CHAOS THEORY AND SYSTEMIC DYNAMICS IN EDUCATIONS
Professor Djordje Nadrljanski, PhD
University of Split
Mila Nadrljanski, PhD - docent
University of Split
Marko Tomašević, PhD
University of Split
Apstract
Chaos theory ought to be comprehended as very important for social dynamical field. It gives a new, a little
bit twisted perspective to the notion of complexity and holistic reflection. The analysing of nonlinear system
conduct is like walking through labyrinth whose walls are moving with every step (in other words, playing
a game changes it). The systems precomplex for traditional mathematics can be studied with simple laws,
e.g. fractal geometry and fuzzy logic. Complexity laws hang on hierarchial scale (scalar, self –
resemblance) and they are not caused by the conduct of constituing parts. It is less probably that one can
explain conduct of parts studying entirety conduct.
Key words: Chaos theory, Education, Systemic dynamics

New time understanding comes to the
comprehension that time is not really defined by
clock, but by rhythm and iterations. Four sorts of
attractors define the nature of patterns:

Dotted attractor (attracted or rejected by
certain activity)
 Cyclical attractor (organised among one
or many activities)
 Torus attractor (organised complexity
that repeats)
 Strange
attractor
(unpredictable
complex patterns appear through certain
time).
When chaos theory is used in social
context, four contractors, observed from
perspective, can be explained with the next:


Torus
attractors
(equally
final/negatively-entropical/aim
searchers) reflect behaviour of
socially cultural systems with
choice of outcomes and means;
unpredictable patterns come out of
stylistic aptitudes of appropriate
factors.
Education is being performed more in
surroundings of constant changes and
uncertainties. Fact development of new
technologies, changeability of structures and
conditions, information accessibility, are only
some of factors that intensify changes in
educational surroundings that becomes more
complex and dynamic. Faster reaction of
education system on changes is claimed, time
together with information becomes key factor for
successfulness. In order to react on changes
quickly and efficiently, education system must
be able to foresee what will happen next year,
and the very fact that education is being
performed in changeable surroundings, makes it
yet more difficult, and even makes it impossible.
Dotted attractors (dichotomy/oneway direction) present behaviour of
social beings in search for their
natural instincts fear, love hatred,
desire for division or self-interest.
Cyclical attractors (dialectic/selfpreserved) would refer onto the
principle of multidimensionality,
search for openly opposite but
complementary
tendencies:
stability and change, safety and
freedom, and even differention and
integration. Cyclical or periodical
emphasis from one orientation to
another,
is
the
result
of
suboptimism.
Ten years ago it was relatively easy to predict
what would happen if education was led on the
right track, desired results would be realised.
Meanwhile, it is today in doubt and more
difficult to be done. Exactly here appears the
need for redefining traditional concepts of
controlling and inventing new ones. To this
effect nowadays in the world are more being
used the knowledge and achievements of
deterministic chaos theory, the science although
relatively young one, gives high and original
1
contribution. Chaos theory had originated and
has developed at first in natural sciences, just in
the field of physics and meteorology, and after
that is being applied in biology, chemistry,
medicine and during the last 10-12 years more
and more goes through researches of
organisation
and
education.
It
is
a
multidisciplinary science that changes the way of
observing and comprehending the world,
discovers new legalities and universal behaviours
that cross limits of various scientific fields. As
the systems studied by chaos theory are
unpredictable and seemingly chaotic, in other
words act as it is impossible to find order in
them, one of central concepts is the
strengthening of total conduct of the system,
instead of the strengthening of the precise state.
geometry, which one, in contrast to the
traditional Euclid’s geometry, deals with the
description of irregular forms in nature, which
ones are more numerous than regular and linear
forms.
Significant fractal characteristic is its selfresemblance, that indicates that fractal will look,
almost, even completely identically, no matter in
what size or proportion it is observed, as it can
be seen in the picture 2, where is presented the
fractal form of snowdrop, than fractal dimension
which is not integer, but fraction.
The basis feature of chaotic systems is fine
sensitivity in starting conditions where quite
little input change may cause great changes in
output value. It is known in literature entitled
”Butterfly’s effect“ and it may be expressed
already by undoubtedly known words “The
butterfly in China flies up from a flower, and
next week the sky in Split becomes overcast.“
There are two substantial elements in
deterministic chaos; attractors and fractals.
Attractors present an area in field of results
toward which system dynamics is directed.
Attractors enable transforming seemingly
unexpected data into noticeable data, i.e. they
present results model of system conduct. Their
size and form depend on key parameters and
dynamics to which system is by such parameters
directed. Key parameter is any factor influencing
system conduct. The most famous attractor is
Loren’s attractor presented in the picture number
1.
Picture 2. Fractal appearance
Fractals are used for organizational structure
forming, so in literature are already being cited
new forms of organization called fractal
organization, which is one of substantial forms
of education adjustment to changes in
surroundings. Often is raised the question of
significance of chaos theory for self-education,
and how at all the theory, whose origins are
based on physical systems, can be applied on
education. The chaos theory in fact raises the
question of traditional ways of understanding
and education control. According to such
concept education is a system that requires
unyielding check up. Meanwhile, it is a part of
mechanistic and deterministic view on the world
which was in power up to the beginning of the
20th century, when begin doubts in determinism,
so then appear new theories and knowledge, and
among them as well chaos theory.
In this way, in traditional manner comprehended,
education is inert, slowly reacts on changes and
it may badly be adapted to incentives coming
from complex and chaotic surroundings. The
very application of chaos theory to education
would not remain on pure assumptions, there are
cited education characteristics identified by
chaos researchers and practitioners, and which
are analogous to the ones in complex physical
systems, what has enabled theory chaos
application onto education. Those characteristics
are:
Picture 1. Loren’s attractor
Another important element is fractal, which is
defined as irregular geometrical form. Fractals
are significant for chaos theory, because the very
conduct of chaotic systems is founded on fractal
1.
2.
2
Non – linearity
Non – predictability
3.
4.
5.
6.
7.
8.
Interdependency
Synergetic conduct
Autopoesis
Limits
Reversed connections
Self organisation
better understanding of changes in system and
possibilities of influence on factors that cause
changes through time and get to successfulness,
in other words unsuccessfulness of certain
system. The first systems studied with the help
of systemic dynamics, were business systems, so
the systemic dynamics was in the beginning
considered as managerial discipline. The
originator of that method was Jay Forrester. As a
manager of various projects and member of
management in different companies, he has
observed that basical obstructions for company
progress do not come from method or means of
work, but from politics and decisions made by
management on the basic of feed-back
information. Feed-back information coming
from environment and from system conduct. It is
a cyclical process in which decision is result of
change, and change again influences on
decisions that will be made later. Decision that
solves a certain problem in company is based on
politics of that company, which then directs
company toward success or failure.
Chaos theory enables better understanding of
operations of organisation. Applying its concepts
education prepares itself for fast reaction in view
of the changes in extern surroundings. Its
signification reflects in the identification of key
parameters directing system from one dynamic
state into another, it takes care of unpredicted
circumstances and supports assimilation of
flexible strategies for unexpected events. It is
considerably to emphasize that the modification
of the education aims is beforehand foreseen, so
it does not cause shocks and surprises.
Substantial characteristics of management in
chaotic conditions are early identification of
events, fact answer with mix of strategy and
acceptance of the uncertainty in which education
takes place. Understanding of events dynamics
enables consideration about actions and
consequences coming from it.

During defining of management conception one
ought to respect some rules substantial for
chaotic systems, and they are as follows:




Managers ought to give attention to
communication at all levels in
education
Little changes can cause great
effects in chaotic conditions
Unforeseeable and unexpected
characteristics can be expected in
systems
Systemic dynamics has a great part in
understanding influences of various
politics on system conduct because it
enables creating computer models
which simulate system conduct in view
of the applied politics.
On the basis of logical simulatory
model of dynamical system, it is
possible to put certain mathematical
equitations and with their help to
present arithmetically and graphically
system movement in future.
The using of simulatory models in discovering
consequences of conduct of certain system, has
stimulated Forrester and his followers to research
also other dynamic systems except business
ones. Nowadays systemic dynamics applies in
various fields as ecology, micro and macro
economy, medicine, engineering, researches of
social trends, education etc.
Using positive feed-back, education receives
information from surroundings about state in
surroundings, so it sets up field of order by
control over events or changes. Because of
feature of educational surroundings, on account
of faster adjustment, the very education must be
changed, so there appears the need for
strengthening of surroundings that learns.
Short history of systemic dynamics
Systemic dynamics as a new methodology of
modelling complex dynamic systems appeared in
the middle of the fifties of the twentieth century,
and is connected with the works of professor Joy
Forrester from Massachusetts Institute of
Technology. Forrester was then interceded with
several projects connected with testing army
equipment (supervisory mechanisms, flying
simulators), so as a member of management
Systemic dynamics
System conduct, especially the conduct of
complex dynamic social and natural systems is
already fifty years the object of research of the
method known as System Dynamics. The
method has developed in case of the need for
better understanding and control of the system
with feed-back, primarily of social systems, i.e.
3
observed that the problems with projects are not
exclusively of engineering nature, but that they
come from politics set by management itself.
Forrester published the book „Industrial
Dynamics“ in 1961. He had presented in it the
basis of systemic dynamics and its use in solving
business-management problems. A little later he
observed that that method can be applied also in
other fields, for example in system as town, so
he presented in the book „Urban Dynamics“ how
the strategy of town development can influence
solving different problems. Forrester had been
researching during last twenty years two fields:


construction of simulatory model of dynamical
system through the next steps:



Forming economy model of U.S.A with
the help of systemic dynamics and
Applying systemic dynamics in school
system (from kindergarten to faculty)


Basic notions of systemic dynamics
Systemic dynamics is a methodology that
combines theory, philosophy and methods for
analysis and management of conduct in complex
systems with feed – back, such as business and
other social systems. Feed – back supposes that
an activity X results by activity Y which again
influences upon X etc. It is erroneously to
observe independently and linearly the
connection between X and Y, as well between Y
and X, but it is necessary to observe the whole
system as a cyclic feed – back loop, so that
system conduct could be predicted. The world
around us is not linear, it is cyclical. Man lives in
surroundings where a state leads to decision
making that causes changes and brings to a new
state on which basis he makes new decisions. It
can be graphically presented by cyclic feed –
back loop as it is in the picture 3.

Systemic dynamics methodology
Systemic dynamics is based on System Thinking
concept. Systemic thinking enables observing
system structures, so that system conduct could
be understood, but it does not imply building a
simulatory model. Modelling of system
structural enables system dynamics. System
structure includes four hierarchical components.


Action,operation
decision

Information about
problem
problem identification – defining of
the point for system observation,
defining system limit, aim of model,
etc.
system
conceptualization
–
strengthening of elements and structure
of system, diagram making, setting
dynamical hypotheses that explain
cause of problem,
model formalisation – building
computer simulatory model of system
that presents essence of problem.
analysis of model conduct – testing of
model that ought to copy conduct of
real system,
evaluation and validation of model, as
well testing of alternative problem
solutions,
model application
Result(change)

Picture 3. Cyclic feedback loop
Systemic dynamics as methodology used for
apprehension of system conduct supposes

4
System limits – limits have to be
defined so to include only those
elements that cause certain system
conduct
Cyclic feed – back loop as basic
system presentation – system conduct
is defined by elements structure in
closed loop; feed – back loop is
responsible for changes in structure
during course of time.
Levels and rates – within system there
are levels or states and rates or changes
of state. Levels or states present
quantity of an element, e. g. number of
employees, and level of water in water
tank, etc. feet or rates present unit of
change of state (increasing or
decreasing of level).
Aims, observed state, disharmony
between aims and conditions, wanted
action
Aim is system state which one wants to
achieve through change, and observed
state is momentary state of system.
Disharmony between the two states
leads to action in order to decrease
discrepancy between them.
System structure can be graphically presented
with two kinds of diagrams:


Learning
-
Free time
Casual Loop Diagram, Feed back
Diagram and
Stock / Flow Diagram. Every diagram
has its symbols.
Casual Loop Diagram, Feedback Diagram
presents cyclical cause consequential feed – back
loop. Cause leads to consequence which
influences cause etc.
Picture 6. Negative feed – back loop
Models of systemic dynamics
Models and modelling are in detail processed in
the second chapter of this book, and here are
exposed only the texts that are necessary for
comprehension of systemic dynamics. When
people consider various occurrences of real
world, e. g. family, town, growth / fall of
population and the like, they form mental models
about those occurrences. Systemic dynamics
enables formalisation of mental models of
dynamic systems forming computer simulatory
models. With simulatory model it is possible to
simulate conduct of very complex dynamical
system.
Learning
Mark
Teaching
Average mark
Picture 4 . Feed – back loop in teaching
Mental model is a network of various facts and
concepts that contain human comprehension of
social and physical phenomena. It is being made
during man’s interaction with certain
phenomenon i.e. with the observed system.
Mental model is in fact mental perception of a
system, connections in system and in comparison
of systems as well conducts produced by system
structure. Mental model is formed on the basis of
man’s previous knowledge and experience with
similar systems and on the basis of his abilities
for data processing. It is flexible, rich with
details and often very complex. Mental models
are used in daily life, they are used with various
decisions making in learning process during
which they change, correct, improve or are
dropped. But human mind often can not make
mental model of complex system on the whole,
but only of one of its part. Therefore mental
models are incomplete, imprecise, insufficiently
correct, unstable. Because of unreliability of
mental model comprehension and conduct
foreseeing of a system need not be correct.
Solution of such a problem can be found in
applying systemic dynamics. Transformi.
Learning
+
Mark
Picture 5. Positive feed – back loop
mental model into computer simulatory model is
possible easier disclosing, learning and
comprehending of conduct resulting from certain
system structure. Computer simulatory models.
The basic characteristic of systemic dynamics is
making computer simulatory models that copy
mental structure model of a system. Simulatory
model copies system conduct on the basis of in
advance given system variables.
5
Changing system variables it is possible to
change system structure and analyze system in
the course of time. Such simulations enable
modelling and comprehension of system which
is otherwise difficult or impossible to follow in
real world (e.g. ecosystem change in longer time
period or in a remote geographical region ), and
likewise influences learning process, which it
makes easier, and decision making process,
which it supports.
Wickens, C. D. (1992) 2nd Ed. Engineering
Psychology and Human Performance, Harper
Collins Publishers, 551 p.
Zhang, D., Zhao, J. L., Zhou. L. & Nunamaker,
J. F. Jr. (2004) “Can e-Learning Replace
Classroom Learning?” Communications of the
ACM, Vol. 47, No. 5, pp. 75-79.
Good simulatory model of systemic dynamics
has to show how and why dynamic process
behaves in certain way at certain time. It is
efficient if it improves mental model of real
system and makes easier connection of various
kinds.
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