Tuesday, 10 March 2020

determinism - What are the *necessary* conditions to deterministic chaos?


What are the necessary conditions (not saying sufficient conditions) in mathematical terms that a deterministic dynamic system can transit to deterministic chaos?


We collected yet:



  1. A positive feedback loop


  2. Non-linearity

  3. Minimum of three instable eigenmodes

  4. ...???



Answer



According to Nonlinear Dynamics and Chaos by Steven Strogatz The requirements for chaos are:




  1. Deterministic system (only one future for each state)





  2. Irregular spatial, temporal, or spatiotemporal patterns (a qualitative feature)




  3. A positive maximum Lyapunov exponent.




3) is pretty much the quantitative standard in journals of chaos, assuming you meet the conditions of 1). 2) is subjective and there's things like "stable chaos" and there can be periodic behavior that appears irregular but just has a really long period before it repeats itself, so you have to be careful with 2).


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