Non-Linear Dynamics (Chaos Theory) and its Implications for
Policy Planning
F. David Peat
I Introduction
Non-linear dynamics, of which Chaos Theory forms an important
part, is currently an active and fashionable discipline that is having
a profound effect on a wide variety of topics in the hard sciences.
Its combination of novel mathematics and high speed computing,
often using full color visual displays, has produced new insights
into the behavior of complex systems and reveals surprising results
even in the simplest non-linear models.
Recently these ideas, particularly those of chaos theory, have found
applications in economics, ecology, populations dynamics, the
health sciences (including the dynamics of processes in the human
body) and sociology. It is also possible that these new ways of
thinking will also become important in the whole field of policy
planning.
This paper presents a brief overview of non-linear dynamics and its
implications for policy planning.
II History
Non-linear dynamics has its origins in the famous "three body
problem" and the attempts, at the turn of the century, by the great
French mathematician and physicist, Henri Poincare, to calculate
the motion of a planet around the sun when under the perturbing
influence of a second nearby planet or moon. In many cases, as
expected, the presence of the third body acted to modify the
original orbit. But there were also situations in which the planet
moved in a highly erratic way, even to the extent of behaving
chaotically.
To have discovered chaos at the heart of an apparently stable solar
system came as a considerable surprise. However, further
exploration of these ideas had to await the development of new
mathematical techniques (major contributions coming from
mathematicians and theoretical physicists in the Soviet Union) and
the development of high speed computers capable of displaying
their complex solutions visually, on a screen, or graphically.
Early applications of this new approach include chaotic swings in
insect populations, extreme sensitivity of weather patterns, stable
waves (solutions) in water, non-linear behavior of electronic
systems, vibrations in mechanical structures, brain waves, heart
beats, coupled chemical reactions and a host of other applications.
Today the application of non-linear dynamics can be found in
almost every branch of science. It includes systems in which
feedback, iterations, non-linear interactions, and the general
dependency of each part of the system upon the behavior of all
other parts, demands the use of non-linear differential equations
rather than more simple and familiar linear differential equations.
Its particular sub-disciplines and key concepts include:
- Chaos Theory (deterministic chaos)
- Rene Thom's Catastrophe Theory
- structural stability
- Ilya Prigogine's dissipative systems
- systems with complex feed-back loops
- feed-forward systems
- solitons
- fractals
- bifurcation theory
- strange attractors.
While these terms may sound highly technical it should be realized
that, in addition to the mathematical framework of non-linear
dynamics, new informal concepts, paradigms and approaches to
the problems must also be taken into account. It is these new ways
of thinking that will have a major influence on policy planning.
III. The Newtonian Paradigm and Public Policy
Current ideas about public policy and the ability to exercise control
or give direction to a organization or sector of the economy are, to
some extent, derived from the outstanding success of the
Newtonian model of physics. And since the very assumptions of
classical physics are called into question by non-linear dynamics, it
is worth examining them in detail.
i. A system.
Public policy, like classical physics, assumes that it is possible to
focus upon a well defined system. This implies that the system can
be conceptually isolated from its surroundings and that its
characterization should not change radically with time. For, if the
whole nature of the system were to change in a bizarre or
uncontrolled fashion, how would we know if we were studying the
same system or something entirely different? Moreover it must be
possible to separate the internal behavior of the system from
external fluctuations. Otherwise how would we know what was
the result of internal decisions within the system, our planned
interventions or the product of external contingencies? When we
enter the non-linear domain we discover that many of these
assumptions are no longer valid.
ii. System description.
In physics the essential features of a natural system can be
identified and quantified. It is then possible to associate them with
mathematical variables. Indeed a system in physics is normally
associated with a point in phase space. Moreover it is assumed
that, in principle, it is possible to obtain a full, experimental
description of the system in terms of the numerical values of all its
variables, with any errors or uncertainties having a negligible
implication.
In the case of policy, although certain important factors will be of a
qualitative nature, it is usual to employ some degree of quantitative
measure such as progress, productivity, efficiency, value, return on
an investment, etc. Clearly it is vital that the meaning of such
variables should not suddenly change at some point in time.
iii. System Dynamics.
Newton's laws show how it is possible, given an initial point in
phase space, to plot out the trajectory of a system for all future
times. In other words, given the full specification of a system, it is
possible to determine its future behavior. Any external force or
perturbation will produce a predictable change while tiny external
fluctuations have a negligible effect.
Moreover it is assumed that the behavior of the system is orderly,
and does not fluctuate erratically or totally changes its qualitative
nature.
As regards policy, it is assumed that sufficient data can be
collected in order to predict the future of the system. Or if not
predict, then at least to show general trends. Moreover
externalities can always be taken into account as perturbations. As
regards the implementation of policy, it is assumed that a well
defined intervention will produce a well defined and predictable
result. And when the system begins to deviate from its preassigned
or nominal behavior it should be possible to exercise control and
dampen any unwanted oscillations.
iv. Deviations
Where deviations from this well defined scheme occur, where any
sudden qualitative changes of behavior, chaotic or wild oscillations
are found in a hitherto well behaved system, it is assumed that they
can be tracked down to an external factor and action can be taken
to modify or steer the system back in the right direction. And,
while major external shocks can disturb a system, it is assumed that
vanishingly small chance fluctuations will produce only very minor
changes.
v. Non-linearities
Wherever non-linearities occur in a system, wherever the output of
one cycle period iterates or feeds into another, wherever parts of a
system depend every sensitivity on each other, then a situation
arises in which one or more of the above assumptions becomes
invalid. While this happens, the whole framework upon which
traditional policy making is based must be called into question and
some new approach developed. (While general system's theory has
made great progress in describing the dynamics of complex self-
interacting systems even a "system" must be defined within a
context, and this context is always limited in some way and liable
to future change.)
IV. Non-linear systems.
Let us examine specific ways in which non-linearities can frustrate
an attempt at policy planning.
i. Butterfly effects
It may not always be possible to pin down a system exactly. There
may be, for example, certain unknown or uncertain factors. The
boundary to a system may not be well defined or the very act of
observation and measurement may introduce uncertainties.
To give a technical example, B. Mandelbrot has pointed out that
the distribution and number of weather stations has a "lower fractal
dimension" than that of any real weather system. This means that,
in principle, we can never gather sufficient information to
characterize the world's weather. A tiny degree of uncertainty in a
linear system does not really matter-- it simply results in a small
degree of uncertainty in its future. But for some non-linear systems
these uncertainties can increase exponentially; such systems are
infinitely sensitive to their initial conditions so that the smallest
initial fluctuation soon swamps the system.
Other systems may be infinitely sensitive to externalities -- the
butterfly effect - so that a tiny fluctuation or perturbation arising in
some nearby system will swamp the system. Another aspect of the
butterfly effect is that a small periodic effect, operating over a long
enough time, may end up dominating the system while large
external "shocks" are damped out.
Not only will the future of such systems be uncertain but attempts
at control or corrective measures will give unpredictable results.
ii. Sudden changes
Non-linear systems are characterized by having "bifurcation-
points", regions where the system sits on a knife edge, as it where,
and may suddenly change its qualitative behavior. A system that
has been well behaved for a long period may suddenly act
erratically. A company that has been growing steadily for several
years may unexpectedly enter a period of uncontrolled oscillations
of its economy. Other systems may become self-organized and
settle down into a relatively stable period of well defined economic
behavior. Attempts to steer this behavior into new directions
during this period will be surprisingly difficult.
Over its life, a non-linear system can enter a series of quite
different economic regimes and behaviors. And, it must be
stressed, these changes need not always be the result of external
perturbations or "shocks" but are the natural unfolding of the
internal dynamics of the system. Policy makers would therefore
have to take into account that a system may, at one time, be
insensitive to control, and at another infinitely sensitive and that
major changes in a system may not always be the result of external
factors for an apparently negligible effect may, given time, swamp
the behavior of the system.
iii Exogenous or Endogenous Change?
When a system, steered by a particular policy, undergoes a sudden
dramatic change one normally looks for some external cause. Has
something changed in its environment, has some unforeseen
demand surfaced, or is it the result of the development of a new
technology? But what if this major fluctuation or qualitative
change has nothing to do with external circumstances but is
endogenous - the result of purely internal dynamics? A small
regular, periodic internal fluctuation can suddenly swamp the
system; and the iteration of an output into the next cycle will, in
time, result in qualitatively new behavior. It is of obvious
importance to be able to distinguish endogenous from exogenous
factors.
iv Chaotic behavior
Systems sometimes enter regions of highly erratic and chaotic
behavior. In such cases it becomes impossible to predict the future
behavior of the system even when based on its entire past history.
From moment to moment the system jumps violently in its
behavior, moreover, it may be infinitely sensitive to any external
change of fluctuation.
But is a chaotic system totally devoid of order? A chaotic system
appears totally unpredictable in its behavior, moreover its behavior
may be impervious to corrective measures. But scientists are now
finding that what is called "deterministic chaos" exhibits certain
regularities. For example, erratic swings, while entirely
unpredictable, may nevertheless be confined to a particular limited
region -- called a chaotic or strange attractor. So while the moment
to moment behavior of the system is unpredictable, uncovering the
geometry of the strange attractors give information about the
overall range of behavior. It is also a matter of debate as to
whether a chaotic system should be spoken of as totally devoid of
any order, or as exhibiting a highly complex and subtle order.
Moreover such systems may also exhibit "intermittency", periods
of simple order which emerge again and again out of chaos. When
faced with the alternation of order and chaos one may ask: "Does
this represent a break down of good order, a failure of policy? Or
is the order itself a temporary breakdown of a more general chaos -
or infinite complexity of behavior?"
That there can be order within chance can be seen in the following
way: Suppose someone has tossed ten "heads" in a row. Most
people would bet that the next throw must be tails. But knowing
that the system is truly random indicates that there is a 50:50
chance that the next throw will be "heads". In this way an
experienced gambler will, on the average, win over a gullible
opponent. In a similar fashion, knowing the range of chaotic
behavior enables one to hedge policy bets and come out marginally
ahead over a long period of time.
v Self similarity
Chaotic systems have much in common with fractals, indeed their
strange attractors have a fractal structure. Likewise there may be
detailed fractal patterns in their dynamics that repeat at different
scales of time. Having knowledge of such patterns would make it
possible to, on the average, make better micropredictions. I.e. one
computer analysis of stock market data suggests that there are self-
similar patterns at 14, 5 and 2 yr. periods and in 5 month periods
and that the same patterns may be present within each day.
vi Feedforward
Where two or more products compete for a given market a process
of feedforward takes place. The effect of a tiny initial fluctuation
may cause one particular product to eventually dominate the
market. An example of this is the competition between VHS and
Betamax videocassettes.
V. Examples
The manifestation of non-linear effects can be discovered in a wide
variety of examples, from sociology, population dynamics,
economics and ecology. In each case mathematical models can be
built that have the potential for a wide range of behavior from
stability, gradual growth, persistent oscillations, self-organization,
rigidity to change, infinite sensitivity to externalities, all the way to
chaotic and unpredictable swings. Of course mathematical models
are far from the real world but the possibility that a well behaved
system could, at some point, engage in a radically different, and
uncontrollable, form of behavior gives food for thought.
Moreover, as more and more examples are found in the real world
of qualitative changes in behavior, of chaos, sensitivity or rigidity,
it becomes important to take them into account wherever policies
are being made and the implications of actions contemplated.
i. Ecology
Take an obvious example where non-linear effects occur. There
has been much debate about the greenhouse effect. Suppose,
therefore, we ask what will be the effect of increasing carbon
dioxide on plant growth? The whole question of the effects global
warming, increased humidity and carbon dioxide on vegetation is a
highly complex issue. Not only will growth rates change but the
whole balance of a region will be modified, with some species
being favoured over others. For example, what may be good
conditions for the growth of a certain crop may be even better for
weeds and predators. In turn, the effects of these changing
vegetation patterns will feed back into the atmosphere, both
directly - in terms of the amount of carbon dioxide that is fixed by
plant-life - but also indirectly, for as the mixes and yields of
different vegetation changes so too will the economics and even
the lifestyles of a given region. As the economy and social
structure of a region changes so too does its energy demands,
which results in different amounts of carbon dioxide being released
into the atmosphere. Moreover, there will be a variety of lags in
the various feedback loops of such a system, so that attempts to
control variations in one part of a cycle may have the effect of
magnifying another. Even the attempt to isolate a single variable in
this whole complex system becomes incredibly complex. A single
variable will exhibit the whole range of behaviors from extreme
sensitivity to extreme stability as well as limit cycles, bifurcation
points, large oscillations and possibly even chaotic behavior. Yet
this system, by itself, is part of a much wider system that is
embedded in global and local politics, attitudes towards agriculture
and population density. Each of these elements is, in turn,
dependent upon yet other factors which even include religious and
ethical values - of key importance in population growth and
attitudes to the environments.
This single example shows how complex a system may be. It
shows that a given problem may be sensitive to a wide range of
externalities, each of which is linked to a variety of other factors.
No single policy, no rigid plan is capable of meeting the subtleties
and range of possibilities within natural and social systems. Clearly
a whole new philosophy is demanded.
ii. Economics
Economics is currently under the scrutiny of experts in the field of
non-linear dynamics and a variety of analyses of short and long
term stock market trends have been made. There are deep
questions to be answered about the very definition of economic
systems and about the meaning of their variables, such as value.
Chaos theory and non-linear dynamics have been added to those
voices that are questioning the whole basis of economic theory.
The concept of money, for example, is highly complex and analysts
are questioning the idea of economic equilibrium and of an
intrinsically stable market. As Richard Day of the University of
Southern California puts it: "An economic world in which money
enters in a nontrivial way can be highly complex in its behavior in
theory, just as in reality". Day himself has shown that even the
simple models, in which expenditure and income lag behind each
other, can give rise to chaotic fluctuations.
The Systems Dynamics Group at MIT have a variety of models in
which a human economist or policy maker can "drive" the
computer model. In one of these, a seasonal variation in the
demand for beer is passed on to the main supplier and its
distributors. When a human operator attempts to smooth out
fluctuations the system tends to move towards ever more
uncontrolled oscillations. (In essence a non-linear iteration is
dominating the system.)
Dr. Ping Chen of Univ. Texas at Austin has made an extensive
analysis of monetary data from the Federal Reserve and argues that
economics contains inherently chaotic behavior. Even if external
shocks could be totally eliminated the economy will not run
smoothly, for wild fluctuations are inherent in its very dynamics
and recessions and downturns are often independent of external
shocks. One analyst has suggested that the 10 October 1987 crash
arouse out of the non-linear dynamics of the market and not
through a combination of external causes.
R.H. Day has made an analysis of a variety of situations, such as
investment in competing new technologies which, over time, shows
a shift from steady expansion and economic health into one of
financial crisis. Day's agricultural model. in which a variety of
factors such as market, prices, supply, investment in new buildings
etc., are included, shows an initial set-up period, followed by a five
year cycle. In time the system's internal dynamics change again and
move into a period of irregular oscillations. During the former
period the economics of the situation are relatively stable but in the
latter period they are highly sensitive to any new trend.
In a variety of analyses of different industries and technologies,
quite distinct regions of behavior have been discovered, some of
these are quite stable, other chaotic, some oscillate violently, or are
extremely sensitive to an external trend or perturbation.
iii Order in Chaos
The notion that it is the inherent non-linear dynamics of the market
that produce fluctuations rather than a combination of externalities
suggests to some analysts that it may be possible to carry out
"micro forecasting". ( A more careful analysis may indicate that it
is impossible to separate out endogenous from exogenous causes. )
Some claim to see the characteristics of deterministic chaos - i.e.
strange attractors - within economic data. If this is true it suggests
that while economic fluctuations are unpredictable they will always
lie within certain bounds.
In addition, there are suggestions that a degree of self-similarity
holds. Self-similarity is associated with fractal structures and would
suggest that a certain range of behavior patterns repeat at various
scales of time, from years, months, days and even hours. If this is
true then micropredictions will take into account that a random
fluctuation will fall within a particular range. A number of
investment houses are currently developing sophisticated computer
models to investigate this chaotic behavior.
Other analysts are looking for "co-operative effects", for example,
the manifestation of decision processes that are made in a
collective way. Often the behavior of a crowd is simpler to predict
than that of an individual. So where people respond to the news
and other externalities in a collective way it may give rise to
predictable results. Or the market itself may exhibit a degree of self
organization.
Economics is only one factor in which public policies are
concerned. The above brief overview suggests a variety of ways in
which non-linearities may be effecting the market. If this is true
then it would mean that many economic policies are ineffective and
are attempting to change what is inherent in the dynamics of the
market itself. Major swings may not be the result of externalities
and oscillations may be purely chaotic.
Conclusions
Policy has always been more of an art than a science. Why then
should the new developments of non-linear dynamics be of interest
to those working in the fields of sociology and policy planning? In
fact the conclusions generated through the analysis of non-linear
systems confirm what many policy analysts has suspected - the
inherent limitations of their own subject. Indeed computer models
and other analyses provide objective evidence for the inherent
complexity of systems and may help to convince those who have a
more naive approach to policy making.
The results presented in this paper demonstrate the limitations in
describing any non-linear system and placing faith in its variables
and parameters. Policies aim to describe a system and make
predictions about general trends. But what if the whole nature of a
system changes unexpectedly or if its well defined variables loose
their meaning? As the physicist Richard Feynman put it, "nature
cannot be fooled" and it is absurd to suppose that simplistic plans
and policies can cover the wide range of behavior possible within
natural systems- including social and economic systems.
Hand in hand with policy making go criteria for taking action and
for steering a system back on the correct track. Yet we have seen
that some systems can infinitely sensitive to externalities while
others are highly resistive to change. In the former case the effect
of attempts at control may be totally unpredictable - they may even
precipitate the system into some totally unexpected new mode of
behavior. In the latter case they may be no more than blowing in
the wind.
Clearly no single, global policy will work for a natural system.
What is called for is constant flexibility, for a continual
watchfulness in which information is constantly being gathered and
the existing description modified. What may be needed is
something similar to the propreoception of the human body, in
which tiny signals are continuously being sent out to, and fed back
from, the muscles so that the body can become aware of its
position and orientation in space. In this way the human body can
maintain its equilibrium in a rapidly changing world. Organizations
also require their own propreoception to learn where they are and
can they maintain their balance in an every changing environment.
Attempting to control and correct a natural system will only work
within a limited context and any preconceived plan of action is
bound to encounter contexts that lie outside its domain of validity.
What may be called for is a more gentle and globally coordinated
form of action; something that takes into account the ever
changing dynamics of the system and acts in a gentle way to
coordinate all its parts. Attempting to solve problems in traditional
ways often causes new problems to surface in remote locations.
The gentle action called for by a non-linear system involves an
understanding of its whole context and dynamics and must be
applied, not locally where the particular problem appears to
originate, but over its whole domain. Policy makers must also learn
to tolerate fluctuations and deviations from equilibrium as being
inherent to the health of all natural systems. Indeed their very
robustness may lie in the system's ability to support its fluctuations.
Moreover where any local oscillation appears, its ultimate origin
may lie within the dynamics of the whole system. So attempting to
"control" or prevent local deviations from prescribed behaviour
may give rise to yet more problems. What would be required
would be a very gentle steering of the whole system.