general system definitions
0.1 General dynamic systems descriptions as stable space-time 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.
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:
components or subsystems;
mutual interactions, relations or links;
the specification of the system’s environment;
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 super-system of (sub)systems, as it is the case of emergent super-complex systems or organisms).
Point (5) claims that a system should occupy either a macroscopic or a microscopic space-time region, but a system that comes into birth and dies off extremely rapidly may be considered either a short-lived 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 inter-dependence of parts/components/sub-systems or variables.”
Inter-dependence consists in the existence of determinate relationships among the parts or variables as contrasted with randomness or extreme variability. In other words, 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 stable attractor or else it should occupy a stable space-time domain, which is generally expressed in closed systems by the concept of equilibrium.
On the other hand, in non-equilibrium, open systems, such as living systems, one cannot have a static but only a 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 self-maintenance will also be generic, or structurally-stable: 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, multi-stable 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.
A general dynamic system is a quintuple , where:
and are, respectively, the input and output manifolds of the system , ;
is a category with structure determined by the components of as objects and with the links or relations between such components as morphisms;
is a topological groupoid specifying the boundary, or boundaries, of .
|Title||general system definitions|
|Date of creation||2013-03-22 18:19:02|
|Last modified on||2013-03-22 18:19:02|
|Last modified by||bci1 (20947)|
|Defines||general dynamic system|
|Defines||configuration space algebraic topology|
|Defines||dynamic configuration functors|