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Resonant pattern formation in achemical system

Nature volume 388, pages 655–657 (1997)Cite this article

Abstract

A periodic force applied to a nonlinear pendulum can cause the pendulum to become entrained at a frequency that is rationally related to the applied frequency, a phenomenon known as frequency-locking1. A recent theoretical analysis showed that anarray of coupled nonlinear oscillators can exhibit spatial reorganization when subjected to external periodic forcing2. We present here experimental evidence that reaction–diffusion processes, which govern pattern evolution and selection in many chemical and biological systems3, can also exhibit frequency-locking phenomena. For example, periodic optical forcing of the light-sensitive Belousov–Zhabotinsky (BZ) reaction transforms a rotating spiral wave4 to a labyrinthine standing-wave pattern (Fig. 1). As the forcing frequency is varied, we observe a sequence of frequency-locked regimes, analogous to the frequency-locked ‘tongues’ of a driven nonlinear pendulum, except that in the reactor different frequencies correspond to different spatial patterns. Resonant interactions leading to standing-wave patternshave not been observed previously in chemical or biological media, but periodic forcing (such as circadian rhythm) is abundant in nature and may lead to similar pattern-forming phenomena.

The reaction occurs in a 0.4-mm-thick porous membrane disk that is placed between two reservoirs (I and II) which are continously refreshed with BZ reagents. The region shown is a 13 × 13 mm section of the 25-mm-diameter reactor. The patterns in the membrane are determined from light absorption measurements at 450 nm. In the perturbed region, light of 0.2 W m−2intensity in the spectral range 430–470 nm is periodically switched on for 6 s and then off for 13 s. Spatially uniform illumination is achieved using a video projector (Sanyo PLC-220N) calibrated by measuring light reflected from a scatter plate and then adjusting each pixel to have the same intensity. The concentrations of thechemicals in reservoirs I and II are: [malonic acid]I = 0.22 M, [BrO3−]I = 0.046 M, [BrI−] = 0.2 M, [H2SO4]I = 0.8 M, [Ru(bpy)32+]II = 1.0 mM, [H2SO4]II = 0.8 M, [BrO3−]II = 0.184 M (here bpy indicates 2,2′-bipyridine). The volume of each reservoir is 10 ml. The flow rate in reservoir I is 20 ml h−1; in reservoir II, 5 ml h−1. The system is maintained at a temperature of 23 °C.

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Figure 2: Time series of pattern intensity at the points A (solid line) and B (dashed line) in Fig. 1.
Figure 3: Bifurcation diagram showing different frequency-locked regimes observed as a function of fp/f0 where fp is the perturbation frequency and f0 is the natural frequency of the system.

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Acknowledgements

We thank A. Lee for discussions and G. Li for help in conducting the experiments. This work was supported by the US Department of Energy Office of Basic Energy Sciences and the Robert A. Welch Foundation.

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  1. Center for Nonlinear Dynamics and Department of Physics, University of Texas, Austin, 78712, Texas, USA

    Valery Petrov, Qi Ouyang & Harry L. Swinney

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  1. Valery Petrov
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  2. Qi Ouyang
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  3. Harry L. Swinney
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Correspondence to Harry L. Swinney.

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Petrov, V., Ouyang, Q. & Swinney, H. Resonant pattern formation in achemical system. Nature 388, 655–657 (1997). https://doi.org/10.1038/41732

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