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A set of rules often invoked to explain or justify the evolution of intentions and the consequent behavior of competitive systems consists of the following: 1. The friend of my friend is my friend. 2. The friend of my enemy is my enemy. 3. The enemy of my friend is my enemy. 4. The enemy of my enemy is my friend. These rules have been modeled as a set of nonlinear, coupled differential equations from which predictions can be derived as to eventual alliance building or conflict in the system, predictions which are quite ominous for the behavior of simple three-body systems. This paper presents illustrations of the application of the rules to international behavior and discusses the relationship between the rules as usually verbally applied and the associated mathematical models. Not all of these rules are equally desirable: behavior in accordance to the fourth rule has led to major difficulties in the "real world." This fourth rule cannot just be dropped, since the four rules are not independent of each other in the mathematical model. Hence the model must be altered. A simple mathematical modification is suggested which implies increased flexibility in the verbal statement and application of the rules; e.g., the fourth now reads "The enemy of my enemy may be my friend." The altered model leads to altered predictions of the evolution of intent which are much less ominous for the outcome of three-body competitions.

Public consideration of nuclear-security matters, such as what to do with surplus fissionable materials from retired nuclear weapons, is often overwhelmed by the pro- or anti-nuclear energy biases of the discussants. Clearly a ground for such discussion is required which is explicitly independent of such biases. The Richardson Model and its linear and non-linear extensions link the onset of war to the "interactions" between the participants in the international system. These "interactions" include competitions for resources as well as military and economic components. The disposal of surplus nuclear weapons materials contributes to both the resource and the military aspects of these "interactions". Hence the proper disposal method might be linked to international security: the best option is that which maximizes the stability of the system, as determined by a Richardsonian model. Several simple models are developed and investigated in this paper. Admittedly, these models do not do full justice to the complexities of the real world problem.. But they do allow simple answers which should provide clues to the eventual development of more realistic models as well as some possibly unbiased insights into how to deal with the real problem. The simple models of this paper indicate that it may be better to "burn" rather than "bury" the fission components of surplus nuclear weapons.

Mathematical models-static or dynamic-of international systems arc often used to predict outcomes of specific policy choices. The focus of this paper is not such predictions themselves but the question of whether or not such predictability is meaningful. Non-predictability in a deterministic dynamic model is usually referred to as "chaos" and is used in this paper as a paradigm for crisis instability and the possible outbreak of war in a system of competing states. The use of the paradigm is illustrated via initial explorations of three international security questions: 1.) Which is more stable, a bipolar or a multipolar system? 2.) Which are more war prone, democracies or autocracies? 3.) Which leads to more stability, seeking security via independent acquisition of arms or via a balance of power? A simple model-not empirically founded but manifesting some dominant aspects of the situation-is formulated for each question. The stability regime of each model is then determined. Using the postulate that a larger stability range indicates a less war-prone world, and vice-versa, answers to these questions are found, thus providing an alternate approach to the investigation of pressing theoretical and practical questions in international relations.

An important international security question is whether nations in an anarchic competitive world system are more likely to ensure their security by pursuing policies of complete independence or of alliance building. The question is addressed using the paradigm that "strong chaos" in an arms-race model is equivalent to crisis instability and war in the international system being modeled. Nonlinear recursion relation models of competitive arms procurement are constructed for independent arming and for alliance building policies in a world system of three symmetric nations. The results are that alliance building always preserves a peaceful world from perturbations in the relationships between the competing member states whereas independent courses of action may allow such perturbations to build up to strong chaos and presumably war.

One of the reasons advanced for the absence of a major war between the two superpowers during the forty-five years of their enmity since World War II is that the world system in which they functioned was essentially bipolar and hence, presumably, inherently more stable than previous multipolar worlds. Given the recent decline in the power of the U.S. and the U.S.S.R. relative to the rest of the world, it is important to test the validity of this presumption. A nonlinear mathematical model of international competition is presented in which the transition from predictable laminar to unpredictable turbulent flow is the model manifestation of the transition from cold to hot war in the world system being modelled. The model is a tripolar competition which arises continuously from a bipolar system as a coupling parameter is varied. Thus the realm of nonchaotic, stable, competition can be examined as a function of the coupling parameter. It is found that the regime of stability decreases as the system complexity increases. Thus the simple model lends credence to the presumption that a tripolar world is inherently less stable than the corresponding bipolar world.

ONE OF THE REASONS ADVANCED FOR THE ABSENCE OF A MAJOR WAR BETWEEN THE TWO SUPERPOWERS DURING THE FORTY-FIVE YEARS OF THEIR ENMITY SINCE WORLD WAR II IS THAT THE WORLD SYSTEM IN WHICH THEY FUNCTIONED WAS ESSENTIALLY BIPOLAR AND HENCE, PRESUMABLY, INHERENTLY MORE STABLE THAN PREVIOUS MULTIPOLAR WORLDS. IN THIS ARTICLE, A NONLINEAR MATHEMATICAL MODEL OF INTERNATIONAL COMPETITION IS PRESENTED IN WHICH THE TRANSITION FROM PREDICTABLE LAMINAR TO UNPREDICATABLE TURBULENT FLOW IS THE MODEL MANIFESTION OF THE TRANITION FROM COLD TO HOT WAR IN THE WORLD SYSTEM BEING MODELLED. IT IS FOUND THAT THE REGIME OF STABILITY DECREASES AS THE SYSTEM COMPLEXITY INCREASES. THUS THE SIMPLE MODEL LENDS CREDENCE TO THE PRESUMPTION THAT A TRIPOLAR WORLD IS INHERENTLY LESS STABLE THAN THE CORRESPONDING BIPOLAR WORLD.