Explaining the Fibonacci order in the period-doubling cascade to chaos.

In their paper, Linage et al. [Phys. Lett. A 359, 638-639 (2006)] reported the appearance of Fibonacci ordering in the Feigenbaum scaling of the period-doubling cascade to chaos. Although their results demonstrated this connection numerically (verifying up to the 22nd term), the theoretical origin of the Fibonacci sequence in this context was not fully developed. Here, we provide a theoretical explanation showing that Fibonacci ordering arises naturally from the combinatorial structure of branching generated by period-doubling cascading. This provides the theoretical connection between the period-doubling route to chaos and the Fibonacci sequence.