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John Baez (Dec 26 2022 at 19:44):I don't know what it means to say attractors are a "better" concept than equilibrium or fixed point. They're both important concepts in dynamical systems theory. It's like asking "what's better, groups or abelian groups?"
I usually try to avoid ranking important math concepts by "goodness". As soon as we start talking about that, any hope of a really precise conversation is gone.
Peiyuan Zhu (Dec 27 2022 at 21:33):I think mathematicians talk about "better" in the sense that it is more general. For instance, it looks like fixed points are always attractors but attractors are not necessarily fixed points. What are the cases when attractors that are not necessarily fixed points? Do they corresponds to some physical / biological reality?
Spencer Breiner (Dec 28 2022 at 01:41):Think about a ball free to roll on a surface.
If the surface forms a bowl (e.g., ), then the point at the bottom is a fixed point, and it will be an attractor if there is friction b/c small oscillations around the point will be dampened.
If the surface forms a hill (), then the top point will be a repelling fixed point b/c any small decision from the (flat) top will start the ball rolling down the hill.
John Baez (Dec 28 2022 at 12:17):"Better" is not a good synonym for "more general". "More general" has the great advantage of being fairly precise. But depending on your taste, sometimes more general concepts may feel "worse" rather than "better". For example, a lot of people think semigroups are "worse" than groups, even though they're more general.
(Personally I would not say semigroups are worse than groups, any more than mammals are worse than horses.)
John Baez (Dec 28 2022 at 12:18):But on to the more important issue: as Spencer pointed out, fixed points are not always attractors.
John Baez (Dec 28 2022 at 12:19):A fixed point that's an attractor is called an attractive fixed point.
John Baez (Dec 28 2022 at 12:20):There are also repulsive fixed points, and there are various kinds of fixed points that are neither attractive nor repulsive. There's a theory of this stuff.
Peiyuan Zhu (Jan 05 2023 at 02:26):Is there any well-known applied context that an attractor is more suitable than fixed point?
Peiyuan Zhu (Jan 05 2023 at 02:26):When people say "equilibrium" I suppose they're referring to fixed-point no?
Peiyuan Zhu (Jan 05 2023 at 02:44):In this paper it says “This note proposes a definition for the concept of “attractor,” based on the probable asymptotic behavior of orbits.”: https://link.springer.com/article/10.1007/BF01212280
Nathaniel Virgo (Jan 05 2023 at 03:48):Peiyuan Zhu said:
Is there any well-known applied context that an attractor is more suitable than fixed point?
A classic example of an attractor that is not a fixed point is the Lorenz attractor. That originally came up in simulating the Earth's atmosphere, and they're pretty important in that context.
Peiyuan Zhu (Jan 05 2023 at 15:43):I see. From the graph it's like saying for some parameters the model converge to one of the two, which is classical, but for some parameters it's neither, it goes back and forth in-between. image.png
Peiyuan Zhu (Jan 05 2023 at 15:44):Is there any good reference on chaotic systems? How are they usually handled numerically? Is there a curse of dimensionality in the numerical methods?
Peiyuan Zhu (Jan 05 2023 at 16:07):Does the idea of chaos ever interface with probability?
Nathaniel Virgo (Jan 06 2023 at 01:22):Yes to all three! But it's a huge field and I'm not sure I really know the best reference to get started - it probably depends a lot on what you're interested in. I first learned this stuff from Hofbauer and Sigmund's book 'Evolutionary Games and Population Dynamics', which is very specifically about applications in ecology, but it's a readable introduction and quite good for telling you what you need to know in a reasonably precise way without getting bogged down in technical details, if that's the kind of style you like.
Peiyuan Zhu (Jan 07 2023 at 02:45):Hmm so you're saying there's a curse of dimensionality problem.
Peiyuan Zhu (Jan 07 2023 at 02:47):If it's a big field, is there anyone else working in this direction besides Sigmund and Hofbauer? I've tried reaching out to them but didn't get a response
Peiyuan Zhu (Jan 07 2023 at 02:47):Also is there any conferences in this area that I can follow?
Peiyuan Zhu (Jan 07 2023 at 03:03):In what sense is it interfacing with probability? I didn’t expect that it has to be made sense in the context of ecology, but it looks interesting to me.
Nathaniel Virgo (Jan 07 2023 at 05:02):I don't know what you mean by a curse of dimensionality exactly, but there is a big difference between "high-dimensional chaos" and "low-dimensional chaos", and plenty of people studying that. It connects to probability in several different ways, but I'm not any kind of authority on that. I'd say the best place to start is to pick a textbook you like and go through it.
Peiyuan Zhu (Jan 07 2023 at 15:49):"curse of dimensionality" means it's computationally intractable for higher dimensions
Ryan Wisnesky (Feb 10 2023 at 20:31):Please stop me if this is the wrong thread, but does anyone have any pointers to literature about encoding runs of dynamical systems as solutions to lifting problems? I'm trying to encode a chaotic system (e.g. the weather) in datalog, basically
Spencer Breiner (Feb 11 2023 at 00:37):You might want to look at [[cofunctor]]
Ryan Wisnesky (Feb 11 2023 at 18:12):A relationship to cofunctors had not occurred to me; thanks for the reference!
Let me clarify the exact question. We know how to write Petri nets in prolog, CQL, etc so it's easy to see how their simulations can be formulated as lifting problems for copresheaves (in fact I remember Statebox doing stuff like this; another reference is "Simulation of Petri nets in Prolog" by Palshikar). But I understand that because of the various decidability properties of Petri nets, they can't be chaotic in the same sense as the weather. So I'm looking for a dynamical system formalism that 1) can be expressed in prolog/cql/lifting problems etc and 2) admits a chaotic system with undecidable long term behavior that 3) would make for a cool demo. I've been toying around with encoding the Collatz conjecture, for example; its behavior is chaotic enough that no one can prove it always settles, and easy enough to write just using equations, but I don't usually hear it described as a dynamical system. Examples like this but more 'natural'/'better to demo' are my real ask.
John Baez (Feb 11 2023 at 23:50):But I understand that because of the various decidability properties of Petri nets, they can't be chaotic in the same sense as the weather.
I don't know what you mean by that. Undecidability and chaos are different things.
There are various semantics for Petri nets, but if you want chaos you should pick a semantics that involves differential equations. Plain Petri nets are not very good for describing differential equations, but "Petri nets with rates" are great - this idea is familiar to chemists, I wrote a book that explains it, and this idea has been programmed into AlgebraicJulia.
A Petri net with a rates is a Petri net where each transition is equipped with a positive real number called a rate constant. There is a standard way to convert such a thing into a system of first-order nonlinear ODE.
These ODE are sufficiently general to model the "rate equations" in chemistry, including some that exhibit chaos.
Ryan Wisnesky (Feb 12 2023 at 00:02):that's good advice, and I'm conflating chaos and undecidability as both tend to make for flashy ("unpredictable") demos (my goal here); the crux the problem could really be about how to implement ODEs in prolog, which I was hoping to avoid, but I'll take another look, thanks!
Peiyuan Zhu (Feb 18 2023 at 06:31):(deleted)