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Blue_Morpho@lemmy.world 1 month agoNo closed form solution is one property. It’s not wrong, only incomplete. But if a system of equations had a closed form solution, it wouldn’t be called chaotic. For example any exponential equation like x^y is extremely sensitive to initial conditions yet it isn’t chaotic.
vrighter@discuss.tchncs.de 1 month ago
oh really?
Blue_Morpho@lemmy.world 1 month ago
'Robert L. Devaney, says that to classify a dynamical system as chaotic, it must have these properties:[22]
it must be sensitive to initial conditions, it must be topologically transitive, it must have dense periodic orbits. " en.m.wikipedia.org/wiki/Chaos_theory
f(x)=x^y doesn’t satisfy those 3 conditions. Nor does the paper you link say that either.
vrighter@discuss.tchncs.de 1 month ago
and again, in the definition you just pasted in there does not say anything about closed form solutions. You keep contradicting yourself in trying to die on that hill
Blue_Morpho@lemmy.world 1 month ago
It’s implicit in the method. There also isn’t a definition of computability in the papers or Wikipedia because it assumes you have a basic understanding.
Chaotic functions require that you iteratively step through them because they aren’t closed form.
“For chaotic systems the evolution equations always include nonlinear terms,5 which makes “closed-form” solutions of these equations impossible—roughly, a closed-form solution is a single formula that allows one to simply plug in the time of the desired prediction into the equation and determine the state of the system at that time.”
www.sciencedirect.com/topics/…/chaos-theory#%3A~%…
I last wrote a paper on chaos in a mechanical system 35 years but I haven’t forgotten the basics.