Chaos  

• Two fundamental questions: (refer to Stephen Wolfram)
  • Given a state equation that characterizes the evolution of a system, can you decide whether the system is chaotic or not, by using theoretical analysis only, without doing experiments?
    • No.   Doing experiments is the only way to decide whether the system is chaotic or not, according to Stephen Wolfram's book "New kind of science".
  • Given a sequence that looks random (e.g., data analysis shows that roughly, it is a sequence of independent, identically distributed random variables), can you find a state equation that generates the sequence, using theoretical analysis only? In other words, can you compress the data by analyzing the sequence only?
    • No.   Doing experiments is the only way to find such a state equation, according to Stephen Wolfram's book "New kind of science".  Such a state equation may not even exist.   It may be an undecidability problem.  It is related to computational irreducibility.
• Iterative Function Systems (IFS), chaotic IFS