. Several groups [18, 19, 29–32] have investigated the stable stationary

Re(A)

Im(A)0

0Re(A)

Im(A)0

0Re(A)

Im(A)0

0FIGURE 2. The complex Amplitude of the stable stationary solutions of the forced CGL equation areplotted in the complex plane which shows the phase-shift among the stable oscillations (purple dots).Stable front solutions (blue lines) connect these states. From C. Elphick, A. Hagberg and E. Meron, Phys.Rev. E 59 5285 (1999).

solutions of this system in which an oscillating field

C r t C0

C1A r t expiω0t

c c (3)

is modulated by the complex amplitude A r t ∂tA r t µ

iν A 1

iα Axx 1 iβ A

2A γnAp 1 (4)

where p is a particular p:1 resonance. The last term in the equation is added to break thetime-translation symmetry which is the affect of the forcing.

The stable stationary solutions of the CGL for p=1,2,3,4 are represented as dots in thecomplex plane plots shown in Fig. 2. The stable front solutions that connect these phase-shifted solutions are shown as lines. Arrows on the lines indicate the fronts are traveling.No arrows indicate stationary fronts. These phase-shifted solutions and the stable frontsolutions that connect them provide predictions of stable resonant patterns, i.e. whetherthey will be stationary or traveling patterns and what phase-shifts in oscillation may beobserved. We use the results of this model to guide our investigation of resonant patternsin the Belousov-Zhabotinsky reaction as described below.

Laboratory experiments: the Belousov-Zhabotinsky reaction

We have examined the effect of external periodic forcing in laboratory experimentson a spatially extended, quasi-two-dimensional reaction-diffusion system. The reaction-diffusion process occurs within a 22 mm diameter by 0 4 mm thick porous membrane,which is continuously fed at its two faces with the chemical reagents of the light-sensitive BZ reaction [5]. The chemical concentrations in the two reserviors were,in Reservior I: 0 22 M malonic acid, 0 20 M sodium bromide, 0 264 M potassiumbromate, 0 80 M sulfuric acid; and in Reservior II: 0 184 M potassium bromate, 1 10 3

M Tris(2,2’-bipyridyl)dichlororuthenium(II)hexahydrate, 0 80 M sulfuric acid. Eachreservior volume is 8.3 ml and the flow rate of chemicals through Reservior I was20 ml/hr while through Reservior II it was 5 ml/hr. Chemicals were premixed beforeentering each reservior; a 10 ml premixer and a 0.5 ml premixer fed Reservior I and II,respectively. The experiments were conducted at room temperature.

A B

in flow

unperturbed spiral

periodicperturbation

time

=

spiral pattern in membrane = concentration gradients of Ru(III)

γ 2

membrane

out flow

perturbationlight

FIGURE 3. Schematic of chemical reactor and light forcing. The spiral traveling wave pictured is froma snapshot of a 13 mm by 13 mm region of a 0 4 mm thick membrane reactor. Orange regions have higherRu(II) concentration while green regions have lower Ru(II) concentration (False color).

Light with a wavelength of 430-470 nm is absorbed by the ruthenium catalyst, Ru(II).The resulting excited state Ru(II)* affects the reaction rates of both the activator reactionset and the inhibitor reaction set of the BZ system [33, 34], thus providing the mechanismby which external perturbations are applied to this naturally oscillating chemical system.Spiral patterns are observed in the unperturbed reactor, as seen in Fig. 3. Periodicforcing is achieved by illuminating the reactor periodically with square wave pulses.The perturbing light is spatially uniform and has a wavelength of 430-470 nm, which isstrongly absorbed by the ruthenium catalyst.

As the perturbation frequency was varied, a sequence of resonance patterns wasobserved, each persisting over a range of perturbation frequencies and amplitudes. Fig. 4shows the patterns observed at the frequency-locked ratios, or winding numbers, p

q

ω f

ω0= 1, 3

2, 2, 3 and 4 for a fixed forcing intensity. In the tongue corresponding to

ω f

ω0

1, the entire membrane is synchronized with the perturbation and oscillatesbetween light and dark. In the nearby ω f

ω0= 3/2 regime, bubble-shaped structures

appear and disappear. The pattern evolves in space and slowly in time, but the temporalpower spectrum of any point in the pattern has well-defined peaks at multiples of ω f

3.

We will discuss other p:q resonant patterns we observed in the following sections, afterwe have introduced some data analysis tools.

FIGURE 4. Diagram showing the different frequency-locked regimes observed in an experiment on aperiodically perturbed ruthenium-catalyzed Belousov-Zhabotinsky reaction-diffusion system. The light-sensitive reaction was perturbed periodically with pulses of light 6 seconds in duration (the natural oscil-lation period is 36 s). The patterns were examined as a function of ω f ω0, where ω f is the perturbationfrequency and ω0 is the natural frequency of the system. In the absence of external perturbation, thepattern is rotating spiral traveling waves (Figure 3). Patterns are shown in pairs, one above the other, attimes separated by ∆t 1 ω f , except for the 1:1 resonance where ∆t 1 2ω f . Striped boxes on thehorizontal axis mark perturbation frequency ranges with the same frequency-locked ratio. Each image is13x13 mm. Taken from [5].

A

CB

D

E

0

0 AC

B

0

0

E

D

FIGURE 5. Reactor images and the corresponding complex Fourier amplitude plots for (a) an unforcedrotating spiral pattern (ω f 0, γ2

0) and (b) a standing wave Ising front pattern (ω f 0 0380 Hz,

γ2 412 W m2). The reactor images are 4 5 4 5 mm2 and 9 9 mm2, respectively, and chemical

conditions are given in the text. Taken from [6].

Determining temporal resonance

Chemical pattern data are collected as a series of snapshots. The natural oscillationfrequency is roughly 60 s and our sampling rate is 0.5 Hz. Each snapshot is 100x100pixels. We measure the temporal response of a pattern by calculating the temporalFourier transform of the time series of each pixel in the pattern. We then calculate thepower spectrum for each pixel and then determine from this the average power spectrumof the entire pattern. If the peak of the strongest mode subharmonic to the forcing iswithin 3% of the forcing frequency, we designate that the pattern is frequency locked tothe forcing. We vary the forcing frequency and intensity in the experiments and explorethe temporal resonant response as we move through the parameter space.

To identify bifurcations in patterns a quantitative measure of the resonant patternsneeded to be developed. Two-dimensional Fourier transforms and analysis methodsusing 2D FFT’s, such as autocorrelation functions did not differentiate our data wellbecause patterns are often comprised of multiple wavelengths and orientations.

Instead, we again made use of the temporal information in the patterns. We usedthe temporal Fourier transform calculated at each pixel in the pattern but this time donot calculate the power spectrum, which throws away phase information. We also donot spatially average the data. Instead, we use a finite width frequency filter to extractthe complex Fourier amplitude a of the temporal sub-harmonic response of the pattern

mode. This is the experimental analog of determining the complex amplitude of theappropriate amplitude equation. Graphs of the complex Fourier amplitude coefficient(at ω f

p) yield information about the relative phase-locked-angle and oscillation mag-

nitude of adjacent discretized oscillators in the different patterns. Fig. 5 illustrates theinformation that can be read from the complex Fourier amplitude plots (phase portraits).The real space images in the top row show a portion of an unforced rotating spiral wavepattern (Fig. 5(a)) and a sub-harmonic standing wave pattern (Fig. 5(b)). The points ineach real space image labeled A B C and D E span the dynamic range of the patterns.The plot below each real space image is a corresponding phase portrait. The point la-beled A in the complex plane is the complex Fourier amplitude coefficient a of the spiralfrequency for the pixel labeled A in the real space image; similarly for points B C.Through the distribution and connectivity of the Fourier coefficients, the phase portraitshows the distribution of oscillation phases and magnitudes along the dashed line in thereal space images. The phase portrait of the unforced spiral pattern in Fig. 5(a) is a cir-cle, indicating that the phase-angles of the discretized oscillations in one wavelength ofthe unforced traveling spiral wave are distributed monotonically from 0 to 2π and havea uniform magnitude. In contrast, the 2 : 1 standing wave pattern shown in Fig. 5(b), thephase portrait obtained after filtering the data at ω f

2 shows that the oscillations remain

π out of phase on either side of the zero amplitude oscillation node, and the magnitudeof the oscillations decreases monotonically as the node is approached.

2 : 1 resonance

Unlike the 1 : 1 resonant response for which we observed only a single qualitativepattern, i.e. completely phase-synchronized oscillations, several qualitatively differentresonance patterns were observed inside of the 2 : 1 tongue. Fig. 6 shows the 2:1resonance tongue as a function of the applied light intensity γ 2 and the perturbationfrequency ω f ; for each data point within the solid lines in Fig. 6 the temporal powerspectrum of the intensity time series for any spatial point in the pattern exhibits a large,sharp response at one-half the forcing frequency. The bending of the 2:1 tongue towardhigher frequencies at low amplitude is a characteristic of the BZ-reaction – the naturalfrequency of the oscillations is γ-dependent. Normalizing the ω f -axis by the naturalfrequency is not feasible because we can not accurately measure the homogeneousnatural oscillation frequency at low forcing amplitudes. There are places in the control-parameter space shown in Fig. 6 where different symbols overlap because of a slow driftin the parameter values over several months; there is no evidence for multiplicity ofpattern states.

Fig. 7 shows the different patterns observed within the 2:1 resonance tongue. Thequantitative phase portrait representation of the data helps identify distinct 2:1 patterns.Pairs of reactor images and phase portraits are shown in Fig. 7 for the different patternsobserved within the 2:1 tongue. A histogram of the phase-angles is shown directly beloweach phase portrait. In Figs. 7 and 8, a is plotted for all pixels in the image, and the linesconnecting adjacent pixels are not shown. The interpretation of the spatial distribution ofthe oscillations in the unforced rotating spiral in Fig. 7(a) and in the Ising front pattern

2

(W/m2)

(Hz)

0

500

1000

0.02 0.06 0.100 f

γ

FIGURE 6. 2:1 resonant tongue in the frequency-intensity plane for the experimental system. Thepatterns (points) within the solid curves resonate at one-half the forcing frequency. The small dots outsidethe curves are non-2:1-resonant. The perturbation is spatially uniform square-wave light pulses of intensityγ2, the square of the light amplitude. See Fig. 7 to connect the symbols and letters to a pattern: + , (b); ,(c); , (d); , (e); , (f); , (g). Taken from [6].

FIGURE 7. (Top row): Reactor images (9 9 mm2) presented using a rainbow (false) color map: (a)unforced rotating spiral wave, (b) rotating spiral wave, (c) mixed rotating spiral and standing wave pattern,(d)-(g) different standing wave patterns. Patterns (b)-(g) exhibit a 2:1 resonance in the temporal powerspectrum of the pattern. (Middle row): Each point in the complex plane corresponds to the temporalFourier amplitude a of a pixel in the image after frequency demodulation at ω f 2. The abscissa is Re(a),the ordinate is Im(a). (Bottom row): Histograms of phase-angles of all the pixels in each image; theabscissa range is [0 2π] radians and the ordinate range is arbitrary. Chemical conditions are given in [35].Parameter values ω f (Hz), γ2 W m2 are: (a) 0,0; (b) 0.1000,119; (c) 0.0625,214; (d) 0.0556,248; (e)0.0417,358, (f) 0.0455,386; (g) 0.0385,412 (see the circled points in Fig. 6). Taken from [6].

FIGURE 8. (Top row): Patterns in the Brusselator model: (a) unlocked rotating spiral wave, (b) two-phase spiral, (c) labyrinth, (c) Ising front pattern. (Middle row): Fourier amplitude complex plane phaseportraits. (Bottom row): Histograms of phase-angle for all the pixels in each image. Parameter values areA = 0.5, B = 1.5. Initial conditions are perpendicular spatial gradients in U and V . Values of ω f /ω0 and γare, respectively: (a) 1.58,0.05 (b) 2.12,0.05; (c) 2.40,0.06; (d) 1.66,0.07. See Fig. 7 caption for ordinateand abscissa axis labels. Taken from [6].

in Fig. 7(g) is the same as that given above for Fig. 5(a) and Fig. 5(b), respectively. Atopological bifurcation occurs between Fig. 7(a) and Fig. 7(b) as the forcing strength isincreased.

Numerical simulations: the Brusselator model

We have also conducted numerical simulations of frequency locking in a reaction-diffusion system with Brusselator kinetics, which is not a model of the BZ reaction butis a simple oscillating chemical system with two chemical species,

∂u∂ t

A B 1 u

1

γ sin 2π f t u2v

Du∇2u (5)

∂v∂ t

Bu u2v

Dv∇2v (6)

where the parametric forcing term is γ sin 2π f t , Du and Dv are the diffusion coeffi-cients of species u and v, and A and B are constant parameters corresponding to initialconcentrations. Two-phase spirals (e.g., Fig. 8(b)), labyrinths (Fig. 8(c), and Ising front(Fig. 8(d)) patterns form within the 2:1 tongue, while unlocked spiral patterns occur out-side the tongue (Fig. 8(a)). The phase portrait in Fig. 8(b) shows no zero crossings at thephase-fronts (no nodes), indicating that the pattern is a traveling wave (a Bloch spiral[29]). The phase of the oscillations varies continuously as one passes from one phase-synchronous domain to the other. In contrast, the phase portraits of the standing wave

patterns in Fig. 8(c) and Fig. 8(d) show that the phase angle remains fixed as the oscilla-tion magnitude monotonically decreases as the node of a phase front is approached, andthe phase angle abruptly changes sign (from π to π) at the node. This type of phasefront is also observed in the laboratory system; see Fig. 7(f) and Fig. 7(g). While wedo not find a one-to-one correspondence between the simulation and the experimentalpatterns, we note that we have not conducted a complete exploration of either system’smulti-parameter space where other patterns may exist.

Oscillatory vs. excitable kinetics

In the BZ reaction illumination of the reacting medium produces an increased in-hibitor concentration. This shifts the chemical kinetics from oscillatory to excitable.Two qualitative features of excitability are a threshold response to stimulus and a re-fractory period, i.e. a relaxation time interval during which re-excitation is not possible.Increased inhibitor concentration increases the refractory period of the medium. As aresult the oscillation frequency [5] decreases. The effect of inhibitory stimuli [5, 35, 36]on excitable behavior has been studied in biological systems such as cardiac musclecells [37] and networks of nerve cells [38, 39].

We examined the effect of periodic light perturbations for two sets of chemical feedconditions in our experiments [40]. One set of the chemical feed conditions we usedhad a low concentration of inhibitor in the reagents fed to the reactor; this results inhigh frequency rotating spirals. The other set had a higher concentration of inhibitor inthe feed, resulting in lower frequency rotating spiral patterns and refractoriness of thechemical medium. The resonant patterns observed for these oscillatory and excitablechemical conditions are shown in Fig. 9. A qualitative difference in the transition fromtraveling to standing wave patterns is seen in the phase portraits of the patterns. Alsonotice in the phase portraits the difference in the orientation (phase) of the distributionswith respect to the forcing phase, which is oriented along the positive Re(a) axis.

The FitzHugh-Nagumo model

To probe the generality of our experimental observations, we examined the resonantresponse of another excitable system, the FitzHugh-Nagumo (FHN) model. The FHNmodel is a derivative of the Hodgkin-Huxley model of nerve response. It is a two variablesystem to which we added a parametric periodic forcing

u u u3 v

∇2uv ε u a1v a0

γ f ω f t δ ∇2v

where the forcing function

f ω f t 12

1 cos ω f t

!#"%$& '

(*)+-,./01,24356()7/ 398:; ,=<>,?@/A0

B /0CA-;D843E/GFH2@56IJ/ 3LKNMOAPHQ4REST8UPHA0VFt = 1885, fc 72 t = 5738, fc 200 t = 4399, fc 140

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

ℜ(ak)

ℑ(a

k)

−1 0 10

1

2

3

4

phaseangle/π

log 10

(# o

f poi

nts)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

ℜ(ak)

ℑ(a

k)

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

φk/π

log 10

(# o

f poi

nts)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

ℜ(ak)

ℑ(a

k)

−1 0 10

1

2

3

4

phaseangle/π

log 10

(# o

f poi

nts)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

ℜ(ak)

ℑ(a

k)

−1 −0.5 0 0.5 10

1

2

3

4

phaseangle/π

log 10

(# o

f poi

nts)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

ℜ(ak)

ℑ(a

k)

−1 0 10

1

2

3

4

phaseangle/π

log 10

(# o

f poi

nts)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

ℜ(ak)

ℑ(a

k)

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

φk/π

log 10

(# o

f poi

nts)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

ℜ(ak)

ℑ(a

k)

−1 0 10

1

2

3

4

phaseangle/π

log 10

(# o

f poi

nts)

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

ℜ(ak)

ℑ(a

k)

−1 −0.5 0 0.5 10

1

2

3

4

phaseangle/π

log 10

(# o

f poi

nts)

FIGURE 9. Transition from traveling to standing patterns in the 2:1 tongue as intensity is increased fortwo sets of chemical conditions and for the FitzHugh-Nagumo model with excitable kinetics. For eachtriplet grouping: (Top row): Reactor images (9 9 mm2) of three different observed patterns, (a) unforcedrotating spiral wave, (b) forced rotating spiral waves, (c) traveling fronts with traces of the spiral core,(d) standing wave pattern. (Middle row): The complex Fourier amplitude a for each image: the abscissais Re(a), the ordinate is Im(a). Each point in the complex plane corresponds to a pixel in the image andis the temporal Fourier amplitude a after frequency demodulation at ω f 2. (Bottom row): Histograms ofphase-angles of all the pixels in each image; the abscissa range is [0 2π] radians and the ordinate rangeis arbitrary. For the oscillatory experimental patterns the parameter values γ 2 W m2 and ω f (Hz) are,respectively: (a) 0, 0; (b) 119, 0.100; (c) 214, 0.0265; (d) 412, 0.0385. For the excitable experimentalpatterns the parameter values γ2 W m2 and ω f (Hz) are, respectively: (a) 0, 0; (b) 164, 0.0384,; (c) 211,

0.0384; (d) 290, 0.0384. For the FitzHugh-Nagumo model patterns the parameter values γ 2 W m2 andω f (Hz) are, respectively: (a) 0, 0; (b) 0.54, 0.240,; (c) 0.64, 0.219; (d) 0.68, 0.180. Chemical conditionsand other model parameters are given in the text. Taken from K. Martinez, Masters Thesis, Univerity ofTexas at Austin, 2002.

oscillates between 0 and 1.The model parameters are fixed at the values a0 = 0.1, a1 = 0.5, ε = 0.05 and δ = 0.2.

At these parameter values the qualitative behavior of the model is in agreement with thequalitative behavior of the BZ experiments. By this we mean that the spiral frequenciesare comparable, and a static forcing increases the observed spiral wavelength, similarto the experiments. The 2:1 resonant patterns observed in the FHN model are plotted inFig. 9. As the forcing strength is increased there is a transition from traveling to standingwaves qualitatively similar to the transition observed in the excitable BZ system. Theorientation of the distribution in the phase portraits of the two excitable systems is alsosimilar.

SUMMARY

We have described our experimental investigations of an oscillatory chemical systemdriven by an external forcing. In our investigations we find qualitative agreement in theresonant pattern formation observed in the light-forced Belousov-Zhabotinsky reactioncompared to that found in numerical simulations of the forced Brusselator reaction-diffusion model and in the complex Ginzburg-Landau equation. This agreement issurprising since the experimental conditions are far from the Hopf bifurcation whilethe CGL equation is valid only close to the onset of oscillations. Qualitatively similarresonant patterns are observed in the BZ system and in the FitzHugh-Nagumo modelwhen the parameters of both systems are tuned so that their dynamics are excitable.

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