general system definitions
0.1 General dynamic systems descriptions as stable spacetime structures
0.1.1 Introduction: General system description
A general system can be described as a dynamical ‘whole’, or entity capable of maintaining its working conditions; more precise system definitions are as follows.
Definition 0.1.
A simple system is in general a bounded^{}, but not necessarily closed, entity– here represented as a category^{} of stable, interacting components with inputs and outputs from the system’s environment, or as a supercategory^{} for a complex system consisting of subsystems, or components, with internal boundaries among such subsystems. In order to define a ‘system’ one therefore needs to specify the following data:

1.
components or subsystems;

2.
mutual interactions, relations^{} or links;

3.
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;

4.
the specification of the system’s environment;

5.
the specification of the system’s categorical structure^{} and dynamics (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 supersystem of (sub)systems, as it is the case of emergent supercomplex systems^{} or organisms).
0.1.2 Remarks
Point (5) claims that a system should occupy either a macroscopic or a microscopic spacetime region, but a system that comes into birth and dies off extremely rapidly may be considered either a shortlived process, or rather, a ‘resonance’ –an instability rather than a system, although it may have significant effects as in the case of ‘virtual particles’, ‘virtual photons’, etc., as in quantum electrodynamics and chromodynamics. Note also that there are many other, different mathematical definitions of systems, ranging from (systems of) coupled differential equations to operator formulations, semigroups, monoids, topological groupoid^{} dynamic systems and dynamic categories. Clearly, the more useful system definitions include algebraic and/or topological structures 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 interdependence of parts/components/subsystems or variables.”
Interdependence consists in the existence of determinate relationships among the parts or variables as contrasted with randomness or extreme variability. In other words, interdependence 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 stable attractor or else it should occupy a stable spacetime domain, which is generally expressed in closed systems by the concept of equilibrium.
On the other hand, in nonequilibrium, open systems, such as living systems, one cannot have a static but only a dynamic selfmaintenance in a ‘statespace region’ of the open system – which cannot degenerate to either an equilibrium state or a single attractor spacetime region. Thus, nonequilibrium, open systems that are capable of selfmaintenance will also be generic^{}, or structurallystable: their arbitrary, small perturbation from a homeostatic maintenance regime does 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 determinate, multistable pattern rather than random variation relative to the starting point.
0.2 General dynamic system definition
A formal (but natural) definition of a general dynamic system, either simple or complex can also be specified as follows.
Definition 0.2.
A general dynamic system ${S}_{GD}$ is a quintuple $([I,O],[\lambda :I\to O],{\mathcal{R}}_{S},[\mathrm{\Delta}:{\mathcal{R}}_{S}\to {\mathcal{R}}_{S}],{\mathbb{G}}_{B})$, where:

1.
$I$ and $O$ are, respectively, the input and output manifolds of the system , ${S}_{GD}$;

2.
${\mathcal{R}}_{S}$ is a category with structure determined by the components of ${S}_{GD}$ as objects and with the links or relations between such components as morphisms^{};

3.
$\mathrm{\Delta}:{\mathcal{R}}_{S}\to {\mathcal{R}}_{S}$ is the ‘dynamic transition’ functor^{} in the functor category^{} $Au{t}_{S}$ of system endomorphisms^{} (which is endowed with a groupoid^{} structure only in the case of reversible, closed systems);

4.
$\lambda $ is the output ‘function or map’ represented as a manifold homeomorphism;

5.
${\mathbb{G}}_{B}$ is a topological groupoid specifying the boundary, or boundaries, of ${S}_{GD}$.
Remark. We can proceed to define automata and certain simpler quantum systems as particular, or specialized, cases of the above general dynamic system quintuple.
Title  general system definitions 
Canonical name  GeneralSystemDefinitions 
Date of creation  20130322 18:19:02 
Last modified on  20130322 18:19:02 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  31 
Author  bci1 (20947) 
Entry type  Topic 
Classification  msc 93A30 
Classification  msc 93A05 
Classification  msc 93A13 
Classification  msc 93A10 
Synonym  dynamic system 
Synonym  dynamic structure 
Synonym  general system 
Related topic  SimilarityAndAnalogousSystemsDynamicAdjointness2 
Related topic  CommutativeVsNonCommutativeDynamicModelingDiagrams 
Related topic  GroupoidCDynamicalSystem 
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Defines  general dynamic system 
Defines  complex system 
Defines  system dynamics 
Defines  configuration space algebraic topology 
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Defines  dynamic configuration functors 
Defines  qualitative biodynamics 
Defines  supercomplex system 
Defines  generating class 
Defines  dynamic multistability 
Defines  biodynamics 
Defines  q 