Define dynamical systems.

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}
+}

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.
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@@ -88,6 +88,8 @@
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+\DeclareMathOperator{\Ima}{Im}
+
 \begin{document}
 
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