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Stability of Fixed Points of High Dimensional Dynamical Systems (adipandas.github.io)
66 points by adipandas 43 days ago | hide | past | favorite | 12 comments

Some of the conclusions in the post are misleading. For systems that are uniformly expanding the statistical description of the dynamical system is very stable, and such systems can have sense periodic orbits that are necessarily all unstable! By statistical description I mean (at least) the long term frequency that the system is in any particular region of state space (i.e. the systems physical invariant measure).

It is actually one of the nice facts of life that as you dial up the local instability of a dynamical system you get a corresponding increase in the statistical stability. As systems become more and more chaotic they act more and more like iid coin tosses, which we understand quite well.

Very interesting. Do you have any suggestions for reading more about this?

"It is actually one of the nice facts of life that as you dial up the local instability of a dynamical system you get a corresponding increase in the statistical stability."

I am a mathematician working in a tangentially related area and have never come across this before.

The most accessible introduction (in my opinion) is 'Laws of Chaos - Invariant Measures and Dynamical Systems in One Dimension' by Boyarsky and Gora. For a short survey on just stability of the invariant measure you could look at 'Linear Response, Or Else' by Baladi. I also like the lecture notes 'Statistical properties of uniformly hyperbolic maps and transfer operators' spectrum' by Carlangelo Liverani.

Does this perhaps have some relation to Monte Carlo methods where the qualities of the noise (white, pink blue etc) and/or stratification that you use - can strongly affect the qualities of the integral that results? Could be analogous to the local instabilities in the section you quoted? (reminds me a lot of how information moves around in a reaction-diffusion simulation too)

It seems likely to. In MCMC this is known as the mixing/convergence speed. If one valid transition kernel decreases the mutual information between samples faster than another it mixes faster. Meaning the sampled chain has better statistical properties (effective number of independent samples).

Doyne Farmer et al have written about the connection between spectra and stability in reaction-diffusion.


This is fantastic. :)

That's a very cool connection I'd never thought of before, that IID random variables correspond to "maximamal local instability". What's this perspective called so I can read more about it for my interest?

The study of dynamical systems from a probabilistic perspective is called ergodic theory. Have a look at my other comment for some specific references for more on this.

And yes it is very interesting. One can actually model iid coin flips (or dice rolls) as a type of expanding dynamical system called a full-shift.

Is that not accommodated by the following point from the "Important points to note regarding this article" section?

"This approach of interpreting the stability of the system by linearizing it near the equilibrium does not tell much about a system’s asymptotic behavior at large."

Yes, I understand this. The system's global stability is unaccounted for when we linearize the system near a fixed point.

@hgibbs, This is cool. I will go through this in more detail. Thank you.

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