Dynamic Instabilities Observed in the Belousov-Zhabotinsky System

Abstract

Since the work of RUELLE and TAKENS [l], which established that turbulence may occur in systems with few degrees of freedom, a lot of effort has been devoted to study the so-called weak turbulence. Theory and experimentation (numerical as well as bench experiments) have both contributed to a new insight into the onset of turbulence. Let me briefly recall some salient results of the experiments performed on real hydrodynamical systems. Two geometries have been more thoroughly studied than any others, namely the circular Couette flow [2] and the Rayleigh-Benard instability. The more comprehensive results belong to the latter, having given rise to a great number of observations. For large aspect ratios, turbulence occurs at, or very near to, the threshold of convective instability [3.a]. On the other hand, several bifurcations lead to turbulence when cells with a low aspect ratio are used. Amongst other factors, the detailed sequence of bifurcations depends on the Prandtl number of the fluid, i.e. on the ratio of its kinematic viscosity to its thermal diffusivity. Although the instabilities involved are not the same, there are strong similarities in the behaviour of liquid helium [3c], silicon oil [3b] and water [4]. On the different routes leading to turbulence, three phenomena may be encountered: a cascade of period doubling bifurcations (sometimes named the Feigenbaum cascade), a quasi-periodic regime involving 2 or even 3 independent frequencies which, eventually, “lock in”, and an intermittency phenomenon, that is to say, bursts of noise emerging from time to time in a coherent regime. In agreement with the basic prediction of RUELLE and TAKENS, it has been observed in all cases that the transition to chaotic flow always takes place through a small number of bifurcations. Other experiments on the circular Couette flow, and numerical integration of sets of differential equations, such as the celebrated Lorenz model, have also led to the same general conclusion.