| To measure how rapidly the diameters of the
period-doubling components shrink, we
look at the ratio of successive diameters. That is, |
| |cn+1 - cn| / |cn+2 - cn+1| |
| The values of the first few ratios are |
| 4.233738275 ... |
| 4.551506949 ... |
| 4.645807493 ... |
| 4.663938185 ... |
| 4.668103672 ... |
| 4.668966942 ... |
| 4.669147462 ... |
| 4.66916224 ... |
| 4.66919003 ... |
|
| This sequence approaches a limit, the
Feigenbaum constant. |
| The same limit is obtained for the
period-doubling cascade
of the logistic map, and also for the analogous
sequence of many, many other functions. |