Systems based on numbers
Salient points:
- lots of 'simple' sequences of numbers look 'complex' if you write them in binary and draw them...
- the shift map generates random chaotic sequences, but this is purely down to complexity in the initial condition, since we can see the shift map is merely reading off successive digits of the number we give it to start with.
- i.e. rational initial conditions give static or repeating patterns eventually, while reals gives randomness
- continuous cellular automata behave similarly, so it's not the discreteness of values that matters
- one can think of ODEs as limit case of continuous CA
- lots of work on PDEs in 1D has concentrated on 3 equations that give pretty simple behaviour:
- diffusion equation
- wave equation
- sine-Gordon equation
- traditional maths isn't much help in finding DEs that give complex behaviour
- nearly all sampled DEs diverge to infinity
- but Wolfie found some suggestive examples (pg 165-6) - (give solitons, )
- speculation: all the interesting behaviour seen with discrete systems (CA) could occur even in continuous systems
Comments:
- A question: Is Wolfram saying that the the randomness in so-called chaotic systems is really just complexity in irrational numbers themselves? (marcus)
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