1) Chaos Theory
a) "A chaotic system is one in which nearly nearly identical initial conditions can lead after a while to entirely different outcomes."
The presence of chaos in a system means that for any given accuracy with which we specify the intitial conditions, there will eventually come a time at which we lose all ability to predict how the system will behave, but it is still true that however we want to be able to predict the behavior of a physical system governed by Newton's laws, there is some degree of accuracy with which a measurement of the initial conditions would allow us to make this prediction."
i) What this means is that a chaotic system is dependent upon the initial conditions
b) According to Nicholls and Tagarev, there is evidence that warfare is chaotic.
First, strategic decision making, an integral part of war, has been found to be chaotic. Second, nonline- arity, which is a requirement for chaotic behavior, appears to be a natural result of Clausewitzian friction. Third, some computer war games and arms race simulations have been found to exhibit chaotic behavior. Fourth, previous work ... applied several tests for chaos to historical data related to war. Those tests demonsrated that warfare is chaotic at the grand strategic, strategic, and operational levels.
What this means for us is that is warfare is chaotic at the strategic level, it is very difficult for statesmen to control what they have begun. More importantly, the initial period of the war is critical. Chaotic systems are extremely dependent on their initial conditions.
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.
c) What this means for warfare is that while it cannot be reduced to a set of equations, chaos theory can at least provide bounds.
i) Problem is that equations which govern chaotic systems are nolinear equations and therefore not analytically soluble.
ii) A theory of warfare must be based on obsertvations, hypothesis and testing. A model of warfare would require a structure of the model, the determination of the number and type of variables, and the determination of the form of the equations. Chaos theory can be used to define the minimum number of variables required in our computer model.
The rate of information loss can be calculated for a chaotic system. This quantity is related to how far into the future predictions can be reasonably be made.
For example if we found ourselves in conditions of great unpredictability, we could determine what conditions could bring us to a new position where the outcome is predictable and desirable (i.e., controllable). Additionally, a warfare model can be used to determine the initial conditions and whcih variables have the most effect on the predictions. This would aid in identifying the center of gravity of the enemy.
All chaotic systems are nonlinear. Among other things, nonlinearity means that a small effort can have a disproportionate effect. If warfare is chaotic, then chaos theory suggests COGs may be found where there is a nonlinear process in the enemy's system ... Because you can't predict future behavior of a chaotic system based on initial conditions, chaos theory suggests that the campaign planner should concentrate on process in an emeny system rather than data on its current condition.
iii) Sources of nonlinearity in warfare:
(1) Feedback - Col Warden suggested massing for a few blows rather than many minor blows as a result of attrition analysis.
(2) Psychology associated with interpreting enemy actions. Clausewitz stated that in strategy everything is very simple, but not on that account very easy. Maneuvers, such as flanking movements are simple in concept, they are difficult to accomplish because there is always the danger of what the enemy might be doing. "In this environment, small actions on the part of the enemy often assume larger significance in a commander's mind than they deserve."
(3) There are a number of processes within warfare that appear to be inherently nonlinear. Mass for example being one. Warden showed that airpower losses vary disproportionately with the ratio of forces involved.
(4) Clausewitzian friction demonstrates that there are events in war which, whether through chance or not, have a disproportionate effect out of their apparent importance. Though difficult to predict, it can be taken advantage of once it happens. German doctrine of Auftragstaktik, allowing initiative on part of junior commanders, was designed to do precisely this.
(5) The process of decision making is itself nonlinear. Sometimes decision clear-cut, but other times the decision can depend upon relatively minor circumstances at the time.
iv) If warfare is chaotic, then aspects of it must be fractal
(1) The attracto of a chaotic system is fractal and therefore infinitely complex. Efforts to analyze every aspect of an enemy's system are bound to be in vain as there will always be some finer level to analyze.
(2) Behaviors at the tactical, operational, and strategic level are linked. One thing that succeeds at the tactical level can succeed at higher levels. Sun Tzu implied fractal nature of war "Generally, management of many is the same as management of few." This means that the principles and processes in war are essentially the same regardless of the scale of the fight.
d) Multiple attractors are possible in war. That is the difficlty that we encounter, because chaotic systems can have multiple quasi-stable states.
In war, this means that the enemy can change the organization and means of fighting a war. Example, North Vietnamese and Viet Cong actions. As warfare is chaotic, enemy systems can exist in different states. We must be capable of changing our own state in response. 
e) Some terms associated with Chaos Theory
i) Strange attractor - paths around which chaotic trajectories occur. The space paths never coincide and indeed the longer one looks at a chaotic system the more paths are taken and the more messier the phase space plot of the attractor appears.
ii) Poincaré Map - Two-dimensional slice through attractor, makes its structure more obvious.
iii) Fractals - Objects with fractional dimensions. Ex. object wtih 1.5 dimensions is more than a line but less than a plane. An example of such a figure is the Koch snowflake (and equilateral triangle with one-third scale triangle added to each side of the resulting figure, ad-infinitum). Benoit Mandelbrot calculated the dimension of the perimeter of the Koch Snowflake to be 1.26 (between a line and a plane). These geometries are central to chaos theory because strange attractors are fractal. Strange attractors are infinite curves that never intersect within a finite area or volume.
If system is chaotic it will have a strange attractor and the Poincaré map will show fractal characteristics.
 Steven Weinberg, Dreams of a Final Theory: The Scientist's Search for the Ultimate Laws of the Universe, (New York: Vintage Books edition, 1994), p. 36.
 Maj David Nicholls, USAF and Maj Todor D. Tagarev, Bulgarian AF, "What does Chaos Theory Mean for Warfare?" Airpower Journal, Vol. VIII, No. 3, Fall 1994, p. 49.
 Ibid., p. 53.
 Ibid., p. 55.
 Ibid., p. 55.
 Ibid., p. 56.
 Ibid., p. 57.
 Ibid., p. 51.
 Ibid., p. 52.1)
Labels: chaos, fractals, military operations, warfare