Are hash functions chaotic? I'm looking into "bitcoin like" mining processes. Say a security process uses a hash of data as proof that the data was known at some point in time. 

From what I understand, hash functions are pseudorandomly generated based on the initial seed (the raw data). 

If the hash function is chaotic, then the seed data should have strange attractors. If so, then I may be able to map a seed to its attractor and "pre-mine" using attractor buckets.  

Are these pseudorandom functions "deterministic chaotic systems"?
No.  That's not going to help you.  The cycle structure of a random function is well-studied, and it doesn't help you break hash-chaining, "proof-of-work", Bitcoin, etc.  Crypto schemes already take into account the structure of (the graph of) a random function.  It's not going to help you "pre-mine" and get some speedup on breaking Bitcoin (for instance).

For instance, the graph of a random function $\{0,1\}^n\to \{0,1\}^n$ typically has the following form: there is a single large cycle of length $\Theta(2^{n/2})$, such that most inputs eventually lead to the cycle.  Starting from a random input, the length of the path that leads to that large cycle is typically $\Theta(2^{n/2})$.  This does not break standard schemes for hash-chaining, proof-of-work, Bitcoin, etc., as the parameters for their hash function is chosen to take this structure into account.

Whether you want to count this as "chaos" depends upon your precise definition of "chaos", which you haven't provided in the question.  Anyway, from a technical/scientific/engineering perspective, what label we assign it (chaos or not) is less important than the technical properties it provides and the security goals it can help to protect.