The Feigenbaum Constant and Universality

Let us denote the critical value of $r$ at which the logistic map bifurcates into a period-$2^k$ orbit as $a_k$, so that for $a_{k} < r < a_{k+1}$ the map has a stable period $k$ orbit. Look at the bifurcation diagram in Fig.(3.7) and Fig.(3.9). Notice how the distance, $a_k - a_{k-1}$, between period doublings decreases as the control parameter is increased.

The first few values of $a_k$ are

$$a_1 = 0.75 ,$$

$$a_2 = 0.862372...$$

$$a_3 = 0.886022...$$

$$a_4 = 0.891101...$$

$$a_5 = 0.892189...$$

$$a_6 = 0.892423...$$

$$a_7 = 0.892472...$$

$$a_8 = 0.892483... .$$

Mitchell Feigenbaum noticed that succesive differences appear to converge geometrically (see Fig.(3.9)) and that the ratio of successive separations tends to a constant as k goes to infinity,

$$\delta \equiv \lim_{k-> \infty} { a_k - a_{k-1} \over a_{k+1} - a_k} = 4.669201609... .$$ (3.13)

Although the discussion so far has been strictly limited to the logistic map, the constant $\delta$ is the same for other smooth one-dimensional maps with a single hump. This is an example of universality, a concept which we will encounter more of when studying phase transitions in the next chapter. In general systems fall into different universality classes, so that systems within each class have the same behaviour. For the present discussion, one says that all 'unimodal' (smooth, concave downwards, with a single hump) maps belong to the same universality class, that is bifurcate at a rate leading to the universal Feigenbaum constant $\delta$. The actual proof of this statement is quite involved, but briefly stated, it uses the concept of the renormalisation group that was developed to deal with critical phenomena in statistical mechanics. Although Eq.(3.18) is defined in the limit $k \to \infty$, it can be used to estimate the point at which a system becomes chaotic, $a_{\infty}$, if the first few bifurcations of the system are known.