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Sophie Libkind (May 04 2022 at 03:32):Sam Tenka said:
Sophie Libkind you mentioned something about fast/slow decompositions in chemistry... I'd love to hear what you had in mind!
My ears just perk up when I hear fast/slow systems! I haven't studied them very closely, and I don't know anything about the applications in chemistry. The thing that I'm interested in is how the Van der Pol oscillator feels like it "computes" something like ...01010101... . I'm curious if other, more complicated fast/slow systems compute more complicated sequences and if there's a nice mathematical story there!
Sam Tenka (May 04 2022 at 15:29):My ears just perk up when I hear fast/slow systems! I haven't studied them very closely, and I don't know anything about the applications in chemistry. The thing that I'm interested in is how the Van der Pol oscillator feels like it "computes" something like ...01010101... . I'm curious if other, more complicated fast/slow systems compute more complicated sequences and if there's a nice mathematical story there!
I love Van der Pol! That example (as found in Landau&Lifschitz and/or Abraham&Shaw) was what convinced me that classical physics was interesting haha. @Sophie Libkind Could you say more about 010101?
John Baez (May 04 2022 at 15:31):Sophie Libkind said:
Sam Tenka said:
Sophie Libkind you mentioned something about fast/slow decompositions in chemistry... I'd love to hear what you had in mind!
My ears just perk up when I hear fast/slow systems! I haven't studied them very closely, and I don't know anything about the applications in chemistry.
They're used a lot in mathematical chemistry and other dynamical systems. I think the key buzzword is center manifold. And the key idea is this: a dynamical system can have a submanifold that states rapidly approach, while the move more slowly on that submanifold.
Before reading anything, check out this animated gif of a dynamical system with a center manifold:
https://en.wikipedia.org/wiki/Center_manifold#/media/File:CentreMfld.gif
Unfortunately this gif is not looped so you have to reload it to watch the states approach the center manifold again!
Sophie Libkind (May 04 2022 at 22:09):Thanks John for the pointer and the intuition for center manifold!
Sophie Libkind (May 04 2022 at 22:22):Sam Tenka said:
My ears just perk up when I hear fast/slow systems! I haven't studied them very closely, and I don't know anything about the applications in chemistry. The thing that I'm interested in is how the Van der Pol oscillator feels like it "computes" something like ...01010101... . I'm curious if other, more complicated fast/slow systems compute more complicated sequences and if there's a nice mathematical story there!I love Van der Pol! That example (as found in Landau&Lifschitz and/or Abraham&Shaw) was what convinced me that classical physics was interesting haha. Sophie Libkind Could you say more about 010101?
This my very loose story: In the Van der Pol system there are two regions of the critical manifold near which the dynamics spends most of its time. I'll call these regions "0" and "1". You start out near region 0 and slowly move across it until you reach a special point at which time you fly off the center manifold and zoom towards region 1. Then the reverse happens. You slowly move across region 1 until you get to a special point and then you fly off towards region 0. And this repeats! Hence ...010101...
John Baez (May 04 2022 at 22:31):The general idea of trying to turn continuous dynamical systems into strings of bits or other symbols is called symbolic dynamics, and there's been quite a bit of work on it.
John Baez (May 04 2022 at 22:38):By the way, anyone interested in what chemical reaction networks can compute can get a quick overview of that field in Chapter 25 of my book - the chapter called "Computation and Petri nets".
Sophie Libkind (May 04 2022 at 22:40):I'll take a look! I thought about symbolic dynamics briefly a few years ago but it would be fun to go back to it
John Baez (May 04 2022 at 23:17):Btw, my book chapter talks about 5 or 6 concepts of what it means for a Petri net, or Petri net with rates, to compute something. But it doesn't say anything about symbolic dynamics!
