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Talk Abstract
Meandering and resonant spirals in a reaction-diffusion system

Institute for Mathematics and Its Applications
Harry L. Swinney, University of Texas

We examine spiral patterns in a reaction-diffusion system for two cases: the autonomous system and the system driven by external periodic perturbations. Experiments are performed in a reactor consisting of a thin gel layer sandwiched between two continuously refreshed reservoirs of chemicals; the layer is thin compared to the spiral wavelength so the patterns are quasi-two-dimensional. In the absence of external perturbations, for a wide range of control parameters the spirals are found to simply rotate -- the spirals are stationary in a co-rotating frame. Variation of two control parameters reveals a critical curve at which there is a transition from the simple rotating (temporally periodic) spirals to meandering (quasiperiodic) spirals. There are two types of trajectories of the tips of the meandering spirals: inward-petal (epicycloid) trajectories and outward-petal (hypocycloid) trajectories. These two types of meandering regimes are separated in the phase diagram by a line of traveling spirals that terminates at a codimension-2 point [1]. The observed unfolding of the bifurcation about a codimension-2 point is in accord with theory [2]. The rates of the chemical reaction can be changed by shining light on the gel layer since the reaction (a ruthenium-catalyzed Belousov-Zhabotinsky system) is photosensitive. We examine the effect of periodically modulating the intensity of light incident on a pattern of simple rotating spirals. The modulation frequency is varied from one to four times the intrinsic frequency of the chemical system. As the modulation frequency is varied, the system passes through resonant regimes in which it becomes locked to the driving frequency (that is, the ratio of the driving frequency to the natural frequency is given by a ratio of small integers). These frequency-locked regimes are like the Arnold tongues of periodically forced low dimensional systems. However, in the spatially extended system, bifurcations can also occur between different spatial patterns that lie within the same frequency-locked region [3]. Some aspects of the observations are found in a periodically forced reaction-diffusion model with two species (Brusselator kinetics) [4].

Acknowledgments: This research was conducted with V. Petrov, Q. Ouyang, G. Li, and M. Gustafsson, and was supported by DOE.

[1] G. Li, Q. Ouyang, V. Petrov, and H. L. Swinney, Phys. Rev. Lett. 77, 2105 (1996).

[2] D. Barkley, Phys. Rev. Letter 72, 164 (1994); M. Golubitsky, V. LeBlanc, and I. Melbourne, to appear; C. Wulff, to appear.

[3] V. Petrov, Q. Ouyang, and H. L. Swinney, Nature 388, 655 (1997).

[4] V. Petrov, M. Gustafsson, and H. L. Swinney, to appear.

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