Maj David Nicholls, USAF
Maj Todor Tagarev, Bulgarian Air Force
For the last 30 years, the study of chaos has intrigued investigators, prompting many to see a great future for the study and application of chaos theory. In science and engineering, chaos theory has significantly improved our understanding of phenomena ranging from turbulence to weather to structural dynamics. Chaos theory has even been used to drastically improve our ability to control some dynamic systems. In the social sciences, there has been considerable interest in whether social phenomena, previously thought to be random, have an underlying chaotic order. Several mathematical tests for chaotic behavior have been applied to historical data from both the stock market and cotton prices. These tests indicate that these economic phenomena are chaotic and so have a deterministic basis (i.e., are governed by rules) as opposed to being random. Naturally, this has received some business attention, and at least two firms are now using chaos theory to guide their financial advice.
There is evidence that warfare might also be chaotic. First, strategic decision making, an integral part of war, has been found to be chaotic. Second, nonlinearity, which is a requirement for chaotic behavior, appears to be a natural result of Clausewitzian friction. Third, some computer war games6 and arms race simulations have been found to exhibit chaotic behavior. Fourth, previous work by the current authors applied several tests for chaos to historical data related to war. Those tests demonstrated that warfare is chaotic at the grand strategic, strategic, and operational levels.
An Overview of Chaos Theory
In this paper, we will discuss some important implications of chaos theory in the context of warfare. First, however, we will briefly summarize some important aspects of chaos theory.Nonlinearity
If a system is linear, it means that the output of the system is linearly related to the input. In other words, if the input is doubled, the output will be doubled; if the input is tripled, the output will be tripled, and so on. In nonlinear systems, however, the output might be related to the square or the cube of the input. Such systems are often very sensitive to input. All chaotic systems are nonlinear.Predictability of Chaotic Systems
Dynamic systems can differ from one another in how they change with time. In random systems, future behavior is independent of the initial state of the system and can be characterized only in terms of probabilities. For example, unless the dice are loaded, the next roll of the dice is totally independent of the previous roll. On the other hand, periodic systems return regularly to the same conditions, as exemplified by the pendulum clock. Such systems are totally predictable because once one period is known, all others must be identical. Chaotic systems are neither random nor periodic. They are not random because the future of a chaotic system is dependent upon initial conditions. They are not periodic because their behavior never repeats.
Chaotic systems never repeat exactly because their future behavior is extremely sensitive to initial conditions. Thus, infinitesimal differences in initial conditions eventually cause large changes in system behavior. An often-used example of this sensitivity is weather. Weather is so sensitive to initial conditions that there is a belief that the flap of a butterfly's wings in America could eventually cause a typhoon in China. It is inconceivable that conditions on the earth could ever duplicate an earlier time to the point where even all butterfly flights are duplicated. Therefore, the earth's weather will never be periodic.
In addition to making chaotic systems aperiodic, extreme sensitivity to initial conditions means that it is not possible to determine the present conditions exactly enough to fully predict the future. Figure 1 illustrates this point. In figure 1, successive values for x are plotted resulting from the nonlinear equation xi+1 = 4xi - 4xi2. For one plot the initial value of x was 0.7. For the other plot it was 0.70001. Initially, they are indistinguishable from one another, but as time goes on, even such a small difference between the two is magnified until their behavior appears totally unrelated. Short-term predictions are still possible because small influences will not have had time to grow into large ones. However, what is short-term depends on how sensitive the system is to small changes at that point in time.
The importance of this concept is that it explains how a system can be governed by a set of equations and yet still be unpredictable. We cannot know the initial value of a system, such as that illustrated in figure 1, precisely enough to predict which path the system will follow. If warfare is chaotic, this tells us that we cannot make perfect predictions even if we could reduce war to a mechanistic set of equations. Fortunately, as is also illustrated by figure 1, there are bounds to the unpredictability of a chaotic system. Furthermore, chaos theory provides tools that can predict patterns of system behavior and can define bounds within which the behavior is unpredictable http://www.airpower.maxwell.af.mil/airchronicles/apj/apj94/nic1.jpgPhase Space
The construction of a phase space plot is often used to better understand chaotic behavior. A phase space plot is a plot of the parameters that describe system behavior. It is useful because it provides a pictorial perspective for examining the system. An example of a phase space plot for a simple pendulum is shown in figure 2. At point A in figure 2, the pendulum is the maximum positive distance from the bob's neutral point but its velocity is zero. This is shown as point A on the phase space diagram. At B the distance of the bob from its neutral position is zero, but its velocity is at a maximum (in a negative sense). The other points of the phase space plot show the relation between the velocity and position for other pendulum positions. In this case, where there is no friction, the motion of the pendulum is constrained to remain on the elliptical path shown in the phase space plot. The technical term for this ellipse is the attractor for the system. One can see that this attractor is periodic because the path of the system exactly repeats itself in each orbit around the origin. http://www.airpower.maxwell.af.mil/airchronicles/apj/apj94/nic2.jpg
In contrast, figure 3 shows an attractor for a chaotic system. This attractor is a tangled mess of trajectories. The complexity of this attractor has led to its being dubbed a strange attractor. Although there are still constraints as to how the system behaves, there are a lot more possible states for the system. It is important to note that the phase space paths of a chaotic system will never coincide. If this were to happen, the system would become periodic. The longer a chaotic system is observed the more paths are taken and the messier the phase space plot of the attractor appears. Superficially, the attractor may appear to be completely disorganized. Closer examination of the phase space, however, reveals that the attractor is organized but in an unconventional manner.http://www.airpower.maxwell.af.mil/airchronicles/apj/apj94/nic3.jpg
It is possible to simplify the portrayal of the attractor by taking a two-dimensional slice through it (shown in the lower half of figure 3). This also makes the structure of the attractor more obvious. This two-dimensional section is called a Poincar?ę map.10Fractals
We generally define things dimensionally in terms of integers. Lines are one-dimensional, planes are two-dimensional, and solids are three-dimensional. Fractals are objects with fractional dimensions. This concept appears at first sight to be nonsense. An object with a fractional dimension of 1.5, for example, would be more than a line but somehow less than a plane. Nevertheless, such things are not only thought to exist, but such geometries are central to chaos theory. One example of such a geometry, although it is not chaotic, is the Koch snowflake.
The Koch snowflake starts as an equilateral triangle. A one-third scale equilateral triangle is added to each side. A one-third scale triangle (of the new, smaller triangle) is then added to each side of the resulting figure. This process is continued ad infinitum as illustrated in figure 4. http://www.airpower.maxwell.af.mil/airchronicles/apj/apj94/nic4.jpg
The perimeter of this shape has several unique features. First, although it is a single, continuous loop that does not intersect itself and that circumscribes a finite area, its length is infinite. Second, Benoit Mandelbrot calculated that the dimension of the perimeter of the Koch snowflake is 1.26.11 This means that the perimeter is between a line and a plane. Third, the shape of the perimeter of a Koch snowflake is self-scaling. That is, the perimeter would look the same whether you looked at it with the naked eye or with a powerful microscope.
These geometries are pertinent to chaos because strange attractors are fractal. Strange attractors, like the Koch snowflake, are infinite curves that never intersect within a finite area or volume. If a system is chaotic, it will have a strange attractor and the Poincar?ę map will show fractal characteristics. That is, the Poincar?ę map will remain similar regardless of scale. Thus, Poincar?ę maps can be used to determine if a system is chaotic by visually depicting the nature of the attractor. The dimension of the attractor can also be calculated. If an attractor's dimension is not an integer, then the system is chaotic.
Implications of the Presence of Chaos in Warfare Previous work examined historical data associated with the grand strategic, strategic, and operational levels of war. That work showed that war is chaotic on all of these levels. If war is chaotic, then it must have the characteristics of a chaotic system. We will now describe some of the characteristics of chaotic systems and define what they mean in the context of warfare.Computer Simulation Can Enhance Understanding
Computer numerical modeling or simulation has greatly increased our understanding of physical chaotic systems. The reason for this is that the equations that govern chaotic systems are nonlinear and therefore are generally not analytically soluble. Chaos theory, however, cannot be used by itself to derive a theory of warfare. As with any other theory that describes a phenomenon, a theory of warfare must be based upon observation, hypothesis, and testing. Specifically, development of a model of warfare would require the development of the structure of the model, the determination of the number and type of variables, and the determination of the form of the equations. In addition, system parameters and control factors, as well as sources for noise, would have to be identified. This is a very difficult task for any particular situation that is complicated by the possibility that different models might apply for different antagonists.
Chaos theory can help us by suggesting ways to develop our model and ways to use the model once it is developed. For example, observation of a chaotic system can be used to determine the dimension of the system. The number of variables needed to describe the system must at least equal the dimension of the system. Therefore, chaos theory can be used to define the minimum number of variables required in our computer model. Chaos theory also suggests that computer models of warfare must contain some nonlinear relationships between system variables so that the computer model is chaotic and thus reflects the chaotic nature of warfare. This may actually prove to be advantageous since the fractal nature of chaotic systems may allow relatively small and simple war games to accurately simulate warfare. Realistic war games that could be run on a desktop computer would have significant educational and operational advantages. Finally, the rate of information loss can be calculated for a chaotic system. This quantity is related to how far into the future predictions can reasonably be made.
The ways in which computers have been used to understand chaotic behavior in physical systems also suggest ways to use the computer to model warfare. For example, although chaos theory explains some aspect of the weather, the reader has probably noted that weather forecasting has not become perfect. This criticism, however, misses one of the most important contributions that chaos theory has made to weather prediction--chaos has given weather forecasters a means to determine if their forecasts are likely to be accurate. Chaotic systems are highly dependent upon initial conditions but they are not always equally so. If a chaotic system is in a portion of its phase space where the initial conditions are critical, then uncertainty in determining the initial conditions makes a large number of outcomes possible. If a chaotic system is in a region of its phase space where the initial conditions are not critical, then only one outcome (prediction) is likely. In practice, weather forecasters use this behavior by inputting small changes in initial conditions into their model. If the small changes produce small variations in the prediction, they have shown that the system is in a portion of phase space where the initial conditions are not critical and their prediction is likely to be true. If the minor changes in initial conditions produce large deviations in future behavior, forecasters know that their prediction is likely to be in error.
The same approach could be taken to understand when predictions in warfare are likely to be accurate. This in itself would be a valuable contribution of computer simulation to understanding warfare. There are, however, two additional reasons why this approach may be even more applicable to warfare than it is to weather. First, unlike weather forecasters, we have some ability to change the initial conditions. Specifically, if we find ourselves in a region of great uncertainty, we could determine which conditions would have to be changed to move the system to a position where the outcome was predictable and desirable. The quantity and type of forces are examples of initial conditions that we might be able to change. Second, we could use our model to determine which initial conditions and which variables had the most profound effect on our predictions. This would aid in identifying centers of gravity (COG) and information that we needed to know precisely. That is, it would tell us where to concentrate our attack and what intelligence information was most critical.