One of my professors mentioned that a standard problem in dynamical systems is to show that if two points in your system have the same coding (ie end up in specific regions of the space you're working on) then this coding is periodic. I'm interested in seeing examples of this type of problem, preferably with solutions.
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In neural networks, a number of invariances exist that can be used to show that NNs in general do not have unique solutions. NNs can be thought of as dynamical systems without feedback -- the inputs act as parameters to the system, and the weights themselves are the system variables.
The book Pattern Recognition and Machine Learning by Chris Bishop has a nice treatment in one of the first few chapters on weight invariances. I won't go so far as to say the invariances occur in a periodic manner across weight space -- that's a bit too many dimensions for me to even fathom periodic behavior -- but it might help you find a better answer to your question.
I recommend you to take a look at Symbolic Dynamics & Coding by Lind & Marcus. Other nice options are Symbolic Dynamics: One-sided, Two-sided and Countable State Markov Shifts by B. Kitchen and Substitutions in Dynamics, Arithmetics and Combinatorics by P. Fogg.