Help in understanding how to apply nonlinear function in hashing
Cryptographers have studied various schemes for using chaos theory to try to construct cryptographic primitives (hash functions, encryption algorithms, etc.).  None of them have been successful.  Chaos simply does not seem useful for designing cryptographic algorithms.

First, many of the chaos-based schemes have been broken.  They didn't have as much nonlinearity as the designers anticiapted.

Second, the chaos-based schemes are slower than the state of the art.  You need to iterate many times to get reasonable mixing/nonlinearity.  By the time you do that, you end up with something quite slow.  In comparison, we know of alternative schemes for hashing and encryption that are very fast and appear to be very secure.  So, chaos-based schemes are simply not competitive.

In my opinion, chaotic maps do not appear to be a promising angle for hashing or other cryptographic algorithms.