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Stream: learning: questions

Topic: any work on dynamical system bifurcations?

André Muricy Santos (Jul 01 2023 at 12:55):

Bifurcations in dynamical systems are discrete changes in the topology of the phase space, NOT the state space of a system. I wonder what this corresponds to categorically... it gives me echoes of categorical systems theory. system dynamics staying constant and parameters being inputs, and for certain input values some kind of "global" change happens. i know ACT folks do a lot of dynamical systems work but even the CSS book doesn't mention bifurcations at all. is there any work on modeling them with CT?

Naso (Jul 02 2023 at 02:40):

André Muricy Santos said:

Bifurcations in dynamical systems are discrete changes in the topology of the phase space, NOT the state space of a system.

What's the difference? Wikipedia seems to say they are the same: In dynamical systems theory and control theory, a phase space or state space is a space in which all possible "states" of a dynamical system or a control system are represented...

André Muricy Santos (Jul 02 2023 at 13:31):

Naso said:

André Muricy Santos said:

Bifurcations in dynamical systems are discrete changes in the topology of the phase space, NOT the state space of a system.

What's the difference? Wikipedia seems to say they are the same: In dynamical systems theory and control theory, a phase space or state space is a space in which all possible "states" of a dynamical system or a control system are represented...

Hm, I might be misremembering some terminology. The state space remains constant for a 2D system (it is $\mathbb{R}^2$) no matter its parameters. But I guess the phase space also does. I believe I'm talking about the vector field that arises from the system's dynamics. I don't know what is the formal correspondence in topology to a bifurcation.

Matteo Capucci (he/him) (Jul 04 2023 at 09:22):

I'm not aware of work on this. It'd certainly be interesting to do more phenomenology of dynamical systems categorically, but I don't know how to attack the problem yet.

André Muricy Santos (Jul 04 2023 at 11:17):

ok, thanks @Matteo Capucci (he/him) . does Poly cross your mind when considering this? I tend to think of it whenever i picture sharp, discontinuous jumps that arise in some formal capacity from continuous things happening (mode-dependence) but that only refers to inputs and outputs...

Matteo Capucci (he/him) (Jul 05 2023 at 10:27):

Maybe? One can surely arrange a dynamical system as a lens $(S, S) \leftrightarrows (1,1)$. This is a map of polynomials $\delta:Sy^S \to y$.
If your dynamics is parameterized by $P$, then it is a map of polynomials $P \otimes Sy^S \to y$. But this is merely repackaging the basic definitions. What can the categorical framework say about the bifurcation diagram?