Stability of Travelling Waves in the Belousov-Zhabotinskii Reaction
As we have seen in Dr. Müller’s talk, the Belousov-Zhabotinskii (BZ) reaction leads to a rich variety of non-equilibrium spatial structures and serves as a paradigm for pattern formation in excitable media1. In this talk, we will focus on the simplest such pattern, the planar travelling wave, in which regions of excited and quiescent reagents move uniformly through space (Fig. 1). Understanding this structure is a necessary first step towards a complete picture of more complex structures such as the target2, the rotating spiral3 or, in three dimensions, the scroll4. This is particularly true as these patterns far from their centers asymptotically approximate planar travelling waves.
KeywordsHopf Bifurcation Travel Wave Solution Real Mode Quasistatic Approximation Spatial Instability
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- 1.For a general review see R.J. Field and M. Burger, “Oscillations and Travelling Waves in Chemical Systems,” Wiley (1985).Google Scholar
- 8.This idea seems to be a fairly common one in the reaction-diffusion literature.Google Scholar
- 9.A. Winfree, unpublished data quoted in .ref 6.Google Scholar
- 12.P. Fife, “Non-Equilibrium Dynamics in Chemical Systems,” C. Vidal and A. Pacault ed., Springer-Verlag (1984).Google Scholar
- 13a.J. S. Langer in “Chance and Matter”, J. Souletie ed., North Holland (Amsterdam, 1987).Google Scholar
- 14.Note that the lower of these two modes must go through zero at q = 0 at precisely the point where the velocity versus wavelength curve turns around. ‘ This is because such a saddle node bifurcation always involves an exchange of stabilities.Google Scholar
- 17a.W. Van Saarloos and J. D. Weeks Physica 12D, 115 (1984).Google Scholar
- 18.It should be noted that the Tyson-Fife Oregonator does not possess travelling wave solutions of the type studied here for c ≥ .8, as can be seen from Fig. 5 of ref. 6. The experimental data, however, are only for c ≥ 1.. This discrepancy bears further investigation.Google Scholar
- 20.D.A. Kessler and H. Levine, to appear in Physical Review A.Google Scholar