nLab dynamical system

Idea

A dynamical system is a space $X$ (often a bare set or a manifold) together with a “law of motion” expressed by the action of a monoid $A$:

$$A \times X \longrightarrow X \,.$$

To model continuous time-evolution one may take $A = (\mathbb{R}, +)$ to be the additive group of real numbers.

For discrete time evolution $A = (\mathbb{Z}, +)$ is the additive group of integers (or just the monoid $(\mathbb{N}, +)$ of natural numbers). In this case the law of motion is often given by an initial value problem for a differential equation “of evolution type”.

In this case, the dynamical system is equivalently a space $X$ equipped with an automorphism (this being the action of the unit element in $\mathbb{Z}$, sometimes called the “shift”).

Sometimes the evolution is only partially defined; this is most often in dynamical systems induced by evolution differential equations which do not necessarily have existence of solutions for arbitrary large time, or the dynamical system is defined only for nonnegative time.

The definition evidently makes sense quite generally internal to various ambient categories: For instance one considers complex dynamics, algebraic dynamics, arithmetic dynamics and so on.

Dynamical systems are used to describe not only physical motions but also the behaviour of parameters of various systems, e.g. in sociological, financial, weather and other models.

Literature

Related concepts

• categorical systems theory

• zeta function of a dynamical system

• arithmetic topology

• cohomology of dynamical systems

• chaos

• Furstenberg-Zimmer structure theorem

• shifts

References

General

• Boris Hasselblatt, Anatole Katok, Handbook of dynamical systems, contents

See also

• Wikipedia dynamical system, autonomous system

Category-theoretic formulations

Discussion of dynamical systems in terms of category theory (see also at categorical systems theory):

• William Lawvere, Functorial Remarks on the General Concept of Chaos, IMA reprint 87, 1984 (pdf)

• George Dimitrov, Fabian Haiden, Ludmil Katzarkov, Maxim Kontsevich, Dynamical systems and categories, [arxiv/1307.8418]

• Mike Behrisch, Sebastian Kerkhoff, Reinhard Pöschel, Friedrich Martin Schneider, Stefan Siegmund, Dynamical systems in categories. Applied Categorical Structures 25, 2017. (link)

• Michael Barr, John Kennison, Robert Raphael, Flows: cocyclic and almost cocyclic, Theory Appl. Categories 25 18 (2010) 490–507 [tac:25-18]

• Patrick Schultz, David I. Spivak, Christina Vasilakopoulou, Dynamical Systems and Sheaves, Appl Categor Struct 28 (2020) 1–57 [arXiv:1609.08086, doi:10.1007/s10485-019-09565-x]

• David Jaz Myers, Double Categories of Open Dynamical Systems, EPTCS 333 (2021) 154-167 [arXiv:2005.05956, doi:10.4204/EPTCS.333.11]

Monograph:

• David Jaz Myers, Categorical systems theory, book project [github, pdf]

Exposition:

• David Jaz Myers, Categorical systems theory, Topos Institute Blog (Nov 2021)

• David Jaz Myers, Double Categories of Dynamical Systems (2020) [pdf]

• David Jaz Myers, A general definition of open dynamical systems, talk at MIT Category Theory Seminar (2020) [video:YT]