this post was submitted on 23 May 2025
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Showerthoughts
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A "Showerthought" is a simple term used to describe the thoughts that pop into your head while you're doing everyday things like taking a shower, driving, or just daydreaming. The most popular seem to be lighthearted clever little truths, hidden in daily life.
Here are some examples to inspire your own showerthoughts:
- Both “200” and “160” are 2 minutes in microwave math
- When you’re a kid, you don’t realize you’re also watching your mom and dad grow up.
- More dreams have been destroyed by alarm clocks than anything else
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No 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.
oh really?
'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. " https://en.m.wikipedia.org/wiki/Chaos_theory
f(x)=x^y doesn't satisfy those 3 conditions. Nor does the paper you linked say that x^y is a chaotic equation.
That function in the paper cannot be solved for an input because of its sensitivity to initial input. He used a computer to simulate the time steps. He couldn't immediately calculate any point on the the plot like y^x.
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
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."
https://www.sciencedirect.com/topics/agricultural-and-biological-sciences/chaos-theory#%3A%7E%3Atext=For+chaotic+systems+the+evolution%2Cthe+state+of+the+system
I last wrote a paper on chaos in a mechanical system 35 years but I haven't forgotten the basics.
but you're still wrong
You still don't understand the links you are providing. Fuck, just read the English words even if you don't understand the math.
"We aim to present reversible systems which lie on the border of solvability/integrability and chaos."
The intro says it's not chaotic but a function that borders on chaotic.
"We adjusted the precision in such a way that the true initial data and the result of this round trip did not differ by more than 10−3."
Their function isn't even reversible but only allows for an approximation of reversibility.
Conclusion:
"We have shown that there exist infinite families of rational maps which, at the same time, have positive algebraic entropy, present features of chaos, and are solvable."
FEATURES OF CHAOS ISN'T CHAOS.