Experimental Tests

Careful experiments have revealed period-doubling bifurcations in a number of real dynamical systems of different types: hydrodynamic, electronic, laser, chemical and acoustic. Details may be found in the book by Cvitanovic. Here we will examine the first of such measurements, the Rayleigh-Benard convection experiment of Libchaber. In that experiment, liquid mercury that was confined in a small box was heated from below to set up a temperature gradient measured in terms of a dimensionless quantity called the Reynolds number, $R$. At low values of $R$ the heat flow is by conduction, but as $R$ exceeds some critical value $R_c$, there is bulk motion of the fluid (convection), leading to the formation of a pattern of cylindrical rolls as the hot fluid rises on one side and cooler fluid descends on another. The temperature is measured at a fixed point of a roll. For $R$ only slightly above $R_c$, the rolls are straight and the temperature constant. But as the heat flow is increased, an instability occurs leading to wave propagation along the roll and oscillations in the measured temperature. Further increases in the heat flow lead to period doublings in the measured temperature oscillations. From the first few period doublings the Feigenbaum number was estimated to be $4.4 \pm 0.1$, remarkably close to the theoretical value $4.67$. The other experiments quoted above give similarly close values. There is a puzzle here. The Feigenbaum constant was derived for unimodal one-dimensional maps. Those maps contain none of the physical details of the real experiments which are dynamical systems with many degrees of freedom evolving in continuos time and three dimensions. So why is there any agreement between the predictions of the one-dimensional maps and the real experiments ? In order to understand how that is possible, it is useful to consider first the simple dynamical system of the next section.