Complex dynamics and control of arms race

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH

ELSEVIER

European Journal of Operational Research 100 (1997) 192-215

Theory and Methodology

Complex dynamics and control of arms race D.A. Behrens ~, G. Feichtinger *, A. Prskawetz Institute for Econometrics, Operations Research and Systems Theory, Vienna University of Technology, A rgentinierstrafle 8, A - 1040 Vienna, Austria

Received April 1995; revised December 1995

Abstract The aim of this paper is to show that asymmetric, nonlinear armament strategies may lead to chaotic motion in a discrete-time Richardson-type model on the arms race between two rival nations. Local bifurcation analysis reveals that 'complicated' dynamics will only occur if neither nation has an absolute advantage over the other one with respect to its level of armament and its capability to keep up the expenditures on armament. The calculation of Lyapunov exponents supports the existence of chaos. Since transitions to chaos can be identified with transitions to war, we use the Ott-Grebogi-Yorke-algorithm to stabilize the arms race model in the chaotic regime and improve the system's performance by making very small time-dependent changes of a parameter under control. © 1997 Elsevier Science B.V. Keywords: Nonlinear system dynamics; Arms race; Local bifurcation theory; Controlling chaos

1. Introduction A lot of research work has been done on mathematical models on the arms race. The majority o f these models is based upon the well known work of L.F. Richardson (see Richardson, 1960). In his opinion international conflicts arise due to the built up of fear from each other. This assumption is based upon a report from the war on the Peloponnesus ( 4 3 1 - 4 0 4 B.C.), which resulted from mutual suspicion of Sparta and Athens. Each city-state assumed that the other's armament was a sign of hostile intentions. In Richardson's arms race model the possible trajectories either converge or diverge monotonically. However, investigations on experimental data show that the expenses on armament mostly oscillate in a periodic manner. If we regard the arms race preceding W o r l d W a r I, we know that the armament budgets grew exponentially. To describe this fact by the means of his standard-type model, Richardson restricted the system parameters drastically by equating the intensities to arm respectively to disarm and, consequently, he got a single equation, describing the growth of the combined budgets of the Central Powers (Germany, AustriaHungary) and the Allies (Russia and France). Being aware of the weaknesses of his simple model, Richardson

* Corresponding author. The author has received the 1993-student-awardof the DGOR for investigations on 'Two- and threedimensional models on the arms race'. 0377-2217/97//$17.00 © 1997 Elsevier Science BN. All rights reserved PH S0377-2217(96)00018-5

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himself suggested several extensions to his system. Though his analysis was mainly restricted to the linear interaction of two competing nations, he already referred to nonlinear extensions of his approach. In this paper we shall incorporate nonlinearities in a discrete-time version of the standard Richardson-type model. Note, that the discrete-time version represents a completely different system from the system-theoretic point of view as compared to the continuous-time model of Richardson, so that the results of Richardson and the present paper cannot be compared with each other. On the other hand, the assumption of a discrete-time model seems inherent to the system dynamics of the arms race since decisions about the amount of expenses on armament are based on the military budget which is reported at discrete time. We use a two-dimensional discrete-time nonlinear dynamical system of the military expenditures to describe the arms race between two rival nations, which behave asymmetrically (see also Holyst et al., 1995, for an economic system exhibiting the same dynamic behaviour as the arms race model). The submissive nation (X) spends the more on arms the 'stronger' it has been in the previous period, while the courageous nation (Y) spends the more on arms the 'weaker' it has been one period before. The nonlinearity and asymmetry of the procurement functions of both nations will turn out to be essential for chaotic system behaviour to occur. The extreme sensitivity to initial states in a system operating in chaotic mode can be very destructive to the system because of unpredictable behaviour which implies the transition to war according to Saperstein (1984, 1990). As a result, it is essential to know when a nonlinear system will get into a chaotic mode so as to avoid it, and in case it does, how to recover from it. We use local bifurcation analysis to distinguish the system behaviour according to the system parameters and the calculation of Lyapunov exponents to evidence chaotic motion. Once the arms race is in a chaotic regime we can control the system trajectories by means of the Ott-Greborgi-Yorke algorithm (see Ott et al., 1990, and Holyst et al., 1995). The successful control of the system does not necessarily result in an optimal solution, but in predictability of the time evolution of the trajectories and, consequently in the predictability of the arms race. The paper is organized as follows. Section 2 introduces the nonlinear discrete-time arms race model. The local stability and bifurcation analysis are the content of Section 3. Numerical evidence of chaos is given in Section 4. Section 5 illustrates the control of chaotic time paths. We conclude the paper with a short summary and suggestions for further research. Mathematical calculations are given in the Appendices.

2. The model Let us consider two nations which make their decisions at moments discrete in time, for instance, each year. The state variables of the arms race model are assumed to describe the time evolution of the devotion of these nations to hostilities. For practical applications the variables are assumed to be represented by expenditures on arms (Rapoport, 1980, p.52). Let x t a n d Yt denote the fractions of the available resources which nation X and Y denote to armament at time t = 0, 1, 2 . . . . . Assuming that both countries' national budgets are of the same order of magnitude, we further restrict the state variables to denote the relative share of the national budget spent on arms; i.e. we restrict the possible values of the state variables to the interval [0, 1]. The sequence of the variables x t and Yt for t = 0, 1, 2 , . . . represents the evolution of the system over time. Each nation's military expenditures depreciate by a constant factor and increase depending on the difference between the size of its own military expenses and those of its opponent, both measured in the year before (see Richardson, 1960, p.35). These coupling terms represent the reaction of each country to the 'challenge' of the other one and are stated under the heading 'rivalry' in Richardson (1960, p.35). Contrary to the assumptions stated in Richardson (1960) we assume that the reasons to procure additional arms are different for each nation. Q Nation X shows s u b m i s s i v e behaviour in the sense that an increase of its competitor's military expenditures has a dampening effect on its own expenses. On the other hand, if its military expenses surpass that of its opponent, nation X attempts to spend even more on arms. This behaviour is that of a coward. If nation X is in the inferior position it tries everything to avoid any kind of irritation to the

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superior nation Y. But if nation X is able to gain an advantage it does everything in its power to enlarge it. • Nation Y attempts to increase its military expenditures whenever it is in the inferior position in the arms race. Richardson's argument simply is that higher expenses on armament of nation X are a signal of hostile intentions to nation Y. On the opposite, if nation Y is the superior country it dampens its military expenditures. The behaviour of nation Y can be summarized as courageous. Corresponding to the nations' armament strategies we choose the following step-like function depending on the parameters ct, 13 E (0, 1), a ~ (0, ct], b ~ (0, 13] and c ~ (0, ~): xt+ 1 = (1 - e t ) x , Y t + l =" ( 1 - -

+a/(1

+e -c(xt-y')) =f(x

13)ytd-b//(1 + e -c(x'-y')) = g ( x t ,

t, Yt), Yt).

(1)

The positive parameters a and 13 are called fatigue coefficients and reflect the percentage of the reduction of military expenses in absence of a hostile nation. The procurement functions (which stand for the 'rivalry term' mentioned earlier) q~a(Xt, Yt) =" a(1 + e-C(x'-Y')) - l and q~o(x~, Yt) = b(1 + e-C(x,-Y')) -I provide the influence of the difference between both nations' arms expenses x t - y, at time t on the military budgets at time t + 1. The positive parameter c measures the steepness of the sigmoidal functions q~, and ~b, i.e. it determines the speed of armament, while a and b determine the level of armament. The parameters a and b are closely related to the defense coefficients in the original Richardson's model (Richardson, 1960, p.15) and will be called defense intensities or armament levels in the following. It is demonstrated in Appendix A that the fatigue coefficients a and 13 yield an upper bound for the defense intensities a and b in order to guarantee 0 ~< x t <~ 1. The function q~a is chosen such that nation X ' s armament expenses, holding nation Y ' s military expenditures fixed, is convex to the left and concave to the right of the point where both nations' military expenditures are the same ( x = y ) . Asymmetrically we choose the shape of the function q~b, holding nation X's military expenditures fixed. I.e. we assume that if nation X increases its superiority and nation Y its inferiority respectively, they will procure at a decreasing rate. On the opposite side, the less inferior nation X and the less superior nation Y become, the faster they will increase their military expenditures. The larger we choose the parameter c, the more the functions q0a and q~b resemble the step function where both nations arm only if x > y while both nations disarm if x < y. 1 Fig. 1 illustrates nation X ' s and nation Y ' s expenditures on armament as a function of their own military expenses one period ago, holding the other nations expenditures fixed at the level 0.5. The other parameters have been fixed at the values c = 100, ct = a = 0.2 and 13 = b = 0.4. (The system parameters are predetermined and may change over time, but it is assumed that the change happens after a long period - compared to the units year of the budget cycle - of remaining fixed; see Saperstein (1990).) For values of z < 0.5 only the courageous nation Y increases its military expenses, while nation X shortens them, i.e. Yt+ ~ > Yt and xt+ j < xt. For z > 0.5 exactly the opposite happens. Nation X increases its relative share on the national budget spent on arms, while nation Y does not. 2 Moreover, the assumption of c being large causes the 'sudden' decline (increase) in g (in f ) once the threshold level of 0.5 is surpassed. Among others, this nonlinearity will turn out to be responsible for chaotic dynamics to occur (see Section 4, Fig. 6a). In our model, asymmetry of the procurement functions means that both nations increase their military expenses provided that x > y and they both dampen their expenditures in opposite situations. Symmetry of the

J For c ~ oc the piecewise linear function

~a (~,,)

= fa (b)

if x, >

Yt,

if x t < y~, is obtained, whereas for c = 0, ~,, (q~b)= 7Ja (~-b) holds. Recalling the definition of a stationary point it can easily be checked that for appropriate parameter values the maps f and g will cross the 45°-line at the equilibrium point defined by a/et = 1 +exp(- c(x-0.5)) and b/f3 = 1 +exp(- c(0.5- y)) respectively.

D.A. Behrens et al. / European Journal of Operational Research 100 (1997) 192-215

o91 1.0

- -

195

-

0.8

..- ,fJ ,.05;

_..--"

0.2 0.0

.--'"

P"

jj 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Z

Fig. 1. Armament function of nation X and nation Y.

procurement functions, i.e. if both nations show the same behaviour (either submissive or courageous), implies that either nation X or Y increases its military expenses, while the other nation (Y or X respectively) shortens its expenditures. In case of two submissive nations, the nation with the higher level of expenditures on arms at the outset will further increase its military expenses, while the other nation continuously 'disarms' until it ends up with no military budget at all. Hence, only in case of asymmetric nations or two courageous nations a reversal of the military expenses of both nations will be possible and 'complicated' solutions might result. In fact, it can easily be proved that the map f is monotonic in x on the interval [0, 1] for a fixed value of y, while the map g might exhibit two critical points (i.e. two points where the map g has a local extremum) for appropriate parameter values. Hence, only the courageous nation can produce 'complicated' dynamics. 3 'Peace and security' is very often used as a common term, though the stress usually lies on 'security'. This is one of the reasons why we identify 'peace and security' with the absence of war - the violent conflict. In particular Saperstein (1992) equates 'stability' with the absence of war. In the system as described by model (1) different types of attractors can occur, such as a fixed point, a limit cycle or a chaotic attractor. A stable fixed point means that the military budgets will settle down at a single value while a limit cycle implies perpetual oscillations of the military expenditures. But in both cases the future budgets remain predictable. Contrary, Saperstein (1990) proposed that chaos in an arms-race model represents 'crisis instability'. " A 'crisis unstable situation' ... is one in which neither party is confident as to its ability to wait out a crisis (not striking out at its partner) because of the fear of losing its ability to respond to or protect itself from the actions of the opponent because of the prior actions, then each is tempted to strike first ... [and] neither side can afford to wait out perceived threat or other ... Thus, the slightest hint of hostile action by an opponent in such a situation implies a massive pre-emptive attack, i.e., a small perturbation leads to a major system-changing response." (Saperstein, 1990, p.171) In this situation there is no possibility to predict the effect of any action of the participants. The fact, that small perturbations of initial conditions lead to large changes of the system is one of the properties of a chaotic attractor. Additionally Saperstein (1992) assumes that in a crisis unstable situation there are always appropriate insulting events, sufficient to drive the crisis unstable system into war. Saperstein (1984) noted, that at the threshold between peace and war predictability is lost. In the period immediately before the outbreak of war, the nations desperately try to avoid what is to come, but the process already has got out of hand.

3 W e are grateful to one of the referees for pointing out this fact.

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Journal of Operational Research 100 (1997) 192-215

Thus, it is very important to know the system’s behaviour in advance in order to prevent the system from behaving chaotically. In particular one has to choose rates of arming and disarming which provide solutions outside the chaotic region of the system. If this is not possible - for instance whenever the chaotic region is too large - the system’s behaviour can be controlled by means of the OGY-algorithm. This method allows to stabilize the system to any one of the periodic solutions, which are embedded in the chaotic attractor. Therefore, if not possible to prevent the system from the chaotic regime, at least predictability can be re-established. In other words, even within the chaotic region of the system, the risk of an outbreak of war can be avoided by following a ‘controlled trajectory’.

3. Local stability

and bifurcation

analysis

In this section we analyse the model outlined in Section 2. In particular we shall investigate the existence and local stability of the stationary solutions depending on the parameters of the system using the interactive LOCal BIFurcation program LOCBIF (Khibnik et al., 1993). The local dynamical behaviour of system (1) around the steady state (see Appendix B) can be examined by means of a linear expansion of (1) at the equilibrium point ( x * , y’ 1. The Jacobian matrix J evaluated at the stationary solution (x * , y * > is defined as J=

wax

af/aY

wax

way

The eigenvalues A 1.2

= T‘(tr

(2)

(+., yIj’

of (2) are given by Jk\/(tr

J)‘-4det

J),

(3)

where tr J denotes the trace of the Jacobian J and det J its determinant. The term (tr Jj2 - 4 det J is called the discriminant of J. As long as the modulus (the modulus of a complex number A = OL+ pi is defined by the Euclidean distance between the origin and the point ((Y, p> in the Gaussian plane, i.e., mod A = 1A 1 = /+pz) of the eigenvalues is smaller than one, i.e., 1Xi 1 < 1, stable solutions will result. The special points, where I Xi I is exactly equal to one are called bifurcation points (see Lorenz, 1993, p.l lO-118 on bifurcations in discrete-time dynamical systems). Since both countries enter simultaneously the same state of stability, we shall illustrate the following bifurcation diagrams only w.r.t. one nation’s military expenditures. Equilibrium

curues

We start our analysis with the following set of parameters: (Y= 0.2, p = 0.8, a = 0.2, b = 0.4, c = 100 and the initial conditions x,, = 0.13, y0 = 0.26. This implies that in the initial configuration the courageous competitor Y spends twice as much on armament as the submissive one. Additionally, we assume that the submissive nation X has a rather low speed of reducing ((;Y< 6) and a lower level of procuring (a < b) its military expenditures as compared to nation Y, but the speed of procurement is quite high in both nations, i.e. c is large (Fig. 6a in Section 4 illustrates the change in the long-run dynamics of the arms race model by varying the parameter c). The above given set of parameter values yields the equilibrium point (x * , y * ) = (1 .OO, OSO), which will provide a starting point to compute equilibrium curves. These curves are defined in the phase-parameter space of our model (1) where all but one parameters are fixed. Note that only for values of (Y> a and p > b it is guaranteed that both state variables will be restricted to the domain [0, l]. In Fig. 2 we illustrate the bifurcation diagram in the whole range of possible fatigue coefficients, i.e. CLE [0, 11. In Fig. 2 the fixed points y * of the first and second iterate of system (1) of the courageous nation have been computed for a changing coefficient 01 of the submissive nation X. 4 The fixed points of the second iterate of

4 One could as well plot the stationary

point x * versus the parameter

a

D.A. Behrens et al. / EuropeanJournal of OperationalResearch 100 (1997) 192-215 0.6

Flip bifurcation of first iterate

stable fixed point of first iterate

/

-

-

unstable fixed point o f first iterate

/ / ~ ,,,-" .

-- - -

-

stable fixed point of second iterate

.|

....

0.5

fJ

0,4

y* 0.3

/

Fold bifurcation of second iterate

197

"

te:tatblefixedpointof second

J tt I

0.2

~ ~

Flip bifurcation of , first iterate i

O.l

0.0 0.0

I

1

0.1

0.2

I ct 0.3

ct2 I

I

I

I

I

I

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Fig. 2. Equilibrium curves of the period 1 and period 2 fixed point if we fix 13= 0.8, a = 0.2, b = 0.4, c = 100 and let t~ be the bifurcation parameter.

system (1) are no fixed points of the first iterate but points of period two. Generally, x is called a point of period n of the map h if x = h~(x) and h i ( x ) ~ x for 0 < j < n holds. The less the submissive nation renews its military expenditures, i.e. the larger ot becomes, the lower will be its own military expenses x* as well as those of its opponent y * . Varying et not only changes the level of the equilibrium solution, but also the stability of the system. In particular, three bifurcation points are located on the equilibrium curves. For ot < ct I = 0.31, the fixed point of the first iterate is stable. At a l one stable and one unstable period 2 point emerges via a fold bifurcation of the fixed points of the second iterate while the stability of the fixed point of the first iterate remains unchanged. In fact, it can easily be checked that these four fixed points of the second iterate constitute a stable and unstable orbit of period 2 of system (1). 5 The fold bifurcation (see Strogatz, 1994, p.45), which is sometimes also called a saddle-node bifurcation, is one mechanism by which fixed points are created and destroyed. To the left of ot 1 there are no fixed points while there are four fixed points of the second iterate of system (1) to the right o f et lFurther increasing the fatigue coefficient ~t leads to a flip bifurcation of the fixed point of the first iterate at a 2 = 0.38. At this point the unstable orbit of period 2 merges with the fixed point of the first iterate, which becomes unstable. F o r ~3 = 0.996 another flip bifurcation of the fixed point of the first iterate takes place, the stable orbit of period 2 merges with the fixed point of period l, which becomes stable. The flip bifurcation, often also called a period-doubling or period-halving bifurcation, describes the emergence of an orbit of period 2 in addition to an already existing fixed point of order 1 and exchange o f their stability behaviour. Depending on the stability of the fixed points one distinguishes between a supercritical flip bifurcation (the fixed point o f order 2 is stable while the fixed point of order 1 changes from stable to unstable) and a subcritical flip bifurcation (the fixed point is stable and the emerging fixed point of order 2 is unstable). In case of Fig. 2 the flip bifurcation at ct 2 is subcritical, while the flip bifurcation at ~3 is supercritical.

5 We have verified these results using the program LOCBIF (Khibnik et al., 1993) as well as DMC (Medio, 1992).

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As Fig. 2 illustrates, the coexistence of two stable equilibria in the interval (O/.1, 0/.2) implies a crucial dependence of the system behaviour on the initial expenses on armament. E.g. moving from the left to the right in Fig. 2, the system will follow the stable equilibrium up to et 2 at which point the system jumps to an attractor with a higher periodicity (orbit of period 2). But if one starts from values of et somewhere to the right of a z, then the system will follow the stable orbit of period 2 up to the point a i and jump to an attractor with a lower periodicity (orbit of period 1) at ct ~. Hence, the long-term dynamical states are affected by the direction in which we vary the parameter. For values of the fatigue coefficient a between etl and et 2 the system will either converge to a stable orbit of period 1 or period 2, depending on the initial conditions of the military expenditures. I.e. two nations with almost the same initial military expenditures can show totally different evolutions over time despite having exactly the same system parameters. Codimension one bifurcation curve Case I: a = 0.2, b = 0.4, (a < b). The values of the system parameters at the three bifurcation points in Fig. 2 can be taken as initial parameter setting to compute codimension one bifurcation curves which illustrate the change in the stability of the stationary solution ( x * , y* ) when all but two parameters are fixed. Additionally to the fatigue coefficient of the submissive nation, a , we assume the corresponding parameter of the courageous nation, 13, to vary. The bifurcation curves (the 45°-line does not correspond to any bifurcation curve, but has been plotted to underline the asymmetry of the system dynamics, while the broken lines just mark off the viable domain et > 0.2 and 13 > 0.4) trace out four regions of particular interest regarding the local stability of the stationary solution (x *, y * ). When we pick out 13 = 0.5 and follow a horizontal line through the parameter-space by increasing ct, the fatigue coefficient of nation X, we first pass a region, where the system converges to a stable equilibrium. By crossing the first bifurcation curve (fold bifurcation), a stable orbit of period 2 emerges, coexisting with the stable orbit of period 1. A further increase of a leads to a flip bifurcation at which point the stable orbit of period 1 becomes unstable. Within the area of the stable orbit of period 2 a small interval exists where the well known period doubling route to chaos (Lorenz, 1993, p. 127) takes place (these dynamics are verified in the next section using numerical tools like bifurcation diagrams, phase diagrams and Lyapunov exponents). By a further increase of ot the stable orbit of period 2 merges with the fixed point of period one, which becomes stable again. As Fig. 3 illustrates, the domain where solutions with higher periodicity occur is mainly concentrated to the left of the 45°-line, i.e. where nation Y has a higher fatigue coefficient compared to nation X ([3 > et). This domain shrinks, the smaller the coefficients et and 13 are, i.e. the less both nations reduce their military expenditures. On the other hand, chaos disappears as soon as both nations heavily reduce their military expenditures. These different stability regions in Fig. 3 can be interpreted as follows. If the submissive nation X heavily reduces its military expenditures each period (et > [3, right to the 45°-line in Fig. 3), X gives no signal of hostile intention to its opponent. In this case stability is not endangered, there is no threat to 'peace and security'. Due to the assumption about the initial conditions (Y spends twice as much on armament than the submissive one in the initial outset) both nations disarm from the very beginning of their 'competition'. On the other hand, if the submissive nation reduces its military expenditures at a lower rate than the courageous nation (ct < 13, left to the 45°-line in Fig. 3), nation X will quite soon surpass the military expenses of its competitor Y. Now, both opponents increase their military budgets, but since the courageous nation arms at a higher level than the submissive nation (b > a), nation Y gains (former or later) an advantage over nation X (assumed that 13 is not too large). Thus, the initial situation recurs and the whole 'game' is repeated. Whether this situation leads to stable periodic or chaotic time paths will depend on the exact values of the fatigue coefficients et and 13. The stable solutions in the lower right hand corner of Fig. 3 - which are the result of continuous disarmament of both nations - will certainly be preferred compared to the stable solutions in the upper left hand corner, which are characterized by higher levels of arms expenditures of both nations.

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199

10 09 08

CLLrve

0.7 06

~6o.5 0.4 0.3 0.2 0.1 O.O 0.0

0.1

0.2

0.3

0.4

0.5

06

0.7

0.8

0.9

10

stable period 1 orbit coexistence of stable period 1 and stable period 2 orbit stable period 2 orbit R

period doubling route to chaos

Fig. 3. Stability domains in ( a , 13)-parameter space with a = 0.2, b = 0.4 and c = 100.

Case H: a = 0.4, b = 0.2 (b < a). The question remains, whether and how the system dynamics change if the absolute size of the defense intensities a, b is reversed. I.e. if the submissive nation arms at a higher rate than the courageous nation (a = 0.4 > b = 0.2). The resulting bifurcation scenario is illustrated in Fig. 4 (again the 45°-line has been plotted just to underline the asymmetry of the system dynamics, but does not represent any bifurcation curve, while the broken lines just mark off the viable domain et > 0.4 and [3 > 0.2). Left to the 45°-line the submissive nation X has a higher level of armament (a > b) as well as a lower fatigue coefficient ( a < [3). Both properties favour the superior position of nation X. Once nation X surpasses nation Y, the additional increase of the military expenditures takes place. But since nation X is superior in procuring as well as in maintaining its arms expenditures, Y will never outweigh nation X's military expenses, such that the system monotonically converges to a stable equilibrium point. On the contrary, if we start right to the 45°-line ([3 < ~), the submissive nation X renews its military budget at a lower rate than the courageous nation Y, such that the higher armament level (a > b) of nation X is counterbalanced by the higher fatigue coefficient. In this case a situation similar to the left of the 45°-line in Fig. 3 occurs. Since neither nation has an absolute advantage over the other one (as to the left of the 45°-line in Fig. 4 and to the right of the 45°-line in Fig. 3) periods of armament and disarmament interchange resulting in either stable periodic or quasi-periodic (no cycle repeats itself) system dynamics as indicated by the black region in Fig. 4. Quasi-periodic dynamics is a type of motion which, in some sense, lies between periodic and chaotic behaviour; i.e. almost all time paths are aperiodic (that is, they are not periodic and they do not converge to a periodic time path), but there is no sensitive dependence on initial conditions, which is characteristic for chaotic systems.

200

D.A. Behrens et a l . / European Journal of Operational Research 100 (1997) 192-215 10 0.9 0.6 03 0,6

t~ o.5 0.4 0.3 13.2 0.1 0.13 0.0

0.I

D m

0.2

0.3

0.4

0.5

O.G

0.7

0.8

0.9

1.0

stable period 1 orbit stable periodic and quasi-periodic orbits

Fig. 4. Stability d o m a i n s (or, [ 3 ) - p a r a m e t e r s p a c e with a = 0.4, b = 0.2 and c = 100.

Contrary to Fig. 3 quasi-periodic solutions emerge via a Neimark-Sacker bifurcation defined by the condition that the eigenvalues o f the equilibrium are complex conjugate and cross the unit cycle, i.e. I k il = 1, i = 1, 2, at equilibrium. The stability of these solutions will be investigated in the next section.

Regions of orbits with higher periodicity in (a, t~)-parameter space (see Fig. 5) Up to now we have assumed a constant difference o f the defense coefficients between nation X and nation Y that is equal to I a - b I = 0.2. The question remains whether the system dynamics and in particular the regions o f quasi-periodic and chaotic dynamics are robust to changes in the difference between the armament levels a and b. Assuming that nation Y ' s military expenses depreciate at a faster rate (13 > or) but that new arms are added at a higher level ( b > a) compared to nation X, an increase o f the difference between the defense intensities a and b leads to an outward shift as well as an enlargement of the region where orbits with higher periodicity occur. Right to the 45°-line, where we have exactly the opposite outset (nation X ' s military expenses shrink at a faster rate ~ > 13, while the armament level o f nation Y is lower b < a) an increase in the difference o f the defense coefficients leads also to an outward shift, but different to before to a shrinkage of the region o f orbits with higher periodicity. This change in the position of the regions of orbits with higher periodicity - depending on the difference between the defense coefficients - can be explained as follows. As already illustrated in Fig. 3 and Fig. 4, only if neither nation X nor nation Y has an absolute advantage over the other one, chaotic or periodic dynamics will occur. Hence, if we are to the left of the 45°-line and increase the disadvantage o f nation X w.r.t, the level of armament, i.e. b - a = 0.4 compared to b - a = 0.2, we also have to increase the disadvantage of nation Y w.r.t, its fatigue coefficient 13 o f its expenses to keep the arms race in a situation where no nation has an absolute advantage over the other one. A similar argument holds to the right of the 45°-line. In this case, an

D.A. Behrens et al./European Journal of Operational Research 100 (1997) 192-215

201

0.9 08 07 06

05 04 0.:3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

06

07

0.8

09

1

a=O'l' b=0"5

l

a=0"4'b=0"2 and a=O.5,b=O.1

a=O.2,b=0.4

~

a=O.5,b=0.1

l

a=O.4,b=0.2 Fig. 5. Regions of orbits with higher periodicity in (tx, 13)-parameter space with c = 100.

increase (decrease) in the armament level of nation X (Y) has to be counterbalanced by a higher level of the depreciation factor cx of nation X's military expenditures. Summing up, the assumption that both nations are of similar standard w.r.t, their ability to arm together with nonlinearity in model (1) may lead to stable periodic, quasi-periodic or even chaotic trajectories. Note that the procurement functions represent the only nonlinearity in our model. Therefore the speed at which both nations procure, i.e. the parameter c constitutes the source of chaos in our model. In particular, the smaller c, the smaller will be the 'black regions' in Fig. 3 and Fig. 4 where higher periodic and chaotic dynamics occur. But, according to Saperstein's (1984, 1990) proposal, chaotic motion in the arms race model can be interpreted as indication of the potential outbreak of war. In order to avoid this threat and to know as precisely as possible where control is necessary, it is important to mark off the distinct regions where chaos and solutions with higher periodicity occur. Therefore it is necessary to go beyond the local bifurcation analysis and use specific numerical tools (e.g. the largest Lyapunov exponent) to classify the system behaviour within the regions of Fig. 3 and Fig. 4, where the period doubling route to chaos or the interchange of stable periodic and quasi-periodic orbits happens.

4. Numerical evidence of

chaos

The local stability analysis of Section 3 has revealed that different long-run solutions in the arms race are possible depending on the parameters of the system. In this section we present and classify orbits with higher

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y" [(a) 0.31

0.2:

G.12

9.g

9.9

I 9.24

Q.59

I O. 7 6

c.10 z

Fig. 6a. B i f u r c a t i o n d i a g r a m w i t h r e s p e c t to c ~ [0, 100] a n d f i x i n g the o t h e r p a r a m e t e r s at the v a l u e s a = 0.3, 13 = 0.5, a = 0.2, b = 0.4.

periodicity, which occur via the period doubling route to chaos (Fig. 3) or the Neimark-Sacker (Fig. 4) bifurcation, using the program package DMC (Medio, 1992).

Bifurcation diagrams To confirm that the parameter c, which indicates the steepness of the sigmoidal functions ~ and q~b, is responsible for 'complicated' dynamics to occur, Fig. 6a illustrates the change in the long-run dynamics of the military expenditures of nation Y by varying the parameter c in the range [0, 100]. The other parameters are fixed at ~ = 0.3, 13 = 0.5, a = 0.2 and b = 0.4. Hence, for c = 100 the parameter constellation yields a system behaviour belonging to the small window of the period doubling route to chaos in Fig. 3. In fact, the change in the systems behaviour by varying the parameter c is also described by the well known period doubling route to chaos (Fig. 6a). For low values of c, the system converges to a stable equilibrium. After finitely many period doubling bifurcations for intermediate values of c, chaotic behaviour (indicated by black regions in Fig. 6a) arises after infinitely many period doubling bifurcations for values of c close to 90.

Y

Fig. 6b. B i f u r c a t i o n d i a g r a m w i t h r e s p e c t to c~ ~ [0.29, 0 . 3 2 ] a n d f i x i n g the o t h e r p a r a m e t e r s at the v a l u e s 13 = 0.5, a = 0.2, b = 0.4, c = 100.

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203

Y

Fig, 6c. Bifurcation diagram with respect to c~ E [0.5, 1] and fixing the other parameters at the values 13 = 0,3, a = 0.4, b = 0.2. c = 100.

The succession of asymptotic states of the military expenditures of nation Y along a horizontal line in Fig. 3 with the intercept 13 = 0.5 and restricting the fatigue coefficient of nation X between 0.29 and 0.32 is shown in Fig. 6b. For a = 0.29 the military expenditures of the courageous nation Y are strictly predictable, converging to a stable orbit of period 4. For some reason - maybe to show goodwill - nation X might attempt to disarm faster, i.e. cx increases. But by marginally changing the parameter ot the arms race gets into the chaotic regime (indicated by the black region in Fig. 6b). An action in order to show the willingness to disarm and keep 'peace', causes unpredictability and endangers 'security'. Within the chaotic regime we find windows of periodic behaviour. This is one of the popular features of chaos, but also a dangerous fact for the arms race. It is possible that the system seems to be in a periodic state within the chaotic region, but gets unpredictable by slightly modifying one of the coefficients. Only by a further increase of the fatigue coefficient ~, the nations will return (via a sequence of period halving bifurcations) to stability. As already indicated in Fig. 4, the dynamics of the orbits with higher periodicity change if we reverse the size of the coefficients a and b. Fig. 6c shows the succession of asymptotic states of the military expenditure of nation Y along a horizontal line in Fig. 4 with the intercept [3 = 0.3 and restricting the fatigue coefficient of nation X between 0.5 and 1. We observe a rich class of potential modes of behaviour. Periodic cycles alternate with seemingly irregular time paths indicated by black regions in the bifurcation diagram.

Phase diagrams To get a better impression of the time course of the system variables Figs. 7a-7c illustrate the dynamics in phase space. The long-run evolution of the military expenditures are computed for 10000 iterations. (Since we are just interested in the asymptotic behaviour of our system, we neglect the first 100 iterations.) Fig. 7a (a = 0.2, b = 0.4, c~ = 0.3, [3 = 0.5, c = 100) supports the chaotic behaviour indicated in the bifurcation diagram of Fig. 6a. The military expenditures never settle down at any equilibrium such that predictability will never be possible. Nevertheless the extent of the strange attractor is small compared to the possible changes in the system. If instead we take a time path of the second bifurcation diagram, Fig. 7b (a --- 0.4, b = 0.2, a = 0.6, [3 = 0.3, c = 100), the long-run evolution of the military expenditures seems to follow a cycle. (We have just plotted successive points without connecting lines in between.) But contrary to periodic dynamics, the state variables take on every possible value on the closed invariant curve in Fig. 7b. The dynamics can be described as quasi-periodic, i.e. no cycle repeats itself.

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(a) 0.32

0.25

0.12

0.0

0.0

;.12

;.2.

i0 , 3 8

X

>

F i g . 7a. P h a s e d i a g r a m f o r a = 0 . 3 , 13 = 0.5, a = 0 . 2 , b = 0.4, c = 100.

Finally, Fig. 7c (a = 0.4, b = 0.2, a = 0.81, [3 = 0.3, c = 100) illustrates a stable periodic orbit, where the system undergoes counter-clockwise 14 states step by step. If in the initial setting the military expenses of the submissive nation X are superior (points below the 45°-line in Fig. 7c), both nations will increase their procurement. But due to the higher fatigue coefficient of nation X (~ < a ) , nation Y will finally surpass nation X in its military expenses, from where on both nations will disarm until nation X surpasses nation Y again and the cycle starts over again.

Lyapunov exponents To further classify the dynamical behaviour illustrated in Figs. 7 a - 7 c , we can use the concept of Lyapunov Characteristic Exponents (LCE), which measure the average rate at which initial states, which are close,

(b) Q.?

0.2d

o.o

e.Q

~.a4

~.se

I

0.76 X

F i g . 7 b . P h a s e d i a g r a m f o r c~ = 0 . 6 , 13 = 0.3, a = 0 . 4 , b = 0.2, c = 100.

D.A. Behrens et al. / European Journal of Operational Research 100 (1997) 192-215

205

(c)

0.50

0.24

x

F i g . 7c. P h a s e d i a g r a m f o r et = 0 . 8 1 , 13 = 0.3, a = 0.4, b = 0.2, c = 100.

separate. Let f : ~m ~ ~,~ be a differentiable map. Then the average rate of growth of an initial perturbation vector g x of the initial state x 0 is defined by the following limit:

1

L x = lim --log II Oxf~( n--,~ n

Xo)" gx II,

(4)

where II z II denotes the length of the vector z in R '~ and Dxf"(x o) is the Jacobian matrix of the n-th iterate of x 0. The largest L i, i = 1 . . . . . m, is called the largest Lyapunov exponent of the map f and measures the average rate of divergence of nearby initial states (for more details see Eckmann and Ruelle, 1985). Using definition (4) we can describe the dynamical behaviour of our system (1). A stable periodic orbit is characterized by negative Lyapunov exponents, while a chaotic attractor is characterized by at least one positive Lyapunov exponent. For a quasi-periodic attractor the largest Lyapunov exponent is zero. All numerical values of the Lyapunov exponents have been obtained by using a time path of at least 50000 iterations. For the parameter set chosen in Fig. 7a we get L ~ - - 0 . 3 1 , L 2 -~ - 0 . 1 1 , which confirms that the dynamics on the attractor are chaotic. In case of Fig. 7b the Lyapunov exponents are given by L 1 ~ 0 , L 2 = - 0 . 9 6 . L~ is zero and therefore indicates quasi-periodic dynamics. In Fig. 7c we observe periodic stable dynamics, which yields two negative Lyapunov exponents L l = - 0 . 1 2 , L 2 = - 1.44. Once located in the chaotic region of the attractor marginally different parameter values may cause completely different system dynamics. Continuous control of the parameters is therefore a prerequisite of ' peace and security'.

5. Controlling chaos Saperstein (1984) proposed (cf. Section 2) that chaos in an arms-race model describes a crisis unstable situation. Therefore, we marked off the parameter-regions in which chaotic behavior occurs in the previous section. The stress of course lies on the avoidance of these regions. But whenever these areas are too large to be 'cut off', the successful application of a control-algorithm in the chaotic regions establishes predictability and, consequently, provides 'peace and security'. Additionally, the stabilization to an orbit of period 1 will deliver, as we will show, much better results than the periodic solutions 'around' the chaotic region. The term 'better results' means that, for instance, nation X spends half as much on armament than in the uncontrolled chaotic

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D.A. Behrens et al. / European Journal of Operational Research 100 (1997) 192-215

case and even a quarter of the amount, which is spent in the periodic range close to the chaotic region. By controlling the system we are able to improve the system's performance drastically, without being exposed to the threat of the outbreak of war - what is obviously much more important. Now, what is meant by the term 'controlling chaos'. One approach to control chaotic motion and to establish predictability would be to make some large and possibly costly alternations in the system parameters such that the system behaves regularly. Because of technical reasons or too high costs, this approach need not be practicable. On the other hand, possible regular motions, that can eventually be achieved need not be preferred to the actual chaotic motion since eventually some welfare measures take on worse values on the regular than on the irregular time path (see Kaas, 1995, and Kopel, 1995). An alternative way to control chaotic motion and to establish the predictability without changing the system is provided by the Ott-Grebogi-Yorke-algorithm, the OGY-method (Ott et al., 1990, and Shinbrot, 1993). This method of local control is based upon the observation that chaotic attractors typically contain infinitely many hyperbolic periodic orbits. Any of them can be stabilized as follows (Ott et al., 1990). Determine some of the unstable orbits of low periodicity which are 'embedded' in the chaotic attractor. Examine these orbits and choose one which yields improved system's performance. Finally, tailor small time-dependent parameter perturbations so as to stabilize this already existing orbit, i.e. such that the trajectory is shifted to lie on the stable manifold of the periodic orbit. The presence of chaos is an advantage for this method, since small perturbations cause huge changes in the behaviour of the system and any one of a large number of time-periodic motions can be achieved by making only small time-dependent perturbations of an available system parameter. Moreover, the choice can be made to achieve the best system's performance among these orbits. In the following, we shall use the OGY-method to control chaotic motion in the arms race model. In Holyst et al. (1995) and Hagel (1994) an economic system, exhibiting the same dynamic behaviour as the arms race model, but with the system variables representing the sales of two competing firms, has been stabilized to orbits of period 1 and period 2 using the OGY-method. But different to the paper of Holyst et al. (1995), where the objective of controlling chaos was to improve the system's performance w.r.t, increasing both firms market sales, the ultimate goal in the arms race model is to ensure 'peace and security'. But, of course, we will try to lower the expenses on armament, if it is possible. Assume that each nation can adjust either its fatigue coefficient ( a or 13) or its level of armament (a or b) but not the elasticity c of its procurement function. This seems quite an obvious assumption since changes in the levels of certain parameters can be thought to take much shorter than changes in the elasticity of functional relationships. Additionally these changes in the levels of parameters are very small - in fact, nations are not supposed to ' v a r y ' their coefficients more than 0.01%. Because both countries enter simultaneously the same state of stability, it is sufficient if one of them decides to control the chaotic pattern of its military expenditures in order to stabilize both of them and to reduce the danger of an outbreak of war. 6 The stabilization of the opponent's armament expenses is quite important, because the other nation's behaviour will get predictable, too. In particular, this model enables a nation to stabilize the situation of the arms race without arrangement with its competitor. Reality might be different. Without loss of generality, 7 we assume that nation X attempts to stabilize its military expenditures - and, consequently, the system - by adjusting one of its accessible system parameters. We explain the implementation of the OGY-algorithm for the parameter a below. Assume restricted adjustment, i.e. that the parameter a can be varied in a small range about some nominal value a0: a o - - A a < a < a o + A a . According to treaties,

6 Note, that it is not necessary to know the system's equations in order to control the system. For the application of the control algorithm the experimental time series are absolutely sufficient (see Ott et al., 1990). 7 We assume here that the control is always performed by nation X. However, the same procedure could be followed by nation Y as well.

D.A. Behrens et al. / European Journal of Operational Research 100 (1997) 192-215

207

arrangements and the budget, of course, the defense intensity cannot be changed drastically. Then nation X follows the two steps of the OGY-algorithm:

•

Given a parameter configuration of the arms race model that yields chaotic system dynamics, examine the unstable periodic orbits that are embedded in the chaotic attractor and select the one which gives the best system's performance.

In vector-form model (1) can be written in the following way:

~,+, = * ( ~ , ) = ~, +

(-° 0

0 / - 13 ~' + ' ~ ( ~ ' ) '

(5)

where ~t=(st) Y'

and

q~(~t)=(q~a(~t))=(a/(l÷e-c(x'-y',) ) q~b(~,)

b/(1 +e -'(x'-y'))

"

In order to calculate a period n orbit (recall that a period n point is a fixed point of the n-th iterate of system (1) but not a fixed point of any iterate smaller than the n-th iterate) one has to solve the following equation: ~F = t~n(~F) •

(6)

In particular we are looking for unstable periodic orbits which exhibit at least one stable direction. Since the map + is two-dimensional, the desired fixed point 'embedded' in the chaotic attractor must have the character of a saddle point.

•

Stabilize the trajectories by tailoring a small time-dependent parameter perturbation.

Let h s and h u be the stable and unstable eigenvalues (IX u [ > 1 > [k s [) of the map • at the desired fixed point ~0 which has been chosen in the previous step and let e~ and eu be the unit vectors in the stable and unstable directions of the fixed point. Assuming that the eigenvectors e s and e, are normalized, i.e. [ eu I = I e~ f = 1, one can find a pair of vectors fu and f~ - called contravariant basis vectors - that are perpendicular to the unstable and stable axis respectively: f~ • eu =f~ • e~ = 1,

f~. es = f ~ . eu = 0.

Near the fixed point ~ o we can use a linear approximation for the map t~(~t): ~t+1 ~-~~ '1- f ( ~ t - ~ ) ,

(7)

where f denotes the Jacobian matrix of the function ~ evaluated at ~ ° . Then we change the control parameter a slightly from its original value a 0 to some other value ~. The size of this change is limited by the predetermined value Aa. The fixed point coordinates will shift to some nearby point ~ . For small values of we approximate ~ in the following way:

Assume that ~, falls near the desired fixed point ~ o so that (7) applies. We then attempt to pick a parameter a t such that ~t+ ~ falls on the stable manifold of ~ ° . That is, we choose a~ such that f u ' ~ t ÷ i = 0 holds. This yields the following rule to compute at: 1

a,=

f.-o~

[ a0f~ • to - huf~. (~, - ~ - ° ) ] ,

(9)

which we use when a 0 - Aa < a t < a 0 + Aa. Otherwise, we set a t = a o. As soon as ~ + i falls on the stable manifold of ~ o we can set the parameter perturbations to zero, and, for subsequent time, the orbit will approach the fixed point at the geometrical rate k s.

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D.A. Behrens et al. / European Journal of Operational Research 100 (1997) 192-215

1,0

(a)

Fixed point ~F= ( [ ) = ~( 0J10!0"0918 )

0.$

(~) stable period I orbits

Q

( ~ stable period 2 orbits

0.6

(~) period doubling period 4 orbit for a=0.18

®

~ ) chaotic attractors

0.4

~ ) period halvening

®

0.2 0.0

i

0.1

0.15

0,2

I

I

,

I

0.25

I

I

I

0.3

a Fig. 8a. Mean values ~,, ~, for the expenditures xt, Yt on armament for varying a and fixing the other parameter values at et = 0.3, 13= 0.5, b = 0.4, c = 100.

Summing up, if we use the knowledge of how points move near ~ 0 under a change of the control parameter a, we can determine the parameter change that will move the trajectory to a point located on the stable manifold of the fixed point ~ o . Once placed on the stable manifold the trajectory naturally converges towards the desired fixed point ~ o so that the system can be stabilized near the saddle point ~ o o f the system. To illustrate the OGY-algorithm, outlined above, numerically, we start out from the set of parameter values c~ = 0.3, 13 = 0.5, a = 0.2, b = 0.4, c = 100 (Fig. 7a), where chaotic dynamics can be observed (the largest Lyapunov exponent equals L 1 --- 0.31). Assuming that nation X would like to avoid rapid unforeseen changes of its military expenses, it will try to adjust its armament level a to stabilize the relative share of its national budget, i.e. to settle down the chaotic trajectory on the stable direction of a fixed point embedded in the chaotic attractor. One possibility for nation X to yield predictability of its military expenses is to settle down the chaotic trajectory on the saddle fixed point of period 1 (see Fig. 8a). ~o~(0.0918

0.1101) T.

(10)

The eigenvalues of the Jacobian J at the fixed point ~ o are k s ~ 0.8729 and h u - - - - 2 . 0 4 6 8 with the normalized eigenvectors e'S --- (0.7333, 0.6799) and e'u ~ (0.4206, 0.9072) and the contravariant basis vector f~ -=- ( - 1.7924, 1.9332). To approximate nearby fixed points we calculate to from (8) yielding (0.1376, 0) v ~ v / ~ . Assuming that nation X can adjust the armament level a only by a certain percentage, 8, of its given value a 0 yields A a = 8 - a / 1 0 0 . 8 W e investigate the control of chaotic motion for several values of 8 (see Table 1 and Table 2). Now, as soon as ~t falls near the desired fixed point ~ 0 and a t falls in the interval a t - A a < a t < a t + A a , nation X can change its armament level according to the rule given in (9).

8 The size of the neighborhood and hence the time needed to reach the neighborhood of the desired fixed point depends on Aa, i.e. the maximal allowed perturbation of the control parameter.

D.A. Behrens et al./European Journal of Operational Research 100 (1997) 192-215

1.0

0.8

209

(b) Fixed point/~F=( ~ )= ( e~91e

8.1101 )'

"-'7:

( ~ stable period I orbits

Y

(~) coestlnce of stable period 1 and stable period 2 orbits

0.6

~ ) stable period2 orbits 0.4

",,'®i

®

i

0.2.

0.0

~ ) period doubl/ng period 4 orbit for 0~=0.29

!

,,

0.2

,,,,,

i.

i

®

i:,

( ~ chaotic attractors

(D

{ ~ period halvening period 4 orbit for ¢t=0.33

. . . . . . 2 ...... * . . . . . ":'--.=---,--

0.3

0.4

0.5

0[,

Fig. 8b. Mean values 2r,-Yr for the expenditures xI, Yt on armament for varying et and fixing the other parameter values at 13= 0.5, a = 0.2, b = 0.4, c = 100.

Fixing the parameters at the values of Fig. 7a (et = 0.3, 13 = 0.5, a = 0.2, b -- 0.4, c = 100), Figs. 8a and 8b show the mean-values over 1000 time steps of the expenses on armament of nations X and Y for a varying defense intensity a or fatigue-coefficient a respectively. We have also plotted the desired fixed point towards which the system is stabilized in the chaotic regions. 9 In case of Fig. 8a, the stabilization to the fixed point improves the system's performance in the chaotic regime drastically. Instead of = 17.7% nation X spends only = 9.2% of its national budget on armament, which means a decrease of the expenses of = 48.3%. Nation Y spends even = 57.7% less than in the uncontrolled case (21.2% in the chaotic regime versus 11.0% of the national budget at the stabilized fixed point solution). Possibly more interesting is the fact that the same result, the fixed point delivers, is obtained at a parameter value of a = 0.1785, which already belongs to region 3 (as indicated in Fig. 8a). This means a deviation of the parameter a of = 10.8% from its original value a 0 = 0.2. This change in the system is too huge to be done in practice. We get similar results for the fatigue-coefficient et, where we tried to stabilize the system to the same fixed point as given in (10). 10 Fig. 8b shows that a deviation of --- 13.3% from the original parameter value (a 0 = 0.3 is necessary to ensure 'peace and security' at the same level of expenses as in the controlled chaotic case. Figs. 9a and 9b show the time evolution of the military expenditures of both nations fixing the parameters at the values et = 0.3, [3 = 0.5, a = 0.2, b = 0.4, c = 100 belonging to the chaotic regions 4 and 5 of Figs. 8a and 8b respectively. The gray areas in the rectangles at the bottom of the pictures mark the regions where the control-algorithm is working, hence the parameters a and c( are varied. The width of these rectangles is defined by the m a x i m u m deviation A a = 0.015 and Act = 0.24 respectively, while the actual size of the deviation necessary to stabilze the system is drawn by black lines within the gray areas. In order to get rid of the unpredictable situation and to decrease its expenses on armament, nation X decides to switch on control - using the parameter a as a control parameter - at t = 150 and after 25 time steps the military expenditures are stabilized (Fig. 9a). This event is marked by the vertical dotted line in the gray 'control-area' of the rectangle.

9 Vertical lines in Figs. 8a and 8b mark off different regions of stability of the system dependingon the parameter values a and a. 10Stabilizationto orbits with higher periodicy does not improve the system's behaviour.

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D.A. Behrens et al. / European Journal of OperationaI Research 100 (1997) 192-215

0.5

(a)

xt m rt

...............

0,4

0.3

Xt

Yt

0.2 0.1 0o0

,

0

100

200

300

400

t

Aa

o.ols

0.000 ~

I

,

Fig. 9a. Control o f the chaotic trajectories plotted for the p a r a m e t e r values c~ = 0.3, 13 = 0.5, a = 0.2, b = 0.4, c = 100. Control o f a ( m a x i m a l deviation 0.015) is switched on for t ~ [150, 300].

os[ (b) 0.4

Y'

0.3I

0.0 0

100

...... 200

300

400

t

0.2 ¸ Aa 0.1

0.0 Fig. 9b. Control o f the chaotic trajectories plotted for the p a r a m e t e r values c~ = 0.3, 13 = 0.5, a = 0.2, b = 0.4, c = 100. Control o f o~ (maximal deviation 0.24) is switched on for t ~ [100, 300].

D.A. Behrens et al./ European Journal of Operational Research 100 (1997) 192-215

2l 1

Obviously, we need the maximum deviation only in the beginning of control in order to 'push' the trajectories towards the stable manifold of the saddle fixed point. After this initial period the deviation of the parameters necessary to stabilize the system decays rapidly and becomes almost zero, once the fixed point is reached. Theoretically, the system would now remain at the fixed point for ever and the control parameter would have its original value a 0 = 0.2 again. But since we are only able to determine the values of the control parameter a with finite precision, the nations can only reach a very small neighbourhood of the desired steady state. Moreover, the steady state is unstable, so that the military expenditures will only stay in the neighbourhood of this steady state for a limited time, and will escape afterwards. Therefore, when the control is switched off at t = 300 - the control mechanism stayed activated since t = 150 - the system falls back into chaotic motion after 10 time steps. Only by controlling the defense intensity a again, unpredictability of the arms race can be avoided. While controlling the defense intensity a, both nations spend a lot less on armament than without control as already illustrated in Fig. 8a. Predictability is obtained and the expenses on armament are lowered drastically. On the contrary, the stabilization of the system by control of the parameter c~ does not work. Since a ~ has to hold, we can change c~ no more than A a = 0.1, which is about 33.3% of the original value c~0 = 0.3. But as it is shown in Table lb, the control only works for Ac~ >/0.18, which already exceeds the maximum allowed range of variation. In spite of this limitation, we show what would happen if we were allowed to change the parameter c~ in the extended interval, in order to show that no improvement in the system's performance can be achieved. Fig. 9b illustrates the stabilization of the arms race model in the chaotic regime 5 of Fig. 8b, admitting a maximum deviation of Ac~ = 0.24 The problems are obvious. The time which elapses from the activation of control to stabilization is too long (75 time steeps). Even more serious is the fact, that the demanded deviation from the original value of the parameter a0 = 0.3 is too large in the initial phase of the control mechanism as indicated by the black vertical in the gray rectangle at time step t = 102. In fact, if we added this extreme A a to the original parameter value and created a new system with the fatigue coefficient fi = a + A a , we would yield much smaller solutions (and even solutions of period 1) than in the controlled chaotic case. Table la shows calculations for different maximum deviations 3 from the original parameter a = 0.2. Reducing A a in size leads to a longer time, which elapses from the activation of control to stabilization of the system, while the time until the system falls back to chaos remains the same. This is quite obvious, because a accidentially passing by trajectory can be 'caught' more easily when the permitted interval of deviation is larger, but the return to chaos happens independently from Aa - in fact after about 10 time steps for all values of Aa.

Table 1a Improvement of the s y s t e m ' s performance by control of the parameter a for the parameter values ~ = 0.3, 13 = 0.5, a o := 0.2, b = 0.4, c = 100, permitting 3 different values for the m a x i m u m deviation A a Maximum deviation from the original parameter value

ma

Number of timesteps elapsed from:

Improvement in % 1 0 0 " ( ~ a + ' ' a - ~r:) a

~a+Aa

~CHAOS activated

switch-

3 = - 100. a

control to stabilisat,

offof control to chaos

0.010 0.015

< 2.5 5 7.5

~ 35 25

9 I0

0.050

25

21

10

Aa

Improvement in %

Parameter value after stabilization

fi

( a - ~)

for Y

for X

for Y

controldoes not work in this region 0.1999 0.0065 48.3007 -//-//-//-

57.6528 -//-

69.8605 72.7092

68.2551 70.5711

-//-

-//-

78.3778

72.8882

~=

-//-

for X

100."""""7

-//-

a ~ + a a indicates the fixed point for the parameter ~ = a + Aa, where we have already left the chaotic region (region of period 2 orbits). Contrary, the time series calculated for fi = a - A a still remains in the chaotic region.

212

D.A. Behrens et al./ European Journal of Operational Research 100 (1997) 192-215

>..

+

e-

X 0

II >.,

w'-,

'.D

u~

I

i

~

-x

I

I

0

II o u I~

II .o

i ~J./~ ~

.

X I.

II

II

II t,o

'6¢-

oy,--

~,dYY E

0

0

c,

¢.

8 6 z

~g II "~

.r-

V

>.

~t <1

c5 ,::5 ,:5

D.A. Behrens et al. / European Journal of Operational Research 1O0 (1997) 192-215

213

The case Aa = 0.05 is not really relevant, because a change of 25% from the original parameter value is not realistic. If it was possible to change the parameter about more than 10.8%, we would be in the periodic regime of the system anyway (see Fig. 8a) and the system's performance would monotonically improve by reducing the parameter a in size. The case of Aa = 0.015 is the one illustrated in Fig. 9a. Additionally, the new value of the parameter, ~ = 0.1999, and its deviation of not even ~ = 0.01% from the initial parameter value a = 0.2, is calculated at the fixed point solution (10). One may argue, that, if we are allowed to vary the parameter up toa + Aa, we could simply create a new system with ~ = a + Aa. In this case the system would already be in the periodic range as indicated in Fig. 8a. Therefore we calculated the improvement in the system's performance if we set the parameter constantly equal to ~ = a + Aa. The system's performance in the controlled chaotic case, where a certain small deviation Aa from the original parameter is permitted, is even better than in the periodic case which represents a new system (see Table la, the last two columns). Remember that 'improvement' always means to spend less on armament without giving way to the threat of the outbreak of war. Similarly, Table lb illustrates the results of the OGY-algorithm if nation X tries to control its fatigue coefficient a and assuming different values for A a , which actually lie outside the admissible range of ~ (see discussion of Fig. 9b above). Contrary to Table la, if we created a new system with ~ = c~ + A a , the fixed point solution ~ is a good deal smaller than ~ 0 , because the mean-value-functions (in Fig. 8b) monotonically decrease for increasing a ; i.e. the system's performance in the controlled chaotic case is worse than in the case of the creation of a new system. Summing up, for the parameter set c~ = 0.3, 13 = 0.5, a = 0.2, b = 0.4, c = 100, the only possible way for nation X to ensure 'peace and security' and spend about half the money on armament is to control the defense-intensity a.

6. Conclusions Aim of this paper is to demonstrate that nonlinearity together with asymmetry in a Richardson-type-model, describing the arms race between two competing nations, lead to unpredictable time paths of the military expenses of both nations. In particular, if neither nation has an absolute advantage over the other nation, periodic, quasi-periodic and even chaotic system dynamics are possible to occur. Even if both nations are characterized by exactly the same parameters, slightly different initial levels of the military expenditures might lead to totally different evolutions of the military expenditures over time. Moreover, the level of the stationary values might be affected by the direction in which we vary the parameter of the system. Controlling the arms race model in the chaotic regime by means of the OGY-algorithm not only ensures stability, and hence 'peace and security' according to Saperstein (1992), but also lowers the expenses on armament drastically. Of course, the arms race is not limited to two competing nations only. For a three-dimensional system as well as for two- and three-dimensional models containing more distinct procurement functions see Behrens (1992, 1993). In the three-dimensional case the possibility of the outbreak of war is very high, if a submissive, a courageous and a super-courageous nation interact. 'Super-courageous' means that the country arms, if both other nations are superior to it. See also Behrens (1993) for a related advertising model where the system variables represent the fractions of the profit, which is spent on advertising and public relations by two competing companies. For further investigations it would be interesting to add a separated 'war model' to the arms race model. Of course these two models are related but nevertheless an arms race must not imply war, nor the other way around, though Richardson implicitly assumed that armament is responsible for the outbreak of war. These facts should be taken into consideration, too.

214

D.A. Behrens et al./ European Journal

Appendix 0

100 (1997) 192-215

A

andbE(0,

rfa~(O,al

~1, itholds thatx,, y,~[0,

For c + CC(large), the military expenditures function X

ofOperationalResearch

11 2

x,,,,

y,+, E[O, 11.

x,, ,, y,, , can be approximated by the piecewise linear

,+,=(1-c+,+

The behaviour of the decision-functions in (A.11 is more distinct than the behaviour of the armament functions used in model (1). Hence, if we are able to evidence x,, y, E [O, 11 * x,+,, y,+, E [O, 11for (A.l), then this distinctive feature will count for model Cl), too. 1) Assuming X, Q 1 it follows that (1 --a)x,+a=G(l x,+1 = i

(1 -,X)X,<

-o!)

+a<(1

-o)

-t-o= 1,

X,‘Y,,

(1 --oL) < 1,

==+X,+i
x, < Y,

Analogously assuming y, =G1 it follows that y,, , Q 1. 2) Since x,+ ,, Y,+, are linear combinations of non-negative terms it also holds that x,+ ,, y,+ , > 0. Together 1) and 2) imply x,+ ,, y,+, E LO,11. •I

Appendix

B

Calculation of the fixed point and investigation on its local stability The stationary points of model (1) can be determined by solving the equations x,, , =f(x,, Y,, I = g(x,, Y,) = Y,* This yields the equilibrium point (x * , y * > given by Y* =ba/(up)-x*, a~“(1 +e X*Wc-4W/WV) = a.

y,> = x, and

(B.1)

(B.2) Since the left hand side of (B.2) is continuous with the image ranging at least from 0 to a and since a E (0, al, the equilibrium ( x * , y * ) exists and is uniquely determined. Unfortunately, due to the exponential function, we are not able to calculate the fixed point (x *, y * > in an analytical way. It is only possible to compute (x *, y *> - for fixed parameter values - by means of the Newton algorithm. To figure out the system behaviour in the neighbourhood of the fixed point (X * , y *), we evaluate the Jacobian matrix at the point (x ’ , y * >. uc

Jl (x'.Y*)=

(1 +ebc

.Y')

e-c(x’

(l-o)+

c(x’-y’)

e -c(x'-y')

UC -

)”

(1

+

e-c(x.-Y’))2

e-C(x’-Y’)

(1 fe-

c(x’-y’)

bc

)’

(I-

P)

-

(l+e-

e-C(x’-Y’)

c(x’-y’)

)2

I

The eigenvalues A, and X2 of J Its*, ye) characterise the system-behaviour in the neighbourhood of ( x * , y *>. Therefore we calculate the characteristic polynomial P(A): + e-dX*-Y*))2 E [O, $1. men Let x:=ce -C(x’-Y’)/(l ~(~)=A~-[(1-~)+(1-~)+~(u-b)]X+[(1-ol)(l-~)+~(~(l-~)-b(l-~))].

D.A. Behrens et al./ European Journal of Operational Research 100 (1997) 192-215

215

The e i g e n v a l u e s are the solutions of P ( h ) = 0: hi'2 =

(2-a-13)+x(a-b) 2

_ 27(ct-

13) 2 - 2 x ( a + b ) ( e t -

[3) + x 2 ( a -

b) 2 .

Now, the m o d u l u s of the eigenvalues ( m o d X = I Xt = aV~-2-+ [32 , where X = ct + [3i is the eigenvalue) determines the stability of the stationary solution ( x *, y * ). hi, 2 E R: (or - [3)2 + ×2(a _ b)2 >>,2×(or - 13)(a + b). If h I and h 2 lie within the u n i t circle, ( x * , y* ) is a stable point; if they lie outside, ( x *, y * ) is unstable. If one of the eigenvalues lies within and one outside the unit circle, ( x *, y * ) is a saddle point. h 1,2 E C: ( a - [3)2 + x 2 ( a _ b)2 < 2X(Ct _ 13)(a + b). C o m p l e x eigenvalues imply oscillating trajectories c o n v e r g i n g to the fixed point or m o v i n g away from it - d e p e n d i n g on the following conditions: •

If ~/(1 - ct)(1 - 13) + X( a(1 - [3) - b ( 1 - or)) < 1, the trajectories are decreasing - approaching the fixed point.

•

If ~ / ( 1 - et ) ( 1 - 1 3 ) + × ( a ( 1 from the fixed point.

13)- b(1-e

t ) ) > 1, the trajectories are increasing - m o v i n g away

Acknowledgments C o m m e n t s of C. H o m m e s , M. Kopel, A. Milik and 3 a n o n y m o u s referees are gratefully acknowledged.

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