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However, as sometimes happens with trendy concepts from the natural sciences James Gleick's excellent popular work, Chaos, did much to popularize the concept , chaos theory, often poorly understood, has been stretched and mangled in order to force fit it to social phenomena where its use is inappropriate. Mathematical concepts and formulas, for example, have sometimes been found to be relevant in totally different fields. What is chaos theory exactly? According to one definition, "Chaos theory is the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems.

First, that the system is dynamical, means that it changes over time. Second, that the behavior of the system is aperiodic and unstable means that it does not repeat itself. Third, although chaotic behavior is complex, it can have simple causes. Nonlinearity means that the output of the system is not proportional to the input and that the system does not conform to the principle of additivity, i. A final feature of chaos, although not included in the above definition, is that of iteration or feedback, in which the output of the system is used as the input in the next calculation.

An example is the logistic equation, first devised in , which provides a model for changes in population over time. The element 1-x in the equation establishes a practical limit to population growth, a sensible constraint given the existence of famine, disease, and birth control in the real world.

When the control parameter is less than three, this simple system converges towards an equilibrium point, regardless of the initial population level. When the control parameter has a value between 3 and about 3. Eventually, when the control variable falls between 3. At higher levels of k, the system can display either chaotic or non-chaotic behavior. See Williams, op cit. A complex system is one in which numerous independent elements continuously interact and spontaneously organize and reorganize themselves into more and more elaborate structures over time.

As with chaos, the behavior of self-organizing complex systems cannot be predicted, and they do not observe the principle of additivity, i. Complex systems can naturally evolve to a state of self-organized criticality, in which behavior lies at the border between order and disorder. Again, the same system can display order, chaos, and self-organizing complexity, depending on the control parameters. First, in a metaphorical sense, the theory seems apt in reminding us that actions can have unforeseen consequences and that war can be an unpredictable affair.

This model seems particularly appropriate, given the increasing complexity of the international system since the end of the Second World War resulting from the growing number of independent international and transnational actors. First, as Saperstein13 points out, although chaos by definition cannot be predictive, we can through simulations determine the control parameters under which a system is likely to display chaotic behavior.

Chaotic behavior per se does not mean that war will occur, but the unpredictability of the behavior means that we cannot control the system. As a result, the mechanisms in place to prevent war may fail to work as intended, thereby making an outbreak of war more likely. On the other hand, non-chaotic behavior is more predictable and therefore more controllable.

Again, this does not mean that war cannot break out in a non-chaotic system, only that, at least to some extent, one can predict 10 This section on complexity draws from Williams, op cit. To go back to field that spawned chaos theory, in meteorology, weather forecasters can gain a notion of the accuracy of their predictions by feeding different, but closely similar, sets of data into their computer models. If the output is similar, the forecasters can have some confidence that the system is stable and that their forecasts are likely to be correct.

If, on the other hand, the output varies significantly from one input to the next, then chaos is at play and the forecasters can reduce the confidence intervals of their predictions or do away with them altogether. In each case, he assumes that the greater the range of parameters in which chaos reigns, the greater the instability of the system. Historically, however, deterrence has worked much less well. It may be that the Cold War, with its bipolar simplicity, strong command and 14 Pritchard, op cit. Certainly, deterrence has broken down more often since the end of the Cold War than during it.

That deterrence is working less well now may also be due in part to our willingness to become involved in conflicts, e. Can chaos theory suggest any ways of making deterrence more effective in the future? If we assume the international system to be in a state of self-organizing criticality, then we can consider that war, which is brought about by a breakdown in deterrence, is an instance in which at least parts of the system spill over into chaos.

Deterrence does have characteristics which are consistent with chaotic behavior. First, as political scientist Colin Gray points out, deterrence is nonlinear. Second, as Gray also points out, deterrence is interactive between two or more nation states and characterized by feedback.

And the effectiveness of deterrence has at times proven unpredictable. If deterrence as a system can slide into chaos, we can, borrowing from Saperstein, conclude that we should try to avoid although avoidance may not always be an option those conditions that lead to chaotic behavior in the system since it is under those conditions that deterrence can break down unpredictably and lead to war. Of course, in one sense, if we accept and we should that war is inherently chaotic and hence unpredictable, then the threat of war by a deterring country against a would-be aggressor is itself uncertain.

But can chaos apply to a deterrent system itself? To test this question empirically, we need to first formulate a simple model and to test that model to discover if it displays chaotic behavior and, if so, under what conditions. Naturally, reality will be more complicated than our simple model, but, as Saperstein points out, chaotic behavior does not disappear as we add new variables to a system. We must also quantify the data relevant to our model.

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It is not certain, however, that this can be done in a meaningful way and some trial and error experimentation will be necessary to see if quantification works in practice. A Simple Model of Chaos and Deterrence As one example, of a simple model, we can begin with the classic formulation that deterrence is a function of will and capability. In this case, will does not represent actual will, but will as perceived by one's potential adversaries. We can also posit that capability, beyond a certain minimum level, is relative to the capability of one's potential adversaries'.

Finally, we can assume that our ability to deter potential adversaries is based in part on our adversaries' ability to deter us. In other words, the more deterred we are, the less likely to be deterred is our adversary. For greater simplicity and clarity, we will assume only one potential adversary in our model although, in the real world there are, of course, at least several.

With these assumptions, our model is defined by the following equations: 16 Lecture by Dr. Note the inclusion of the element 1-c.

The chaos cookbook: A practical programming guide

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Chaos Cookbook: A Practical Programming Guide

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