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\begin{document}
\subsection{General dynamic system descriptions as stable space-time structures}
\subsubsection{Introduction: General system description}
A \emph{general system} can be described as a dynamical `whole', or entity capable of maintaining its working conditions; more precise system definitions are as follows.
\begin{definition}
A simple system is in general a \emph{bounded}, but not necessarily closed, entity-- here represented as a \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} of stable, interacting components with inputs and outputs from the system's environment, or as a \htmladdnormallink{supercategory}{http://planetphysics.us/encyclopedia/SuperCategory6.html} for a complex system consisting of subsystems, or components, with internal boundaries among such subsystems. In order to define a {\em system} one therefore needs to specify the following data:
\begin{enumerate}
\item components or subsystems;
\item mutual interactions, \htmladdnormallink{relations}{http://planetphysics.us/encyclopedia/Bijective.html} or links;
\item a separation of the selected system by some boundary which distinguishes the system from its environment, without necessarily `closing' the system to material exchange with its environment;
\item the specification of the system's environment;
\item the specification of the system's categorical structure and \htmladdnormallink{dynamics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html} (a supercategory will be required only when either the components or subsystems need be themselves considered as represented by a category , i.e. the system is in fact a \emph{super-system} of (sub)systems, as it is the case of \emph{emergent super-complex systems} or organisms).
\end{enumerate}
\end{definition}
\subsubsection{Remarks}
Point (5) claims that a system should occupy either a macroscopic or a microscopic \htmladdnormallink{space-time}{http://planetphysics.us/encyclopedia/SR.html} region, but a system that comes into birth and dies off extremely rapidly may be considered either a short-lived process, or rather, a `\htmladdnormallink{resonance}{http://planetphysics.us/encyclopedia/QualityFactorOfAResonantCircuit.html}' --an instability rather than a system, although it may have significant effects as in the case of
`virtual \htmladdnormallink{particles',}{http://planetphysics.us/encyclopedia/Particle.html} `virtual photons', etc., as in \htmladdnormallink{quantum electrodynamics}{http://planetphysics.us/encyclopedia/QED.html} and chromodynamics. Note also that there are many other, different mathematical definitions of systems, ranging from (systems of) coupled \htmladdnormallink{differential equations}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} to \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} formulations, \htmladdnormallink{semigroups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \htmladdnormallink{monoids}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \htmladdnormallink{topological groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} dynamic systems and dynamic categories. Clearly, the more useful system definitions include \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} and/or \htmladdnormallink{topological structures}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} rather than simple, discrete structure sets, classes or their categories. The main intuition behind this first understanding of system is well expressed by the following passage: ``The most general and fundamental property of a system is the
\emph{inter-dependence} of parts/components/sub-systems or variables.''
\emph{The inter-dependence relation} consists in the existence of a family of determinate relationships among the parts or variables as contrasted with randomness or extreme variability. In other words, \emph{inter-dependence} is the presence or existence of a certain organizational order in the relationship among the components or subsystems which make up the system. It can be shown that such organizational order must either result in a \emph{stable attractor} or else it should occupy a \emph{stable space-time \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html}}, which is generally expressed in \emph{closed} systems by the \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of
\emph{\htmladdnormallink{equilibrium}{http://planetphysics.us/encyclopedia/ThermalEquilibrium.html}}.
On the other hand, in non-equilibrium, \htmladdnormallink{open systems}{http://planetphysics.us/encyclopedia/ThermodynamicLaws.html}, such as living systems, one cannot have a \htmladdnormallink{static}{http://planetphysics.us/encyclopedia/Statics.html} but only a \emph{dynamic self-maintenance} in a `state-space region' of the open system -- which cannot degenerate to either an equilibrium state or a single attractor space-time region. Thus, non-equilibrium, open systems that are capable of \emph{self-maintenance} will also be \emph{generic, or structurally-stable}: their arbitrary, small perturbation from a homeostatic maintenance regime does \textbf{not} result either in completely chaotic dynamics with a single attractor or the loss of their stability. It may however involve an ordered process of change - a process that follows a \emph{determinate, multi-stable pattern} rather than random variation relative to the starting point.
\subsection{General dynamic system definition}
A formal (but natural) definition of a \emph{general dynamic system}, either simple or complex can also be specified as follows.
\begin{definition}
A \emph{general dynamic system} $S_{GD}$ is a \emph{quintuple}
$([I,O], [\lambda: I \to O], \R_S , [\Delta: \R_S \to \R_S], \grp_B)$, where:
\begin{enumerate}
\item $I$ and $O$ are, respectively, the input and output \htmladdnormallink{manifolds}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} of the system , $S_{GD}$;
\item $\R_S$ is a category with structure determined by the components of $S_{GD}$ as \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and
with the links or relations between such components as \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html};
\item $\Delta: \R_S \to \R_S$ is the `dynamic transition' functor in the \htmladdnormallink{functor category}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $Aut_S$
of system endomorphisms (which is endowed with a \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} structure only in the case of reversible,
\htmladdnormallink{closed systems}{http://planetphysics.us/encyclopedia/ThermodynamicLaws.html});
\item $\lambda$ is the \htmladdnormallink{output `function}{http://planetphysics.us/encyclopedia/StableAutomaton.html} or map' represented as a manifold \htmladdnormallink{homeomorphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html};
\item $\grp_B$ is a topological groupoid specifying the boundary, or boundaries, of $S_{GD}$.
\end{enumerate}
\end{definition}
\textbf{Remark}. We can proceed to define automata and certain simpler quantum systems as particular, or specialized, cases of the above general dynamic system quintuple.
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\end{document}
