Be Careful: We Live in a Complex, Non-linear World!

by Professor Ted Woodcock

Lecture held at a seminar in the Academy on March 16, 2000.

The remarks contained in this paper reflect the views of the author and do not necessarily reflect the views, opinions, or position of any government, corporate, or academic entity.

Ted Woodcock, born 1943, is Director of The Advanced Mathematics Program at Synectics, a subsidiary of BAE Systems, USA. He is also a Visiting Professor at the Swedish National Defence College and an appointed member of the Royal Swedish Academy of War Sciences.

Introduction

The title of this short paper raises a significant concern. Planners, decision-makers, political and business leaders, and others may not have a fundamental understanding of the real nature of the systems that they attempt to influence or control. Such a lack of understanding can have serious implications when the system is the world energy economy and the global environment of which it is a part. Mathematical research has shown that in many types of system, small changes under critical conditions can lead to sudden, irreversible, changes in behavior. Planners and decision-makers must be made aware of the results of this research, and act with a new and more appropriate level of understanding. The following paper is an attempt to make a small contribution to the necessary dialogue between mathematicians and scientists and the planners, decision-makers, and others involved in the energy economy and elsewhere.

The paper begins with a review of a model of the impact of demand and supply on crude oil prices published by the author in 1978 [1] based on the mathematics of cata-strophe theory. The author believes that this model can illustrate the complete pattern of changes in crude oil prices between 1978 and the present. The paper then suggests how results from research into the stability of non-linear dynamical systems can have great significance for planning and decision-making both for the energy economy and for the global environment.

The Price of Oil

The conference, which took place in March 2000, was clearly concerned with the significant rise in crude oil price to some $30 per barrel and its potential impact on the international economy. The author was invited to present a brief summary of some of his earlier work that had used catastrophe theory to model the impact of the elasticity of demand and level of competition on crude oil price. [2] The author also outlined a model based on catastrophe theory, developed before the incidents at Three Mile Island and Chernobyl had taken place, that showed the impact of competition between the nuclear power industry and the ecology lobby on the level of production of nuclear power stations. He also made a few remarks on the need for planners and decision-makers to understand the behavior of non-linear dynamical systems associated with the energy economy and the global environment. As an introduction to the oil price model and its potential use for describing future changes in oil price, a brief, non-technical, introduction to catastrophe theory is provided.

While an exhaustive description of catastrophe theory is not possible in this paper, it is sufficient to note that the theory provides a series of mathematical landscape-like figures or graphs as models of certain types of system. Catastrophe landscapes describe the behavior (represented by changes in the values of dependent variables) caused by changes in the values of control factors, influences, or independent variables acting on such systems. The control factors specify position on the landscape in much the same way that latitude and longitude specify position on a geographical landscape. The height of the cata-strophe landscape at any location represents the behavior of the underlying system. There are seven elementary catastrophe landscapes. Identification of the number of controls at work in a particular system determines which landscape to use in a particular application.

Figure 1: Catastrophe landscapes demonstrate that gradual and sudden changes in behavior can occur in the same system under different circumstances.

Gradual and sudden changes can occur in the same system under different conditions. These changes can be illustrated by movements that cross flat regions and by others that fall over cliff-like structures that are part of the catastrophe landscape, respectively; see figure 1. In this figure, a catastrophe landscape cliff is shown as an over-folded structure. Changing position on the surface (reflecting a change in the value of an independent variable) from positions (a) to (b) produces a small change in the height of the point (reflecting a change in the value of a system dependent variable). By contrast, a change in that independent variable from positions (c) to (d) to (e) generates a much larger change in the dependent variable. This sudden change is known as a catastrophic change.

Figure 2: Catastrophe theory-based model illustrating the impact of elasticity of demand and number of producers on oil price. [3]

The catastrophe theory-based oil price model developed and published by the author in 1978 is shown in figure 2. The latitude and longitude in this case represent the elasticity of demand and level of competition in the crude oil market. The height of the landscape represents the price of oil. The model illustrates situations involving monopoly, oligopoly, and pure competition. The folded nature of the landscape surface suggests the existence of conditions supporting high and low price ranges. Paths such as (a - b - c - d - e) on the landscape surface illustrate how decreasing competition can lead to sudden increases in price. Paths such as (e - d - f - b - a) reflect sudden price declines due to increasing competition as new suppliers enter the market place. Increasing elasticity of demand can also lead to gradual changes in price (paths (e - h) and (e - g)) under appropriate conditions.

The model suggests that decreasing competition under highly elastic conditions can lead to a relatively gradual increase in price. Subsequent study of this model has demonstrated its ability to describe the actual patterns of changes in the price of crude oil that have taken place since 1978. Recent work by the author has involved development of a more ge-neralized from of model for the prices of other commodities.

Non-linear Systems Can Exhibit Chaotic Behavior

Some political leaders appear to act as if the world was flat and everything could increase forever. Booms are good! busts are bad! Those leaders also appear to be guided by a philosophy that could be loosely expressed as: if a 5 per cent increase was good, a 10 per cent increase would be better. Acting on such assumptions can be shortsighted, and may create dangerous, uncontrollable, conditions. Planners and decision-makers under such conditions face significant risks as what had been good, well-tested, strategies and tactics may actually become inappropriate and may cause great damage if implemented. The problems that such linear thinking can create can be illustrated by the behavior of a very simple mathematical model of the competition between two groups. In this model, a linear increase in the rate of growth of one of the populations can transform an apparently static condition into one that can exhibit sustained oscil-lations and even chaotic behavior. [4]

The use of the model represented by the set of equations (1) and (2) illustrates what can happen when linear changes are made in inherently non-linear environments. The model calculates the sizes of two populations (represented by the dependent variables (x) and (y)) at a future time (t) (re-presented by xt and yt, respectively). The growth of population (x) takes place in a density-dependent or logistic manner at a rate represented by the coefficient (m₁). Losses from that population are equal to the product of the strengths of the populations and the loss coefficient (m₂). The limits to growth are determined by the carrying capacity coefficient (c). Absorption into the second population (y) takes place at a rate described by the absorption coefficient (m₃) and losses occur at a rate described by the loss coefficient (m₄).

xt = m1 x (1 - x/c) - m2 x y	(1)
yt = m3 m2 x y - m4 y	(2)

Figure 3: Equations (1) and (2) can generate (a) s-shaped growth, (b) oscillations, and (c) chaotic behavior; (d) attractors are generated by plotting the simultaneous values of the dependent variables (x) and (y).

Sample model behaviors are shown in figure 3. When the growth coefficient, m1, is set equal to 1.4, and the values of the other coefficients are set arbitrarily at m₂ = 0.2, m₃ = 10.0, m₄ = 0.1, and c = 2.0, population (x) follows an s-shaped growth curve to a maximum value of 2 units; see figure 3a. Increasing m1 to 2.8 generates sustained oscillations in population size; see figure 3b. A further increase to m₁ = 3.8 produces chaotic oscillations; see figure 3c. These changes in behavior have critical implications for planners and decision-makers tasked with affecting changes in the system represented by equations (1) and (2).

In the first (s-shaped growth) case, the system reaches a stationary state where the value of the (x) population remains essentially constant. Under such conditions, it would not matter how long a manager of the system takes to make a decision since the behavior is relatively constant. However, when the system exhibits sustained oscillations, changes can only be affected when they are made in a time interval that is shorter than the period of the oscillation. Faster oscillations demand faster responses and may create the need for more rapid decision-making in order to affect changes. The situation becomes complicated with the onset of chaos. There is no simple way to know the length of the time intervals between chaotic oscillations and therefore no way to know how fast a decision-maker must be to be able to control events. Chaos creates uncertainty and well-intentioned actions can be counter-productive, or worse. Decisions to change from a value of m₁ = 1.4, to m₁ = 3.8 has created conditions under which an initially controlled system becomes uncontrollable.

When the simultaneous output values of both populations (x) and (y) are plotted on a plane, they produce an entity known technically as an attractor. Thus, with m₁ = 3.2, the attractor has a complicated braided pattern when the results of 10,000 iterations of equations (1) and (2) are plotted as shown in figure 3d. Each point in the figure represents a particular pair of values of the dependent variables (x) and (y). This Figure shows that new, com-plicated, and unforeseen structures can emerge from the dynamics of simple systems. The region inside the attractor braid shown in figure 3d is devoid of points, showing that the corresponding values of the dependent variables were never generated by the equations. This also shows, literally, that the system could not get there (inside the braid) from initial starting conditions in the thickness of the braid, or from outside. Some options may not be available to planners and decision-makers. Despite the fact that they could specify the nature of their ultimate goal, the dynamics of the system prevent such a goal from being achieved.

Discussion

Models based on catastrophe theory illustrate how the interplay of several variables can generate both gradual and sudden changes in behavior under different circumstances. These models can show planners and decision-makers that relatively small changes in input can dramatically, and perhaps irrevocably, alter the behavior of systems under their control. Mathematical experiments involving linear changes in parameter values of a non-linear system that had been controllable, create conditions where oscillations and chaos can occur and where the system can become uncontrollable. Thus, assuming that policies that change the rate of use of fossil fuels by what looks to be a linear amount may actually create non-linear, irreversible, responses in the world eco-logy. What might appear to be controllable would become uncontrollable. The complicated behavior of the simple model presented in this paper is a powerful argument for the position that all who want to manage the energy economy and global environment must really understand the detailed properties of non-linear dynamical systems.

References

1. Woodcock and Davis, 1978.
2. Woodcock and Davis, 1978.
3. Woodcock and Davis, 1978.
4. Dockery and Woodcock, 1993.

Bibliography

Dockery, J.T.; Woodcock, A.E.R.: The Military Landscape: Mathematical Models of Combat, Abington Hall, Woodhead Publishing Ltd, Cambridge 1993.

Woodcock, Alexander; Davis, Monte: Catastrophe Theory, E.P. Dutton, New York 1978.