0.2 Dynamics

Dynamics of systems

In this module we will present several definitions and a language that will later be used to model socialsystems. Although a complete and detailed presentation will not be made, the ideas covered are very important for anything other thana superficial understanding of dynamic models. Much of this material grew out of what is called system theory and controltheory. [link] [link]

Definitions

As we noted before, it is sometimes difficult to give clear, precise definitions of some ideas. That is the casefor the definition of a system which sounds a bit vague but seems to be as good as possible.

A system is defined as a set of interrelated entities, variables, or ideas that have some common features orpurpose.

Examples of systems would be a car, a radio, a transportation network, a set of coupled equations, a society, afamily, etc. The system may be physical, biological, social, conceptual, or many other forms.

The dynamics of a system is the way the various variables of the system change and evolve withtime.

The study of dynamics is an important part of physics, engineering, and economics. Indeed, the study of change inhistory, psychology, etc. can be viewed as a study of dynamics, and when anyone makes predictions about the future, he is certainlyusing a dynamic model whether he realizes it or not. There are many studies of systems which are not dynamic models - these use staticor equilibrium models and study relationships where time variations are assumed not important. The mathematics often used in the studyof system dynamics are calculus and differential equations.

The structure of a system is the specification of the components of importance and interest and thedescription of the relations and interconnections within the system.

The choice of structures may be easy or very difficult depending on the system. In many physical systems thestructure is fairly well developed, however, for social systems it is more complicated. The choice of age groupings, economicgroupings, etc. by a sociologist is the choice of structure for a particular system. Indeed, much of the research in the socialsciences has centered around structure with relatively little work being done explicitly on dynamics. For our purposes, we needboth.

Descriptions of systems

There are two rather different but complimentary descriptions that have been used with success insystems analysis. One is an input-output or external approach, and the other is a state variable or interval approach. Both havemerits and will be briefly described.

The input-output description

Here there are three entities considered: the input $x$ , output $y$ , and the system $s$ . Symbolically, this is illustrated by

This has proven a very valuable approach that avoids internal details of the system that are of no interest orare difficult to describe.

There are three problems that can be formulated with this description: