I have been reading about chaos-based cryptosystems. Every designer claims that his design is a secure system without much cryptographic analysis; however, it turns out that this is a false claim in many cases. I do not know if all these systems are weak or inefficient. I do not have the time to perform cryptanalysis on them all.
For the examples of the chaos-based cipher, almost every designer has his style without real proof of why he does this. They take the word "complex" as a synonym for "secure".
For guidelines that rarely is followed (if any, assuming that it is applicable or even sufficient, I can not judge): Some basic cryptographic requirements for chaos-based cryptosystems. Some Hints for the Design of Digital Chaos-Based Cryptosystems: Lessons Learned from Cryptanalysis. Lessons Learnt from the Cryptanalysis of Chaos-Based Ciphers. Framework for the analysis and design of encryption strategies based on discrete-time chaotic dynamical systems.
My question is:
How can I analyze these systems cryptographically? And what subjects in mathematics or computer science do I need?