general system definitions

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.

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

super-system

of (sub)systems, as it is the case of

emergent

super-complex systems

or organisms).

0.1.2

Remarks

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.

Definition 0.2

.

A

general dynamic system

S

G

⁢

D

is a

quintuple

(

[

I

,

O

]

,

[

λ

:

I

→

O

]

,

ℛ

S

,

[

Δ

:

ℛ

S

→

ℛ

S

]

,

𝔾

B

)

, where:

1.

I

and

O

are, respectively, the input and output manifolds of the system ,

S

G

⁢

D

;

2.

ℛ

S

is a category with structure determined by the components of

S

G

⁢

D

as objects and
with the links or relations between such components as

morphisms

;

3.

Δ

:

ℛ

S

→

ℛ

S

is the ‘dynamic transition’

functor

in the

functor category

A

⁢

u

⁢

t

S

of system

endomorphisms

(which is endowed with a

groupoid

structure only in the case of reversible,
closed systems);

4.

λ

is the output ‘

function

or map’ represented as a manifold

homeomorphism

;

5.

𝔾

B

is a topological groupoid specifying the boundary, or boundaries, of

S

G

⁢

D

.

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

2013-03-22 18:19:02

Last modified on

2013-03-22 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

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