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Non-Linear Dynamics (Chaos Theory) and its Implications for Policy Planning

F. David Peat

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A text only version of this essay is available to download.

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(c) 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.

Related Pages:
Chaos Theory | Gentle Action(c) | Science



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