Resonant Chemical Oscillations: Pattern Formation in Reaction-Diffusion Systems

Transcription

1 Resonant Chemical Oscillations: Pattern Formation in Reaction-Diffusion Systems Anna L. Lin Department of Physics, Center for nonlinear and complex systems, Duke University, Durham, NC Abstract. Using the Belousov-Zhabotinsky (BZ chemical system we explore the resonant response of spatially-extended oscillatory and excitable media to periodic perturbation. Resonance in excitable media is particularly relevant to biological systems, where excitable dynamics (threshold response to stimulus and refractoriness are common. Methods to quantify spatio-temporal patterns will be discussed and the resonant patterns in excitable and oscillatory media will be compared. Experimental observations are compared to the results from numerical simulations of the Brusselator and 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 as all living systems, from individual organisms to an entire ecosystem, are examples of driven dissipative systems that exist far from thermodynamic equilibrium. A common, perhaps universal feature of sustained non-equilibrium systems is their tendency to form patterns. Investigations of pattern formation and the transitions between them, as the non-equilibrium analogue of thermodynamic phase transitions, have been pursued to develop far from equilibrium physics. Pattern formation has been experimentally investigated in chemical, fluid, granular, and liquid crystal systems. Amplitude equations provide one theoretical framework to understand the origins of non-equilibrium pattern formation [1]. The equations are based on system symmetries and independent of the microscopic differences among systems. In our experiments we study resonant pattern formation in a spatially-extended oscillatory chemical system periodically forced with light. We use numerical simulations to provide insight and guide our experiments, and discuss our experimental results in the context of theory when possible. RESONANT PATTERN FORMATION IN A CHEMICAL SYSTEM When an oscillatory nonlinear system is driven by a periodic external stimulus, the system can lock at rational multiples p : q of the driving frequency. The frequency range of this resonant locking at a given p : q depends on the amplitude of the stimulus; the frequency width of locking increases from zero as the stimulus amplitude increases from zero, generating an Arnol d tongue in a graph of stimulus amplitude vs stimulus fre-

2 quency. Physical systems that exhibit frequency locking include electronic circuits [, ], Josephson junctions [], chemical reactions [5 8], fields of fireflies [9, 1], and forced cardiac systems [11 1]. Most studies of frequency locking have concerned either maps or systems of a few coupled ordinary differential equations (ODEs. The Arnol d tongue structure of the sine circle map has been extensively studied, and the theory of periodically driven ODE systems has been well developed [1], but there has been very little analysis of frequency locking phenomena in partial differential equations (PDEs, except for a few studies of the parametrically excited Mathieu equation with diffusion and damping [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 systems such as convecting fluids, liquid crystals, granular media, and reaction-diffusion systems. Such systems are often subject to periodic forcing (e.g., circadian forcing of biological systems, but the effect of forcing on the bifurcations to patterns has not been examined in experiments or analyzed in PDE models of these systems. We have conducted an exploratory study of the effect of periodic forcing on a reactiondiffusion system, the Belousov-Zhabotinsky (BZ reaction, which is examined in a regime in which the homogeneous reaction (i.e., the well-stirred system, described by ODEs is oscillatory. Spiral waves form in this reaction-diffusion system in the absence of any external forcing. We can perturb the system using light, since the chemical kinetics are photo-sensitive. As we shall describe, illumination of the spiral patterns with periodically modulated light is found to lead to a change in the pattern, and the particular pattern that emerges depends on the locking ratio p : q and on the amplitude and the frequency of the forcing. We were surprised to find also that bifurcations in the patterns can occur within a single resonant tongue. We have conducted numerical simulations on a PDE model and have found behavior qualitatively similar to that observed in the experiments. We now briefly discuss Arnol d tongues and frequency locking in maps, ODEs, and PDEs. Next we introduce the forced complex Ginzburg-Landau equation, an amplitude equation describing a generic oscillating field, and then present our laboratory experiments and our numerical simulations of two model reaction-diffusion systems, the Brusselator and the FitzHugh-Nagumo models. The sine-circle map Locking has been very well studied for the sine-circle map [ 5]: θ n 1 θ n Ω K π sin πθ n mod 1 (1 where Ω is (in the present context the ratio of the natural unperturbed oscillation frequency to the number of forcing cycles, and K is the strength of the nonlinearity. The sine circle map can be obtained from consideration of two coupled oscillators. Fig. 1 shows the Arnol d tongue structure for the sine circle map. Within a tongue labeled

3 FIGURE 1. Phase diagram for the the sine circle map showing the Arnol d tongues of locking (dark regions associated with labeled rational winding numbers. Above the critical line K 1 the tongues overlap while below the critical line the tongues are uniquely defined. Taken from [6]. p : q, the system is locked, that is, the winding number (where θ n is not mod 1, W θ n θ lim n n is given by a fixed value, p : q (loosely, W is the number of full cycles that the system completes per period q. Between the tongues, W is irrational (the behavior is quasiperiodic. At K, the irrational winding numbers have the full measure of the unit interval (although the rational numbers are dense, but as K 1, the Arnol d locking tongues grow to the full measure of the unit interval; for larger K, the tongues overlap and the behavior is very complicated. Simple periodically forced nonlinear ODEs can yield an Arnol d tongue structure that is in the same universality class as that for the sine circle map [5]. Laboratory experiments of two coupled oscillators have also yielded a similar Arnol d tongue structure [, 7, 8]. We are aware of only one other experimental observation of the structure of Arnold tongues in a spatially extended system; the observation of locked flow regimes near :1 resonance in externally forced Rayleigh-Bernard convection [8]. ( The forced complex Ginzburg-Landau equation We are interested in the generality of our experimental results and so introduce here a generic model of a forced oscillating field with diffusion, the complex Ginzburg-Landau (CGL equation. Several groups [18, 19, 9 ] have investigated the stable stationary

4 Im(A Im(A Im(A Re(A Re(A Re(A FIGURE. The complex Amplitude of the stable stationary solutions of the forced CGL equation are plotted 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 (1999. solutions of this system in which an oscillating field C r t C C 1 A r t expiω t c c ( is modulated by the complex amplitude A r t t A r t µ iν A 1 iα A xx 1 iβ A A γ n Ā p 1 ( where p is a particular p:1 resonance. The last term in the equation is added to break the time-translation symmetry which is the affect of the forcing. The stable stationary solutions of the CGL for p=1,,, are represented as dots in the complex plane plots shown in Fig.. The stable front solutions that connect these phaseshifted 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 front solutions that connect them provide predictions of stable resonant patterns, i.e. whether they will be stationary or traveling patterns and what phase-shifts in oscillation may be observed. We use the results of this model to guide our investigation of resonant patterns in the Belousov-Zhabotinsky reaction as described below. Laboratory experiments: the Belousov-Zhabotinsky reaction We have examined the effect of external periodic forcing in laboratory experiments on a spatially extended, quasi-two-dimensional reaction-diffusion system. The reactiondiffusion process occurs within a mm diameter by mm thick porous membrane, which is continuously fed at its two faces with the chemical reagents of the lightsensitive BZ reaction [5]. The chemical concentrations in the two reserviors were, in Reservior I: M malonic acid, M sodium bromide, 6 M potassium bromate, 8 M sulfuric acid; and in Reservior II: 18 M potassium bromate, 1 1 M Tris(, -bipyridyldichlororuthenium(iihexahydrate, 8 M sulfuric acid. Each reservior volume is 8. ml and the flow rate of chemicals through Reservior I was ml/hr while through Reservior II it was 5 ml/hr. Chemicals were premixed before entering each reservior; a 1 ml premixer and a.5 ml premixer fed Reservior I and II, respectively. The experiments were conducted at room temperature.

5 out flow A B membrane perturbation light in flow unperturbed spiral = periodic perturbation γ FIGURE. Schematic spiral of pattern chemical in membrane reactor and = concentration light forcing. gradients The spiral of traveling Ru(III wave pictured is from a snapshot of a 1 mm by 1 mm region of a mm thick membrane reactor. Orange regions have higher Ru(II concentration while green regions have lower Ru(II concentration (False color. time Light with a wavelength of -7 nm is absorbed by the ruthenium catalyst, Ru(II. The resulting excited state Ru(II* affects the reaction rates of both the activator reaction set and the inhibitor reaction set of the BZ system [, ], thus providing the mechanism by which external perturbations are applied to this naturally oscillating chemical system. Spiral patterns are observed in the unperturbed reactor, as seen in Fig.. Periodic forcing is achieved by illuminating the reactor periodically with square wave pulses. The perturbing light is spatially uniform and has a wavelength of -7 nm, which is strongly absorbed by the ruthenium catalyst. As the perturbation frequency was varied, a sequence of resonance patterns was observed, each persisting over a range of perturbation frequencies and amplitudes. Fig. shows the patterns observed at the frequency-locked ratios, or winding numbers, p q ω f ω = 1,,, and for a fixed forcing intensity. In the tongue corresponding to ω f ω 1, the entire membrane is synchronized with the perturbation and oscillates between light and dark. In the nearby ω f ω = / regime, bubble-shaped structures appear and disappear. The pattern evolves in space and slowly in time, but the temporal power spectrum of any point in the pattern has well-defined peaks at multiples of ω f. We will discuss other p:q resonant patterns we observed in the following sections, after we have introduced some data analysis tools.

6 FIGURE. Diagram showing the different frequency-locked regimes observed in an experiment on a periodically perturbed ruthenium-catalyzed Belousov-Zhabotinsky reaction-diffusion system. The lightsensitive reaction was perturbed periodically with pulses of light 6 seconds in duration (the natural oscillation period is 6 s. The patterns were examined as a function of ω f ω, where ω f is the perturbation frequency and ω is the natural frequency of the system. In the absence of external perturbation, the pattern is rotating spiral traveling waves (Figure. Patterns are shown in pairs, one above the other, at times separated by t 1 ω f, except for the 1:1 resonance where t 1 ω f. Striped boxes on the horizontal axis mark perturbation frequency ranges with the same frequency-locked ratio. Each image is 1x1 mm. Taken from [5].

7 A B C D E D B A C E FIGURE 5. Reactor images and the corresponding complex Fourier amplitude plots for (a an unforced rotating spiral pattern (ω f, γ and (b a standing wave Ising front pattern (ω f 8 Hz, γ 1 W m. The reactor images are 5 5 mm and 9 9 mm, 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 oscillation frequency is roughly 6 s and our sampling rate is.5 Hz. Each snapshot is 1x1 pixels. We measure the temporal response of a pattern by calculating the temporal Fourier transform of the time series of each pixel in the pattern. We then calculate the power spectrum for each pixel and then determine from this the average power spectrum of the entire pattern. If the peak of the strongest mode subharmonic to the forcing is within % of the forcing frequency, we designate that the pattern is frequency locked to the forcing. We vary the forcing frequency and intensity in the experiments and explore the temporal resonant response as we move through the parameter space. To identify bifurcations in patterns a quantitative measure of the resonant patterns needed to be developed. Two-dimensional Fourier transforms and analysis methods using D FFT s, such as autocorrelation functions did not differentiate our data well because patterns are often comprised of multiple wavelengths and orientations. Instead, we again made use of the temporal information in the patterns. We used the temporal Fourier transform calculated at each pixel in the pattern but this time do not calculate the power spectrum, which throws away phase information. We also do not spatially average the data. Instead, we use a finite width frequency filter to extract the complex Fourier amplitude a of the temporal sub-harmonic response of the pattern

8 mode. This is the experimental analog of determining the complex amplitude of the appropriate amplitude equation. Graphs of the complex Fourier amplitude coefficient (at ω f p yield information about the relative phase-locked-angle and oscillation magnitude of adjacent discretized oscillators in the different patterns. Fig. 5 illustrates the information 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 wave pattern (Fig. 5(a and a sub-harmonic standing wave pattern (Fig. 5(b. The points in each 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 labeled A in the complex plane is the complex Fourier amplitude coefficient a of the spiral frequency 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 portrait shows the distribution of oscillation phases and magnitudes along the dashed line in the real space images. The phase portrait of the unforced spiral pattern in Fig. 5(a is a circle, indicating that the phase-angles of the discretized oscillations in one wavelength of the unforced traveling spiral wave are distributed monotonically from to π and have a uniform magnitude. In contrast, the : 1 standing wave pattern shown in Fig. 5(b, the phase portrait obtained after filtering the data at ω f shows that the oscillations remain π out of phase on either side of the zero amplitude oscillation node, and the magnitude of the oscillations decreases monotonically as the node is approached. : 1 resonance Unlike the 1 : 1 resonant response for which we observed only a single qualitative pattern, i.e. completely phase-synchronized oscillations, several qualitatively different resonance patterns were observed inside of the : 1 tongue. Fig. 6 shows the :1 resonance tongue as a function of the applied light intensity γ and the perturbation frequency ω f ; for each data point within the solid lines in Fig. 6 the temporal power spectrum 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 :1 tongue toward higher frequencies at low amplitude is a characteristic of the BZ-reaction the natural frequency of the oscillations is γ-dependent. Normalizing the ω f -axis by the natural frequency is not feasible because we can not accurately measure the homogeneous natural oscillation frequency at low forcing amplitudes. There are places in the controlparameter space shown in Fig. 6 where different symbols overlap because of a slow drift in the parameter values over several months; there is no evidence for multiplicity of pattern states. Fig. 7 shows the different patterns observed within the :1 resonance tongue. The quantitative phase portrait representation of the data helps identify distinct :1 patterns. Pairs of reactor images and phase portraits are shown in Fig. 7 for the different patterns observed within the :1 tongue. A histogram of the phase-angles is shown directly below each phase portrait. In Figs. 7 and 8, a is plotted for all pixels in the image, and the lines connecting adjacent pixels are not shown. The interpretation of the spatial distribution of the oscillations in the unforced rotating spiral in Fig. 7(a and in the Ising front pattern

9 1 γ (W/m (Hz FIGURE 6. :1 resonant tongue in the frequency-intensity plane for the experimental system. The patterns (points within the solid curves resonate at one-half the forcing frequency. The small dots outside the curves are non-:1-resonant. The perturbation is spatially uniform square-wave light pulses of intensity γ, 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]. f FIGURE 7. (Top row: Reactor images (9 9 mm 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 :1 resonance in the temporal power spectrum of the pattern. (Middle row: Each point in the complex plane corresponds to the temporal Fourier amplitude a of a pixel in the image after frequency demodulation at ω f. The abscissa is Re(a, the ordinate is Im(a. (Bottom row: Histograms of phase-angles of all the pixels in each image; the abscissa range is [ π] radians and the ordinate range is arbitrary. Chemical conditions are given in [5]. Parameter values ω f (Hz, γ W m are: (a,; (b.1,119; (c.65,1; (d.556,8; (e.17,58, (f.55,86; (g.85,1 (see the circled points in Fig. 6. Taken from [6].

10 FIGURE 8. (Top row: Patterns in the Brusselator model: (a unlocked rotating spiral wave, (b twophase spiral, (c labyrinth, (c Ising front pattern. (Middle row: Fourier amplitude complex plane phase portraits. (Bottom row: Histograms of phase-angle for all the pixels in each image. Parameter values are A =.5, B = 1.5. Initial conditions are perpendicular spatial gradients in U and V. Values of ω f /ω and γ are, respectively: (a 1.58,.5 (b.1,.5; (c.,.6; (d 1.66,.7. See Fig. 7 caption for ordinate and 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. A topological bifurcation occurs between Fig. 7(a and Fig. 7(b as the forcing strength is increased. Numerical simulations: the Brusselator model We have also conducted numerical simulations of frequency locking in a reactiondiffusion system with Brusselator kinetics, which is not a model of the BZ reaction but is a simple oscillating chemical system with two chemical species, u t A B 1 u 1 γ sin π ft u v D u u (5 v t Bu u v D v v (6 where the parametric forcing term is γ sin π ft, D u and D v are the diffusion coefficients of species u and v, and A and B are constant parameters corresponding to initial concentrations. Two-phase spirals (e.g., Fig. 8(b, labyrinths (Fig. 8(c, and Ising front (Fig. 8(d patterns form within the :1 tongue, while unlocked spiral patterns occur outside the tongue (Fig. 8(a. The phase portrait in Fig. 8(b shows no zero crossings at the phase-fronts (no nodes, indicating that the pattern is a traveling wave (a Bloch spiral [9]. The phase of the oscillations varies continuously as one passes from one phasesynchronous domain to the other. In contrast, the phase portraits of the standing wave

11 patterns in Fig. 8(c and Fig. 8(d show that the phase angle remains fixed as the oscillation magnitude monotonically decreases as the node of a phase front is approached, and the phase angle abruptly changes sign (from π to π at the node. This type of phase front is also observed in the laboratory system; see Fig. 7(f and Fig. 7(g. While we do not find a one-to-one correspondence between the simulation and the experimental patterns, we note that we have not conducted a complete exploration of either system s multi-parameter space where other patterns may exist. Oscillatory vs. excitable kinetics In the BZ reaction illumination of the reacting medium produces an increased inhibitor concentration. This shifts the chemical kinetics from oscillatory to excitable. Two qualitative features of excitability are a threshold response to stimulus and a refractory 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 a result the oscillation frequency [5] decreases. The effect of inhibitory stimuli [5, 5, 6] on excitable behavior has been studied in biological systems such as cardiac muscle cells [7] and networks of nerve cells [8, 9]. We examined the effect of periodic light perturbations for two sets of chemical feed conditions in our experiments []. One set of the chemical feed conditions we used had a low concentration of inhibitor in the reagents fed to the reactor; this results in high frequency rotating spirals. The other set had a higher concentration of inhibitor in the feed, resulting in lower frequency rotating spiral patterns and refractoriness of the chemical medium. The resonant patterns observed for these oscillatory and excitable chemical conditions are shown in Fig. 9. A qualitative difference in the transition from traveling to standing wave patterns is seen in the phase portraits of the patterns. Also notice in the phase portraits the difference in the orientation (phase of the distributions with 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 resonant response of another excitable system, the FitzHugh-Nagumo (FHN model. The FHN model is a derivative of the Hodgkin-Huxley model of nerve response. It is a two variable system to which we added a parametric periodic forcing u u u v u v ε u a 1 v a γ f ω f t δ v where the forcing function f ω f t 1 1 cos ω f t

12 !#"%$& ' (*+-,./1, 56( 7 / 98:;,=<>,?@/A B /CA-;D8E/GFH@56IJ/ LKNMOA PHQREST8UPHAVF t = 1885, fc 7 t = 578, fc t = 99, fc log (# log of points (# of points R(a k R(a k R(a k R(a k R(a k R(a k R(a k R(a k 1 log (# of log points (# of points phaseangle/π 6 φ k /π log (# log of points (# of points log (# log of points (# of points phaseangle/π phaseangle/π φ phaseangle/π k /π phaseangle/π phaseangle/π FIGURE 9. Transition from traveling to standing patterns in the :1 tongue as intensity is increased for two sets of chemical conditions and for the FitzHugh-Nagumo model with excitable kinetics. For each triplet grouping: (Top row: Reactor images (9 9 mm of three different observed patterns, (a unforced rotating 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 abscissa is Re(a, the ordinate is Im(a. Each point in the complex plane corresponds to a pixel in the image and is the temporal Fourier amplitude a after frequency demodulation at ω f. (Bottom row: Histograms of phase-angles of all the pixels in each image; the abscissa range is [ π] radians and the ordinate range is arbitrary. For the oscillatory experimental patterns the parameter values γ W m and ω f (Hz are, respectively: (a, ; (b 119,.1; (c 1,.65; (d 1,.85. For the excitable experimental patterns the parameter values γ W m and ω f (Hz are, respectively: (a, ; (b 16,.8,; (c 11,.8; (d 9,.8. For the FitzHugh-Nagumo model patterns the parameter values γ W m and ω f (Hz are, respectively: (a, ; (b.5,.,; (c.6,.19; (d.68,.18. Chemical conditions and other model parameters are given in the text. Taken from K. Martinez, Masters Thesis, Univerity of Texas at Austin,. 6 1

13 oscillates between and 1. The model parameters are fixed at the values a =.1, a 1 =.5, ε =.5 and δ =.. At these parameter values the qualitative behavior of the model is in agreement with the qualitative behavior of the BZ experiments. By this we mean that the spiral frequencies are comparable, and a static forcing increases the observed spiral wavelength, similar to the experiments. The :1 resonant patterns observed in the FHN model are plotted in Fig. 9. As the forcing strength is increased there is a transition from traveling to standing waves qualitatively similar to the transition observed in the excitable BZ system. The orientation of the distribution in the phase portraits of the two excitable systems is also similar. SUMMARY We have described our experimental investigations of an oscillatory chemical system driven by an external forcing. In our investigations we find qualitative agreement in the resonant pattern formation observed in the light-forced Belousov-Zhabotinsky reaction compared to that found in numerical simulations of the forced Brusselator reactiondiffusion model and in the complex Ginzburg-Landau equation. This agreement is surprising since the experimental conditions are far from the Hopf bifurcation while the CGL equation is valid only close to the onset of oscillations. Qualitatively similar resonant patterns are observed in the BZ system and in the FitzHugh-Nagumo model when the parameters of both systems are tuned so that their dynamics are excitable. REFERENCES 1. Cross, M. C., and Hohenberg, P. C. (199. Rev. Mod. Phys., 65, Pivka, L., Zheleznyak, A. L., and Chua, L. O. (199. Int. J. of Bif. and Chaos,, 17.. Itoh, M., Murakami, H., and Chua, L. O. (199. Int. J. of Bif. and Chaos,, Bohr, T., Bak, P., and Jensen, M. H. (198. Phys. Rev. A,, Petrov, V., Ouyang, Q., and Swinney, H. L. (1997. Resonant pattern formation in a chemical system, Nature, 88, Lin, A. L., Bertram, M., Martinez, K., Swinney, H. L., Ardelea, A., and Carey, G. F. (. Phys. Rev. Lett., 8,. 7. Steinbock, O., Zykov, V., and Müller, S. C. (199. Nature, 66,. 8. Braune, M., and Engel, H. (199. Chem. Phys. Lett., 11, Buck, J., and Buck, E. (1976. Sci. Am,, Mirollo, R. E., and Strogatz, S. H. (199. SIAM J. Appl. Math., 5, Guevara, M. R., and Glass, L. (198. J. Math. Biology, 1, Glass, L. (1996. Physics Today, 9,. 1. Kunysz, A. M., Shrier, A., and Glass, L. (1997. Am. J. Physiol., 7, Sanchez, N. E., and Nayfeh, A. H. (1997. Journal of Sound and Vibration, 7, Rand, R. H. (1996. Mechanics Research Communications,, Rand, R. H., Zounes, R., and Hastings, R. (1997. Nonlinear Dynamics, The Richard Rand 5th Anniversary Volume,. 17. Rand, R. H., Denardo, B. C., Newman, W. I., and Newman, A. L. (1995. Design Engineering Technical Conferences, DE-Vol. 8-1, part A. ASME. 18. Coullet, P., and Emilsson, K. (199. Strong resonances of spatially distrubuted oscillatiors: a laboratory to study patterns and defects, Physica D, 61,

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