POLE PAPER SERIES ISSN 1370-4508 Vol. 4, No. 1, January 1998

Non-linear Dynamics and the Prediction of War

Paper submitted for publication in Global Society
by Prof. Dr. Gustaaf Geeraerts

  1. Introduction

    Despite the vast amount of studies available, our understanding of the principles or mechanisms underlying the recurrence of war is still limited. Increasingly, scholars arrive at the conclusion that a major hurdle on the road towards scientific progress in this regard is the lack of adequate theorising (Bueno de Mesquita 1981; De Vree, 1990, 1991; Geeraerts 1991, 1994).

    How, then, are we to improve our theoretical understanding of the war phenomenon? Saying that the study of war needs improvement is one thing. Quite a different matter is to specify the substance of such an effort. Nevertheless, the assumption is gaining ground that one major reason for poor progress in developing predictive models of war flows from the non-linear nature of international politics (Wolfson, M., A. Puri & M. Martelli 1992; Brown 1995b; De Vree 1991). Obviously, the linear models that are still dominating the field cannot deal with this element of complexity. In the present article I will address the issue of non-linearity and, more specifically, discuss its implications for the predictability of war. Although the argument to follow may be rather programmatic in character, it offers some suggestions, which I suspect to be cardinal to theoretical progress in the study of war. The aim is to stimulate constructive criticism and further efforts at theoretical articulation.

  1. Lack of theoretical progress: A result of non-linearity?

    Illustrative of the lack of theoretical progress in research on international war is the instance that so many mutually inconsistent propositions and findings stand to live side by side. A major example illustrating this point stems from the debate about the question which type of power distribution leads to the highest level of stability in an international system that is anarchical in nature. For one, the 'balance-of-power theory' conjectures that a relatively equal distribution of power among major actors in the international system leads to an equilibrium making war unlikely (Claude 1962; Waltz 1979). This proposition, however, is directly at odds with the 'hegemonic stability theory' as the latter contends that peace (understood here as the absence of war among the great powers) and co-operation between states come about through the stabilising impact of a powerful, dominant state (Organski 1968; Organski and Kugler 1980; Gilpin 1983). To hegemonic theorists, a system containing a preponderant power tends to be more peaceful. In their view the rise and fall of successive hegemonic states dominating their respective international orders shape international politics. Particularly during periods of hegemonic transition war is much more likely to break out.

    Interestingly enough, proponents of each of the above theories appear to make plausible 'empirical' cases for their assertions. Those favouring a balance-of-power perspective mostly refer to the experiences of post-Napoleonic 19th-century as evidence for their views, while those sympathetic with the hegemonic power viewpoint to the long periods of stability during the Pax Romana and the more recent Pax Americana.

    Through the past years, the above two opposing hypotheses have been more systematically tested. In a well-known article Siverson and Sullivan (1983) compare the more important of these critical tests. They ground their comparison on the relationship between power concentration and the probability of war. From their findings both theories turn out to get empirical support. For one, Ferris (1973) obtains results supporting the balance of power theory, whereas empirical research by Organski and Kugler (1980), and Bueno de Mesquita (1981) corroborate, albeit to varying degrees, the power preponderance theory. The picture, however, becomes even more blurred when we take note of quantitative research done by Singer, Bremer, and Stuckey (1979) leading to results that disclose an important anomaly. More specifically, they find that a balance-of-power system is associated with less war in the 19th century but with more war in the 20th, while preponderance is associated with more war in the 19th and less in the 20th century. All this begs the question how these differences in testing results obtain?

    Recently, several analysts have suggested that such anomalies are systemic in nature and result from applying linear methods to phenomena that are essentially non-linear (De Vree 1990, 1991; Wolfson, Puri and Martelli 1992; Saperstein 1994; Geeraerts 1994). Linear models assume that small changes in initial conditions - these may relate to errors in measurement or changes at the level of the variables themselves - cause small variations in the effects. However, as Wolfson, Puri and Martelli (1992:121) remark: "If this were so for our problem, the pattern of events leading up to war would be very similar across history, and statistical inquiries would generate consistent results. But this is not the case." Overall then, this finding intimates international politics to be a non-linear system in which small variations in initial conditions can bring about large or disproportional changes in outcome. Moreover these changes may at times even be chaotic in nature, that is characterised by irregular periodicity, sensitivity to initial conditions and a lack of prediction (Brown 1995a: 10; Saperstein 1994)

    However, before going into the implications for the predictability of war, asking what brings about non-linearity in the first place seems a logical step.

  1. Non-linearity: international politics as a complex adaptive system

    By analogy with Waldrop (1994: 145) one can assume that international politics is an example of what the Santa Fe Institute has come to call "complex adaptive systems". In the natural world examples of such systems are brains, ecologies, cells, immune systems, and ant colonies. In the human universe they comprise cultural and social systems as families, political parties, states, and also the system of international politics. Even if these systems are quite different from each other, the more important thing is that they share several essential characteristics.

    For one, each of these systems constitutes a network of many agents acting in parallel. In a brain the agents are nerve cells, whereas in ecology they are species. In international politics the agents are states, and if we move beyond the realist image, different types of non-state actors come in as additional agents (Rosenau 1990; Geeraerts and Mellentin 1994). No matter how you specify them, each agent acts within a structure produced by its interactions with the other agents in the system. All agents within the system, then, find themselves acting and reacting to what the other agents are doing. As a result basically nothing in their environment is fixed.

    Moreover, complex adaptive systems have many levels of interaction. Agents at any one level serve as the building blocks for agents at a higher level. For example a set of contiguous states linked closely together by patterns of enmity and amity as regards their perceptions of their own security will make up a regional security complex (e.g. East Asia), which taken together with other such complexes (Europe, South East Asia, the Middle East etc.) will form a global security complex involving the great powers of the day. The latter in their turn will be linked to one or several regional security complexes.

    Finally, complex adaptive systems are constantly evolving. They are ever changing and reordering their building blocks as they adapt to variations in their structure or environment. As Waldrop (1994: 146) writes:

      Succeeding generations of organisms will modify and rearrange their tissues through the process of evolution. The brain will continually strengthen or weaken myriad connections between its neurones as an individual learns from his or her encounters with the world. A firm will promote individuals who do well and (more rarely) will reshuffle its organisational chart for greater efficiency.

    In the same way states will enter new trading regimes or realign themselves into new alliances. For example, with the end of Cold War came a fundamental change in the international system. As a result, states must adapt themselves to the emerging structure of international politics if at least they are to survive (Waltz 1993).

    In combination, the three characteristics just outlined make for complex adaptive system to be non-linear in nature. After all, such systems represent complex patterns of different agents interacting and influencing each other mutually, specifically by means of all sorts of direct and indirect non-linear feedback relationships. Accordingly, the state or conduct of any single agent varies with that of (in principle: all) the others. A complex adaptive system, then, is a product of the past behaviour of each agent. Thus the system's evolution is due to the history of the current state of affairs as much as it follows from present interactions. As a result the evolution of complex adaptive systems over any longer stretch of time tends to show all sorts of irregularities and is highly dependent on the system's initial state.

  1. Non-linearity and dependence on initial conditions

    To show in general terms what non-linearity implies for the evolution of a system, specifically how dependent such behaviour is on initial conditions, is not that difficult. Following De Vree (1991: 44) let us assume that the state of a system at moment is described by a simple scalar magnitude , and that the transformation of the system's state at n into n-1 is determined by some scalar function f , so that

    and in general

    in which f(n)(xo) represents an iterated function system describing the evolution of a system over a given period of time (De Vree 1991: 44; Barnsley 1988).

    However, for non-linear relationships, the way in which xn varies with xn-1, i.e. the dependency of successor system states on their previous ones, is itself a function, say g, of the predecessor state. This we can represent as

    and in general

    As for the dependency of the system's subsequent states on its initial state or condition, applying the chain rule and some straightforward notational conventions, we easily get

    and in general

    The above shows that the outcome of a n-step iteration of a non-linear function, or, which is the same, the state of a system after evolving for a period n, must be quite strongly dependent on the function system's initial state. Therefore, extremely small variations in a system's initial state will make a very big difference (De Vree 1991: 45; Nicholson 1996: 43). In meteorology, this is called the butterfly effect: some Japanese butterfly flapping its wings might, in principle, materially influence the weather on the other side of the globe.

    When studying social phenomena we must apparently be prepared for very small variations making for disproportionately large differences in such things as the development of individuals, the working of the economy, the evolution of a society's culture, its legal system, and its political organisation, and in matters of war and peace among nations. Thus, relatively small disturbances in an international system may well set off dramatic and devastating processes of escalation. An example that springs readily to mind is the murder of Archduke Franz Ferdinand and his consort in 1914. Similarly, economic depressions can flow from relatively minor shocks or perturbations, such as might have occurred in the case of the sudden rises in oil prices in the early seventies which were imposed by OPEC, even though in money terms, the share of oil in the Western economies was comparably modest (De Vree 1991: 57-58; Saperstein 1994: 156).

  1. Non-linearity, chaos and predictability

    The preceding has some fundamental implications for the study of complex dynamic systems overall, and thus also for the inquiry into international politics and war. Foremost it means that there is not much sense in conceiving such an enterprise as consisting in the search for observable regularities, that is, of invariant relationships at the level of the empirical phenomena themselves. Due to the dynamic or historical nature of the phenomena under study, meaningful or informative empirical generalisations have proved hard to come by (De Vree 1991, Gaddis 1994: 190-192, Geeraerts, 1994).

    Like it or not, the non-linear nature of international politics allows for few if any valid and non-trivial empirical generalisations. To be sure, what a system does or does not do, how it responds to any given stimulus or event, is to a significant extent determined by its current state or conduct as the product of its previous historical evolution. This implies both that the development of such a system will be governed by a definite dynamism of its own, and that the very same stimuli or events will usually have rather different effects on different systems. Whether an external threat will or will not stimulate internal unity or integration in a political system, or power shifts in a political system will or will not bring about violence or war, or oppression or deprivation will or will not lead to protest and revolution all depend on how the initial conditions are filtered through the current state and structure of the systems under consideration. It also follows that historical analysis of any specific case, system, period, or society, may not simply be projected on other cases, systems, periods, or societies, let alone be used as a basis for present policy-making. To give but one example, most analysts tend to agree that British and French policies of appeasement towards Nazi Germany have contributed to the outbreak of the Second World War. However, can we take this to mean that accommodating to a growing power will invariably lead to war?

    In brief, the development and conduct of no two systems will be entirely the same, not even when the circumstances in which they find themselves appear to be identical. For, even tough they might be practically identical at any given moment, with time they will surely be subjected to somewhat different experiences or disturbances. Such differences will tend to become larger and larger as time goes by, so that, after a certain time has elapsed, they will have grown into quite dissimilar things in which it becomes hard to discover any common origins.

    A major question, then, is what all this means as for predictability. If it is true that international politics is a complex adaptive and thus non-linear system, is it still possible for us to predict a recurrent event like war? Typical of non-linear iterated function systems is that their behaviour may be highly irregular, in the sense that their paths seem to be without reason. In other words, they often show large and apparently arbitrary variations; they may bifurcate at any number of points; and highly erratic paths may alternate with perfectly regular ones. Such chaos or inherent unpredictability is closely connected with the non-linearity of the functions defining the system's state transitions and the ensuing dependence on initial conditions of the system's evolution.

    Does this signify the end of predictability? Not entirely, since non-linear systems may contain well-defined 'attractors', i.e. regions in 'phase space' where the system settles down either to a marginally stable state or to some complex pattern of continual and more or less periodic variation (De Vree 1991: 46; Brown 1995a: 22-23)). Attractors, if well-defined (Saperstein 1994: 161), indicate the states or conditions in which systems can actually work, and assume a more or less definite shape and as such become predictable. Using such insights from chaos theory, Saperstein (1994) has recently developed a paradigm that can be used to illustrate some of the implications of non-linearity for the predictability of war.

  1. Crisis instability, chaos, and predictability: Saperstein's model of the outbreak of war

    In building his predictive model of the outbreak of war Saperstein (1994:150) starts from the following conception of prediction. He represents the present state of the international system by Xo. A set of parameters, lambda, is assumed to change that state into a different state Xn. Furthermore, he presumes that different sets of parameters will result in different futures, i.e. different values of Xn. Prediction, then, implies knowing Xn, given Xo and a specific set of parameters, lambda. In doing so he introduces a dynamical system that can be expressed mathematically as a function:

    where lambda represents the set of parameters that define the form of the function f. Different specifications of lambda, be it sets of policy options or distributions of capabilities, imply different functional relations between present and future states of the international system. Equation (0.4) relates the system state at time n to the next time n+1. Through iterating the function it becomes possible to move into the future. For example, to go from the present, Xo, to the future, Xn, we iterate Equation (0.4) as follows:

    The ultimate outcome of any policy or process is the value of X far ahead in the future: Xn = Xinf as n approaches infinity. This set of values is called the attractor of Equation (0.4), and varies as the parameter lambda is varied. Thus for Saperstein (1994: 150) the problem of prediction becomes: "given Xo and lambda, what is the attractor Xinf ?"

    As a second assumption comes the notion of crisis instability, which is defined as an extreme sensitivity of system behaviour to small changes in input or system parameters (Saperstein 1994: 156). Small perturbations - such as differential growth of power - occur constantly in anarchical settings as the international political system. What we need to find out is when such small variations lead to crisis instability, a system state that Saperstein assumes to be equivalent to a high probability of the outbreak of war. Illustrative of such instability is again the murder of Archduke Franz Ferdinand in the Balkans in 1914, which was a small perturbation compared to the ensuing mass slaughter that World War I turned out to be. Conversely, the downing of a Korean Boeing 747 by Soviet warplanes did not escalate into a major armed conflict, even if it involved the loss of hundreds of lives. So these examples suggest that the Eurasian security complex was clearly "crisis unstable" at the beginning of the century, whereas it was relatively stable in its latter half with small perturbations leading to small changes in the system output. More generally, it appears that some system states may be "crisis stable", whereas other tend to be crisis unstable. In terms of Equation (0.4), this means that for some ranges of lambda small changes in Xo will lead to small changes in Xn, whereas for other ranges of lambda small deviations in Xo may produce highly disproportionate shifts in Xn. The major purpose of the model is to get to know whether a present system state, Xo, and/or set of parameters, 8, will lead to the desired end state, Xinf, i.e. crisis stability (a system state where the outbreak of war is unlikely) or to an undesirable - that is strange - attractor, , i.e. crisis instability (a system state where the onset of war is to be expected).

    Saperstein's notion of crisis instability is clearly inspired by the extreme sensitivity to small changes often displayed by non-linear physical systems, a quality that is now commonly referred to as "chaos" (Waldrop 1994; Brown 1995a; De Vree 1991). Of special interest is that the same non-linear system may show both non-chaotic behaviour and chaotic behaviour. In the first case prediction is possible because the future uncertainty is of the same order of magnitude as present uncertainty. As a result small variations in input, delta(xo) (due to disturbances and measurement errors), lead to commensurately small variations in system output, delta(x(t)) (see Figure 1). Put differently, the paths from closely neighbouring starting points remain close.

    Figure 1

    Figure 1: Predictability

    In the second case prediction it is not possible as the model system is extremely sensitive to small perturbations, delta(xo), resulting in disproportionate shifts in system output, Xn (see Figure 2). More specifically, as the paths separate "exponentially", the rate of increase of the separation is itself proportional to the separation, hence increases as the latter increases, leading to an explosively increasing separation. Consequently, the final outcomes over all possibilities allowed in the system. Therefore, prediction becomes impossible, as every future outcome is possible, meaning that we are in the dark about what is to happen. Such a situation is typically chaotic.

    Figure 2

    Figure 2: Chaos or unpredictability

    In Saperstein's model the outbreak of war is ultimately equivalent to the onset of chaos. As a working hypothesis he assumes that the transition from stability to war in an international system can be represented by a transition from non-chaotic to chaotic behaviour in the non-linear model representing the interactions between the competing elements (i.e. states) of the system. In other words, if the onset of chaos is predicted in the model, then crisis instability and therewith the outbreak of war is to be expected in the system being modelled (Saperstein 1994: 157).

  1. Predictability or unpredictability of non-linear systems: the Lyapunov Exponent

    What we need then to examine a model of competition among states for the absence or presence of chaos, is some measure of chaos. For the sake of argument we use the model as specified in Equation (4), i.e. a dynamical model evolving from the present to the future via discrete recursion relations. The possible starting points, delta(xo), all trace out a path towards the future by means of the model mechanism. If the paths from closely neighbouring points stay close, prediction is possible (Figure 1). In case the paths separate "exponentially", prediction is impossible (Figure 2). A useful numerical indicator of such path separation is the Lyapunov Exponent (see Brown 1995a: 22-26). To give an idea of how this exponent is calculated, let Xo and X'o be any two neighbouring starting points separated by a small distance deltao. Applying the iteration procedure of Equation (0.5) these points evolve respectively to the configurations Xn and X'n, separated by the distance deltan = IX'n - XnI , which can be written in the exponential form

    Equation (6) defines the Lyapunov Exponent L(Xo which (see Equation 4) depends on the initial system state as well as the specific dynamics of the system (represented symbolically by the parameter lambda). Through inversion of Equation (0.6) and considering that the interest is in the long term evolution of the system (n->inf) starting from closely neighbouring points (deltao->0), Saperstein (1994: 160) arrives at

    If L is smaller than 0, closely neighbouring starting points will stay close to each other over time (t). Such systems are defined as stable and allow for prediction of their behaviour (Figure 1). If, on the other hand, L is greater than 0, initially closely neighbouring starting points will drift ever further apart, so that prediction becomes impossible. Thus positive values of the Lyapunov exponent are indicators of chaos or instability.

    So Saperstein's main point is that there exist simple determinate non-linear models, having the form of Equation (4), that shift from non-chaotic to chaotic behaviour as a model parameter is increased. As long as parameter values lambda remain below a critical point lambdac, the models have well defined attractors. All starting points in a given region, delta(xo), evolve towards the same limit surface, delta(xt), within which initially neighbouring points remain close. Under these conditions predicting the evolution of the system is possible. If the critical point is surpassed, no end point exists. Consequently,

      the path originating at any starting value will, in the course of time, range over the entire set of possibilities open to the system; any two neighbouring paths will diverge, making prediction impossible. Such a chaotic system is said to have a "strange attractor" - very different from the simple collection of points, smooth curves, or surfaces, which are the attractors of predictive systems (Saperstein 1994: 161).

    For our purpose the essence is that the critical transition point can be determined within the model. Although the model no longer allows for prediction of a specific path in the future once the critical value lambdac is exceeded, its major point of attraction still is that it makes possible the prediction of this unpredictability.

    In the past few years Saperstein has attempted to demonstrate the heuristic potential of his paradigm. Starting from the assumption that the presence or absence of chaos in a non-linear model indicates stability or instability in the system being modelled, the paradigm has been used to deal with such fundamental problems as (1) the relative stability of bipolar and multipolar international systems (Saperstein 1991); and (2) the relative war-proneness of democracies and non-democracies (Saperstein 1992). Although different models are developed to deal with each problem the method of analysis is basically the same. In each case the appropriate Lyapunov Coefficients are computed to determine the relative regions of model stability: the greater the region, the more likely that the system will be stable. The results obtained are interesting and are suggestive for further inquiring and experimenting, even though it is clear that at present no claims of empirical validity can be made.

  1. Conclusion

    In this paper an attempt has been made to clarify the concept of non-linearity and to indicate some of its essential implications for the predictability of war. Then followed a discussion of one major attempt to deal with the problem of predictability inherent to non-linear systems. The model discussed is Saperstein's model of the outbreak of war, which basically has the form of Equation (4). The major conclusion is that similar non-linear models carry great heuristic potential on condition that they can be made to adequately cover important aspects of the international system. If this potential materialises then such models could be used for exploring conditions of crisis instability, i.e. the conditions under which the outbreak of war is more likely.

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Vakgroep Politieke Wetenschappen, VUB