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Stream: theory: applied category theory

Topic: Dynamical systems: Monoid actions vs "vector fields"

Eigil Rischel (Aug 15 2020 at 09:53):

A discrete dynamical system is a set $S$ with a time-step operator $T: S \to S$ (and optionally an initial state $s_0 \in S$ ). This is obviously equivalent to a set with an action of the monoid $(\mathbb{N}, +)$ .

By a "smooth" dynamical system, one could mean a lot of things, but two natural definitions are

A smooth manifold equipped with a smooth vector field
A smooth manifold equipped with an action of the "Lie monoid" $[0, \infty)$ (This is a smooth manifold with boundary)

Taking derivatives at 0 takes you from 2. to 1, and probably by some general existence and uniqueness theorem, you can go from 1. to 2., although I haven't thought too deeply about that, and there may be some weird corner cases involving finite-time blowup - the equivalence maybe holds only on compact manifolds or something.

This shows us two ways of imagining a dynamical system:

As a set of equations for the dynamics.
As a group/monoid action specifying the dynamics.

In the discrete case, this distinction sort of collapses, but in the smooth case, the two viewpoints are obviously conceptually very different, even if they're close to each other in practice.

Now, there's been a lot of work on "open dynamical systems". When people have thought about continuous dynamical systems in this context, they've generally used perspective 1.

Question: Has anybody written a paper about "open $\mathbb{N}$ -Sets"?
The normal way "open dynamical systems" works is that some of the variables in the equation specifying the dynamics are open, and then when you wire the systems together you plug a function of a different system into that variable.
This seems a bit harder to make sense of from the perspective of a monoid action.

Joshua Grochow (Aug 15 2020 at 21:15):

If you consider algebraic (in the sense of algebraic geometry) monoids, you could always consider it as being defined over R(x, y, z,...) where R is the reals and x, y, z,... are the free parameters.

Morgan Rogers (he/him) (Aug 16 2020 at 11:33):

Indeed, being able to integrate a vector field to give an action of the topological monoid you describe is the content of existence/uniqueness results for solutions of ODEs/PDEs.

Regarding your question, I don't know if this has been done before, but I can sketch what a formalism would look like. Let's think of these systems as we do open Petri nets, with distinguished inputs and outputs. As elements of $\mathbb{N}$ -sets, there is nothing special about the inputs, since anything placed at an input proceeds under the $\mathbb{N}$ -action as that element would. On the other hand, outputs would need to behave as free sub- $\mathbb{N}$ -sets when an open system is being considered in isolation, because the subsequent behaviour is unconstrained. To compose these systems, we assume the existence of a mapping of outputs of one system to inputs of another system, which produces a partial $\mathbb{N}$ -set homomorphism from the free (output) sub- $\mathbb{N}$ -sets of the first system to the second system. Insisting that there are finite numbers of inputs and outputs in the systems and that these must agree, we get a big PROP of these things, but it's easy to see how this would extend to larger numbers/spaces of inputs and outputs.

Morgan Rogers (he/him) (Aug 16 2020 at 11:41):

Incidentally, these systems are expressed nicely as cospans of $\mathbb{N}$ -sets, as you might expect. Take the free $\mathbb{N}$ -set on the inputs and the free $\mathbb{N}$ -set on the outputs; these each have mappings to the system, the latter of which must be monic. It would be nice to check that the canonical pushout composition for cospans gives the composite system I described above.

This formulation immediately begs questions about generalisations (eg if we remove the condition that the output map should be monic, we get open $\mathbb{N}$ -sets with constrained outputs,in the sense that the behaviour of the output in any composite system must be a quotient of that in the original system).

Matteo Capucci (he/him) (Aug 17 2020 at 10:16):

Did you look at Myers and Spivak's open dyanmical systems? I was under the impression that they encompass discrete dynamical systems as well, but it's been a while since I meddled with their theory

Eigil Rischel (Aug 17 2020 at 14:33):

@Matteo Capucci: They do - a discrete dynamical system with inputs $A$ , outputs $B$ and state space $S$ consists of functions $A \times S \to S, S \to B$ .

Jade Master (Sep 09 2020 at 18:22):

@Eigil Rischel I am also interested in dynamical systems from the second approach that you mention. In this embarrassingly titled blog post I described what a category of dynamical systems in the second approach can look like.

You can definitely take the category I describe and naively "openify" it...but it doesn't quite work. Suppose that $F$ is a dynamical system on the manifold $X$ and $G$ is a dynamical system on the manifold $Y$ . Then maybe you want $F$ and $G$ to overlap on some region of space $A$ as specified by nice maps $f :A \to X$ and $g: A \to Y$ . To compose these systems you would want to take the pushout, in the category of dynamical systems, over $f$ and $g$ two maps. The problem is that these maps won't in general match the dynamics of $F$ and $G$ . If $F$ and $G$ don't have the same flow on the image of $f$ and the image of $g$ then they won't be morphisms of dynamical systems. Therefore, composing these systems via pushout over $F$ and $G$ has two cases: either they have the same dynamics in the overlapping region so the pushout is rather trivial...or there are no morphisms of dynamical systems which can facilitate the pushout.

Jade Master (Sep 09 2020 at 18:24):

So it seems like the problem is that morphisms of dynamical systems are too restrictive.

Jade Master (Sep 09 2020 at 18:27):

(deleted)

Jade Master (Sep 09 2020 at 18:31):

I believe that you will run into a similar problem for $\mathbb{N}$ -sets...it will be hard to find pairs of maps which match the dynamics. Although for the discrete case things seem a bit less hopeless. I don't have any evidence to back that up, it's just a hunch.

Eigil Rischel (Oct 04 2020 at 15:01):

Hey @Jade Master, that's a great blog post! Can one compose dynamical systems by pullbacks ("variable sharing") rather than pushout? Like suppose $X,Y,F,G$ are as you say. Then maybe you have maps $f: X \to Z, g: Y \to Z$ picking out some variable in each dynamical system, and I want to build a dynamical system on the pullback $X \times_Z Y$ . But you can't necessarily do that, unless this subset of $X \times Y$ is stable under the diagonal flow. (And also, pullbacks of manifolds don't always exist).

Jade Master (Oct 05 2020 at 20:35):

@Eigil Rischel I think this span approach is actually a lot more productive. The intuition you have sounds right to me...it is variable sharing. Being stable under the diagonal flow means that the variables of your two systems are being synchronized. Two references I know about are this paper: https://arxiv.org/abs/1704.02051

Jade Master (Oct 05 2020 at 20:36):

And this paper: https://arxiv.org/abs/1710.11392

Jade Master (Oct 05 2020 at 20:36):

Check out section 6 of the first one for a category of open dynamical systems which is similar to the one you're describing.