Spiralling to destruction at the edge of chaos |
MANY pattern-forming systems in nature exhibit a transition from an ordered state to a chaotic 'turbulent' state. There are hardly any systems, however, where this transition has been understood in its fine details and where theoretical predictions have been quantitatively tested by experiment. On page 143 of this issue, Ouyang and Flesselles1 demonstrate that in a special parameter range of the Belousov-Zhabotinsky (BZ) reaction, the transition from clock-like spiral-wave rotation to spiral-wave turbulence occurs exactly according to theoretical expectation. Nonlinear dynamics has been quite successful in explaining the transition from periodic to aperiodic motion in systems with a few degrees of freedom. Period doubling and intermittency are well-known routes to chaos that have been seen in numerous experiments and that are now standard textbook material. In non-equilibrium systems that naturally form patterns, however, the transition from order to chaos involves the creation of disorder in both space and time, with many degrees of freedom that increase in number with the size of the system. This transition has remained difficult to describe theoretically, except in a few model examples. One of these examples is the complex Oinzburg-Landau equation (COLE). Despite its simplicity, this equation already exhibits an extremely rich spectrum of dynamical behaviour and for this reason has been an important focus for theoretical studies of spatiotemporal chaos2. In the context of reaction-diffusion systems, it can quantitatively describe oscillatory media close to a Hopf bifurcation (the onset of an instability that oscillates in time). Wave patterns form easily in such media because different regions do not necessarily oscillate in phase with each other, and they are diffusively coupled (the pattern can travel between them). In a spiral wave the phase varies smoothly in space except at the spiral centre, where the contours of constant phase meet at a singular point, or topological defect, and the oscillation amplitude vanishes. Oscillatory media described by the CGLE can exhibit an ordered state, with a single, stable spiral wave rotating at a fixed angular frequency3, or a disordered, turbulent state4'5, with a seemingly random creation and annihilation of defect pairs (each defect being the centre of a small spiral) that have a well characterized average density5. The transition between these two states occurs when the rotation frequency of the spiral is such that the waves propagating outwards from the centre become subject to an instability that causes a long-wavelength modulation. When the amplitude of this modulation becomes sufficiently large, waves break up and defect pairs are generated, leading the system into a turbulent state. Interestingly, tIns instability does not destroy the initial spiral pattern all at once, because of its 'convective' nature6; small perturbations propagate away from the spiral centre faster than they can grow in amplitude, so the central portion of the spiral remains stable. As a control parameter is changed, the growth rate of perturbations increases and this stable region shrinks. Eventually an absolute stability limit is reached where perturbations spread faster than they can propagate away from the centre. This theoretical picture is precisely the one that Ouyang and Flesselles have been able to reproduce experimentally in the BZ reaction, with many details matching theory convincingly. Spiral waves are not new to the BZ reaction. They have been experimentally observed and modelled for decades, in parameter ranges where the spirals are usually stable2 and hence not susceptible to breakup and turbulence (with perhaps some exceptions7). Ouyang and Flesselles, however, have studied the BZ reaction for high concentrations of sulphuric acid and in an open system where the reaction can be accurately maintained near a Hopf bifurcation for a long time, which is precisely the regime described by the CGLE. This is what has allowed them to validate experimentally a theoretical model for the transition to spatiotemporal chaos near a Hopf bifurcation. Their results reinforce our belief that amplitude equations such as the COLE provide an accurate description of spatially extended non-equilibrium systems near bifurcation points,where dynamical behaviour is expected to be universal. Nature,however,rarely operates near such points unless artificially encouraged. In other systems showing defect-mediated turbulence,from fluid convection8,9 to surface catalysis10 to heart muscle11,the transition from order to chaos occurs via what seem to be different wave instabilities or entirely different dynamical mechanisms. In most of these systems particularly in heart tissue,where electrical-wave turbulence directly affects human health,theory and experiment are yet to meet in a clean and convincing way. A major challenge also lies ahead characterising the complex dynamical state that forms after the onset of turbulence. That requires the development of theoretical tools to describe the collective behaviour of defects. Are all these systems really turbulent after an indefinitely long time? Do they behave completely differently,or do they share common behaviours,beyond the superficial fact that they all have topologically similar defects? Can defect-mediated turbulence be controlled? Much research is now being focused on these questions,and pieces of the puzzle are starting to be assembled.
Alain Karma is in the Department of Physics and Center
for Interdisciplinary Research on Complex Systems,Northeastern
University,Boston,Massachusetts 02115,USA 1. Ouyang, Q. & Flesselles, J.-M. Nature 379,143-146 (1996) 2. Cross, M. C. & Hohenberg, R C. Rev Mod. Phys. 65,851-1112 (1993). 3. Hagan, P.S. SIAM J. appl. Math 42, 762-786(1982). 4. Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984). 5. Coullet, P., Gil, L. & Lega, J.Phys. Rev. Lett. 62,1619-1622 (1989). 6. Aranson, I. S., Aranson, L, Kramer, L. & Weber, A. Phys. Rev. E46, R2992-R2995 (1992). 7. Markus, M., Kloss, G. & Kunch, I. Nature 371, 402-404 (1994). 8. Morris, S. W., Bodenschatz, E., Cannell, D. S. & Ahlers, G. Phys. Rev. Lett. 71, 2026-2029 (1993). 9. Assenheimer, M. & Steinberg, V. Nature 367,345-347 (1994). 10. Jakubith S., Rotermund, H. H., Engel, W., von Oertzen, A. & Ertl, G. Phys. Rev. Len. 65,3013-3016 (1990). 11. Winfree, A. T. Science 266, 1003-1006 (1994).
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