> …which we know isn't actually chaotic. It's just a path defined by mathematical functions.
I don't know what function is being presented, so I can't speak to whether it demonstrates chaotic behavior -- but the whole /point/ of chaos is that it's an emergent property of deterministic mathematical functions. Perhaps the author meant "random" or "non-deterministic" rather than "chaotic"?
'actual' or 'true' randomness is a rabbit hole.
https://en.wikipedia.org/wiki/Randomness
It really depends on the exact definition of what you are quantifying 'random' to be.
There is no proof (in the mathematical sense) of real randomness.
There are a number of sophisticated tests for randomness. You can't prove absolute randomness in any Platonic sense, but you can certainly assess a source for different properties that are useful in applications that require randomness.
In this example the path is neither chaotic (in the formal sense) nor random, because a Fourier transform would identify the harmonic components.
I think that makes sense and I can tell that you understand the distinction. Formally I believe they're different concepts and think it may cause confusion in some cases to use them interchangeably.
The chaotic nature of a system is one thing.
Our lack of knowledge of the governing laws, initial conditions, feasibility of simulation forcing us to use the mathematical tools of probability (i.e. randomness) to describe our uncertainty about said system is another thing.
The reason why it matters is that a statement like "a double inverted pendulum behaves randomly" is just wrong as it would imply that you couldn't even do a simulation of one in theory without throwing some dice.
However, it is totally uncontroversial that if someone gave you a measured initial position and velocity of one with 'really good' precision and asked you to predict its state 5 seconds forward you would likely have a big smeared-out probability density function to deal with.
I just did a mental substitution for the word "stochastic" and moved on, since it wasn't the main point of the article. But you are correct. Computing the future states of a chaotic system, given the same initial conditions, will produce the same results every time but change the initial conditions ever so slightly and you have no guarante where you will end up next (unless you picked a state from a previous simulation, that is... But that's cheating;) )
> the whole /point/ of chaos is that it's an emergent property of deterministic mathematical functions.
I believe you. But the colloquial definition (I’m looking at a dictionary right now) is “complete disorder and confusion”; “a state of things in which chance is supreme”; “the inherent unpredictability in the behavior of a complex natural system”. That path fits that apparent definition. The post is written in a way that a relative layman would understand, so it makes sense to speak/write in a way non-mathematicians would expect.
> That path fits that apparent definition.
Then saying it "isn't actually chaotic" is mis-aligned with the layman's understanding, and the disagreement is not explained (either by explaining a more technical definition, or otherwise).
I continue to think that substituting "non-deterministic" or "random" would be more both understandable to the layman, and more correct to the advanced reader.
^^^ This is an extremely underrated nitpick.
I am guessing that the HN audience would be / should be interested in that distinction. Mathematically speaking, chaos is an extreme sensitivity to initial conditions, and is very much still in line with deterministic systems. The resulting output, while seemingly random (since there is no easily identifiable pattern), is mathematically and conceptually different from actual randomness.