Periods doubling due to non linear feedback is standard fare. But now advance you thinking and view the non linear component of des (and any feistel cipher) as composing two digital streams to create a non linear mathematical medium , for fabricating the theoretical Boolean function that maximizes non linearity.
View the outer bits of the nibble widths box as a third input to the medium that is subject to phase inversion, as a mathematical Kerr effect doubles and triple the balancedness … which drives the number of times the inner bits are inverted. That is data driven computation of the symmetric and asymmetric spin similarity and difference, applied to mixing the round function.
It n feistel analysis term, we are amplifying the rate of confusion per bit in proportion to the information content
Tempting to see this as a repelling process. But once you add the wider Hamiltonian cycles of the p&s with the balacedness generator you get more of chaotic attractor for those (theoretical) bent functions.
Now Turing already taught us to generate custom operator norms, that evolve with the plaintext information content. Using the similarity/dissimilarity basis of the quaternion algebra to tease out the information and use it to generate a data-aware averaging norm that is a masks to both linear and differential comparators, one gets self evolving computation of an operator norm that generates a bent function.
With the feistel network and the des era spn the complexity goes beyond the limit available with a simple sequence of rotors with quaternion group wiring. One gets now to tangent spaces of much finer granularity.