From f459d369ef05168e3e0a96ebdf2736173ae0edfc Mon Sep 17 00:00:00 2001 From: Sergiu Ivanov Date: Sat, 30 Mar 2024 10:49:44 +0100 Subject: [PATCH] Define dynamical systems. --- bib/dealb.bib | 53 ++++++++++++++++++++++++++++++++++++++++++++++ deal.tex | 58 +++++++++++++++++++++++++++++++++++++++++++++++++++ hdr.tex | 2 ++ 3 files changed, 113 insertions(+) diff --git a/bib/dealb.bib b/bib/dealb.bib index c466957..df7ff1b 100644 --- a/bib/dealb.bib +++ b/bib/dealb.bib @@ -344,3 +344,56 @@ keywords = {Boolean P systems, Boolean networks, Reachability, Complexity}, url = {https://doi.org/10.1093/oxfordhb/9780199566600.001.0001}, } + +@article{Zeeman1976, + author = {Erik C. Zeeman}, + title = {Catastrophe theory}, + journal = {Scientific American}, + volume = {234(4)}, + pages = {65--83}, + year = {1976} +} + +@misc{wikiDS, + author = "{Wikipedia contributors}", + title = "Dynamical system --- {Wikipedia}{,} The Free + Encyclopedia", + year = "2024", + howpublished = + "\url{https://en.wikipedia.org/w/index.php?title=Dynamical_system&oldid=1208607435}", + note = "[Online; accessed 27-February-2024]" +} + +@article{Thom1974, + title = {Stabilité structurelle et morphogenèse}, + journal = {Poetics}, + volume = {3}, + number = {2}, + pages = {7-19}, + year = {1974}, + issn = {0304-422X}, + doi = {https://doi.org/10.1016/0304-422X(74)90010-2}, + url = + {https://www.sciencedirect.com/science/article/pii/0304422X74900102}, + author = {René Thom} +} + +@book{Brown2018, + title = {A Modern Introduction to Dynamical Systems}, + author = {Richard J. Brown}, + isbn = {978-0198743286}, + publisher = {Oxford University Press}, + year = {2018} +} + +@phdthesis{Riva22, + author = {Sara Riva}, + title = {Factorisation of discrete dynamical systems. (Factorisation de syst{\`{e}}mes + dynamiques discrets)}, + school = {C{\^{o}}te d'Azur University, Nice, France}, + year = {2022}, + url = {https://tel.archives-ouvertes.fr/tel-03937258}, + timestamp = {Wed, 25 Jan 2023 22:01:27 +0100}, + biburl = {https://dblp.org/rec/phd/hal/Riva22.bib}, + bibsource = {dblp computer science bibliography, https://dblp.org} +} diff --git a/deal.tex b/deal.tex index a65ec2a..ee82d1a 100644 --- a/deal.tex +++ b/deal.tex @@ -490,6 +490,63 @@ with Life, the Deal with Life framework can be adapted to characterizing and evaluating the interactions between other living systems, not necessarily involving the human. +\subsection{Dynamical systems} +\label{sec:ds} + +Following the spirit of~\cite{Thom1974,Zeeman1976}, I propose to use +the language of abstract dynamical systems for the general framework +of the Deal. In this subsection I quickly recall the main notions, +and I refer to sources like~\cite{Brown2018,wikiDS} for more general +and more detailed definitions. + +An \emph{abstract dynamical system} over the state space $X$ and time +$T$ is a (partial) function $\Phi : X \times T \to X$ assigning to +a state $s \in X$ the new state $\Phi(s, t)$ in which the system will +be after time $t \in T$. The function $\Phi$ satisfies the following +two natural properties: +\begin{enumerate} +\item $\Phi(s, 0) = s$: the system does not change its state in + 0 time, +\item $\Phi(\Phi(s, t_1), t_2) = \Phi(s, t_1 + t_2)$: evolving the + system for time $t_1$, then for time $t_2$ leads to the same state + as evolving the systems for time $t_1 + t_2$. +\end{enumerate} + +A common shape for a state is an $n$-vector of real parameters of +a system---in which case $X \subseteq \mathbb{R}^n$---but no +particular restrictions are imposed: $X$ may be a more a general +algebraic object. In the most general setting, $T$ is a monoid, but +it is rather customary for the time domain to be a contiguous infinite +subset of either $\mathbb{Z}$ or $\mathbb{R}$. In the first case, +$\Phi$ is usually called a discrete-time dynamical system, and in the +second case $\Phi$ is called a continuous-time dynamical system. +If additionally the state space $X$ of a discrete-time dynamical +system $\Phi : X \times T \to T$ with $T \subseteq \mathbb{Z}$ is +discrete (countable), then $\Phi$ is called a discrete dynamical +system. Finally, if $X$ is a finite set, then $\Phi$ is called +a finite dynamical system. + +In the case of discrete-time dynamical systems, the time variable can +be interpreted as the number of evolution steps. More concretely, for +a $t > 0$, $\Phi(s, t)$ can be seen as the state of the system after +$t$ steps, while $\Phi(s, -t)$ can be seen as the state the system +\emph{was in} $t$ steps ago. When $T = \mathbb{N}$, it is customary +to see the discrete-time dynamical systems as the function +$F : X \to X$, giving the state to which the dynamical system +transitions from state $s \in X$: $F(s) = \Phi(s, 1)$ +(e.g,~\cite[Chapter~1]{Riva22}). + +Given a dynamical system $\Phi : X \times T \to X$ and a state +$s \in X$, it is possible to construct the restricted function +$\Phi_s : T \to X$, $\Phi_s(t) = \Phi(s, t)$, determining the +\emph{trajectory} of $\Phi$ through $s$. +$\Ima \Phi_s = \{\Phi(s, t) \mid t\in T\}$. If $T = \mathbb{Z}$ or +$T = \mathbb{R}$, then the trajectory $\Ima \Phi_s$ can be interpreted +as the set of all states through which $\Phi$ goes before and after +reaching $s$. If $T = \mathbb{N}$, then the trajectory through $s$ is +the iteration of $F$: $\{F^k(s) \mid k \in \mathbb{N}\}$, where +$F^0(s) = s$, $F^1(s) = F(s)$, $F^2(s) = F(F(s))$, etc. + \printbibliography[heading=subbibliography] \end{refsection} @@ -497,4 +554,5 @@ systems, not necessarily involving the human. %%% Local Variables: %%% TeX-engine: luatex %%% TeX-master: "hdr" +%%% reftex-default-bibliography: ("bib/dealb.bib" "bib/sivanov-dblp-mod.bib" "bib/sivanov-extra.bib") %%% End: diff --git a/hdr.tex b/hdr.tex index 9dea2f9..227193d 100644 --- a/hdr.tex +++ b/hdr.tex @@ -88,6 +88,8 @@ \entiredoctrue % Use this if you want to compile the entire document. %\entiredocfalse % Use this if you only want the CV. +\DeclareMathOperator{\Ima}{Im} + \begin{document} \pagestyle{fancy}