As so far developed, the conflict helix¹ is a soft process model interrelating cooperation and conflict, order and disorder, peace and war. Its fundamental concepts should be clear enough; its social field context has been mathematically modeled², and its propositions have been tested successfully against empirical findings in the literature³. But the helix itself is only a conceptual and analytical framework, lacking in sufficient precision as a model of the conflict-peace process to be directly operationalized and tested. This easily will be seen in the description of the helix, below.

My purpose here, therefore, is to provide a mathematical model of the helix that fits within the vector space of social field theory. The vehicle will be catastrophe theory, specifically the butterfly model. This model will be fully developed and then tested against the annual intensity of conflict between India and Pakistan, 1948-1973.

The social field consists of actors (individuals and groups) and their diverse interactions, all located in a sociocultural space-time. Social energy is conveyed throughout this space by a medium consisting of the actor's sociocultural meanings, values, and norms; and through and within this medium, social and psychological field forces are activated and oriented, in equilibrium and disequilibrium, and thus define the major stages of the conflict helix. What field forces affect what behavior in what way is a function of the projection of actors onto the field's space-time components (dimensions).

How then is behavior defined in the field and as part of the conflict helix? First, the disposition of an actor to behave toward another in a certain way within a given situation g is a function of the actor's interests and capabilities regarding another, his situational perceptions, and his unique background and will. That is,

Equation 1:

w_{g,i j} = _gid_,i-j + Q_g,ij,

where

w_g,ij = the behavioral disposition of i towards j in situation g;

_gi = i's perception of the salience in situation g of his interests and capabilities to j;

d_,i-j = i's interests and capabilities regarding j as reflected along the distance vector from j to i on the th common component of sociocultural space-time

Q_g,ij = i's unique interests and capabilities regarding j in situation g and unique will.

Theoretically, situation g is a specific conflict helix between i and j, and a specific equilibrium in the field's social forces.

As a disposition, w, is a psychological pressure toward particular behavior and not manifest behavior itself. The latter depends on an actor's overall dispositions toward another across the various situations involving them, each disposition weighted by his situational expectation of the outcome of actually so behaving. To wit:

Equation 2:

_h,ij = _ghigw_g,ij,

where

_h,ijactor i's specific common behavior h (such as a threat) toward actor j;⁵

_hig = actor i's expectation of the outcome of behavior h in situation g.

And substituting Equations 1 into 2 we then have a fundamental equation of field theory,

Equation 3:

_hij = _ghig(_gid_,i-j + Q_g,ij).

The space-time components are operationalized through super-P factor analysis (on factor analysis, see "Understanding Factor Analysis"); the different situations are defined by the different canonical equations (variate pairs) of canonical analysis; the fit of the equations to data is measured by canonical analysis (canonical and trace correlations); and (expectations) and (situational perceptions) parameters are the left and right side coefficients of each canonical equation.⁶ Time is a dimension of the social field.⁷ These equations are fully operationalizable and have been tested by different people, on different dyadic samples, and for different periods.⁸ They and the associated field conceptualization and philosophy underlie the conflict helix.

To be specific now, and for simplicity to assume a dyadic relationship in being and continuing, there are five key concepts that frame the conflict helix: balancing of powers, balance of powers, structure of expectations, gap, and disruption. The first is that of conflict as a mutual balancing of powers (the plural denotes the family of powers: force, coercion, bargaining, authority, etc.) within a conflict situation. Social conflict serves to communicate and to adjust the parties' opposing interests, capabilities, and wills. And relevantly here, conflict manifests the second equilibrium in the social field--it is that stable state between forces actively trying to overcome each other (as will be seen, one virtue of the catastrophe theory model will be to make this hazy idea mathematically explicit). While momentarily stable (for hours if family quarrels; for many years if large scale internal and international war), the equilibrium is under varying strain to move in one direction or another toward a new stable state.

The outcome of a social conflict is such a new, situationally defined equilibrium consisting of a balance of powers and a structure of expectations. The previous equilibrium grew increasingly unrealistic as a gap formed between expectations and power. In the new equilibrium expectations have been made congruent with power: Conflict has determined a new adjustment--a new balance--between what the parties want and are willing and able to pursue; and based upon this balance of powers, conflict also defines a new structure of expectations governing the situation, a complex of implicit and explicit understandings, agreements, rules, contracts, treaties, and the like, that will regulate the relationship between the parties. This structure of expectations is the base of cooperation; the definition of the social order between two parties in a situation. Overall, the spread and hierarchy of such structures across situations within an overarching structure of expectations defines the social order we call a society, including the international system. Peace is thus a social contract⁹ based on a balance of powers, the second kind of equilibrium in the social field.

Conflict and cooperation, war and peace, disorder and order, are thus two different kinds of (situational) stability in the social field. Each comprises or manifests different types of behavior and a different kind of equilibrium between social forces; and the transition from one to the other is usually a jump--a discontinuity--in social behavior.

In time, a structure of expectations and its underlying balance of powers normally will become misaligned. The structure of expectations, especially that involving the status quo (that delineates who owns and controls what), has a high social viscosity. Once determined through conflict, an agreement over vital interests is difficult to change significantly, although of course reinterpretations of its provisions will evolve. However, the supporting balance of powers can change overnight as the situational interests, will, or capabilities of one party or the other undergoes a radical shift (as through a coup, election, assassination, demonstration of a new weapon, defeat in battle by a third party, etc.). A Marxist coup in Saudi Arabia, for example, would suddenly and radically change its balance of powers with the United States, leaving their mutual structure of expectations hanging in air--without a base in congruent mutual interests and wills.

Even without so radical a change, the balance of powers and associated structure of expectations usually will become more incongruent in time, as what the parties want, will, and can do will diverge from that originally supporting the structure of expectations. Whether sudden or gradual, however, the result is to cause a gap between expectations and the balance of powers, as shown in Figure 1. This gap creates tension, and a force toward conflict; the greater the gap, the more pressure toward restructuring expectations in line with the change in interests, capability, and will; the more this pressure, the more likely some trigger event will disrupt the outmoded structure of expectations, precipitate a jump in cooperative, peaceful behavior to conflict and, if the status quo be involved, possibly social violence or war. Figure 1 illustrates this disruption manifested by conflict behavior and shows the overall process through several conflicts and structures of expectations.

The process of jumps between successive equilibria within a situation, of social contract conflict social contract conflict, is a winding upward in mutual learning and adjustments. As long as the major conditions of a relationship remain fairly constant, this process leads toward longer lasting and deeper peace, interrupted by shorter and less intense social conflict. A sharp change in these conditions can set this process back, or even shear the conflict helix, as shown in Figure 2, and cause the whole process of mutual adjustment and learning to begin anew. What changes will have this affect will depend on the parties involved. Between husband and wife, it might be the first child or the mother-in-law moving in; between business and government, a severe depression or leftist revolution; between two states, a reversal of the military balance or a radical change in the government of one. For example, the process of adjustments between the dictator of Cuba, Batista, and the United States was completely ended by Communist Castro's taking power in 1959. An entirely new process of adjustment had to begin; all evidenced by the great jump in Cuban-American conflict.

To summarize the essential points:

• the behavior of an actor toward another in the social field is dependent on field forces defined by Equation 1;

• this behavior is either peaceful, reflecting social contracts resulting from conflicts; or conflictful, resulting from the breakdown in these contracts;

• behavior thus manifests different equilibria of forces in the social field;

• While stable, within each equilibrium an actor's dispositions change, causing a pressure toward a new equilibrium (from conflict to social contract; or social contract to conflict) to build up;

• the larger the pressure, the more likely a trigger event will suddenly upset the equilibrium, enabling a rapid readjustment among the social forces involved and causing a jump in behavior from peaceful to conflictful and vice versa.

While Equation 3 captures part of the overall variance of these processes across situations in the space-time of nations, I have not described in mathematical detail the conflict helix between two actors. The problem has been that of treating conflict and peace as distinct equilibria, with a jump between them. While there are techniques for handling such discontinuities (such as including in the equation a zero-one interaction term), these are ad hoc. What I required was a model both sufficient to describe the equilibria and jumps of the conflict helix, and consistent with the vector space-time framework of field theory. Catastrophe theory provides this.

There are five aspects of the conflict helix that make catastrophe theory an appropriate model. First, the conflict helix involves different equilibria and jumps between. Second, there is an underlying disposition--power--toward maintaining an equilibrium. Third, the resulting distribution of behavior is bimodal: the same values for the underlying conditions can produce conflict (including violence) or peace, depending on the situation and existence of a trigger. Fourth, under certain conditions, a slight change in an underlying condition can cause a jump from one equilibrium to another, and thereby a radical change in behavior (as from peace to violence). And fifth, the history of the process of conflict is critical to the behavior involved; the same values on the underlying conditions may produce different behavior as the process moves from violence to peace than when it moves from peace to violence. Now, I will try to clarify these aspects and in doing so end up with the cusp manifold of the butterfly catastrophe.

To begin, the aim is dynamically to model a particular conflict helix ij, its forces and equilibrium, and the jumps in behavior _hg,ij. By Equation 2 this behavior is dependent upon the interaction between the associated actor's situational expectation _hig, and disposition w_g,ij toward j. And this disposition is itself a resolution of many different dispositions of ij, each weighted by i's situational expectations of the outcome of manifesting that disposition. Specifically, the static relationship is:

Equation 4:

_hg,ij = _higw_g,ij = _kgikw_k,ij,

where

_hg,ij = i's common behavior toward j in situation g (conflict helix g);

w_g,ij = i's overall behavioral disposition toward j in situation g;

w_k,ij = i's general disposition to behave in k-way toward j (independent of any situation);

_gik = i's expectation of the outcome of behaving in k-way toward j in situation g.

The above equation has been operationalized and evaluated for selected directed dyads.¹¹ Each situation g (and thus each different conflict helix) for an actor i is defined by each canonical equation emerging from a canonical analysis of dyadic behavior's dependence on space-time distances ij. The _gik are then the left side canonical coefficients of the equation; the w_k,ij are the input common behavioral factors of (theoretically) a common factor analysis of manifest dyadic behavior. Therefore, although apparently very abstract, we can operationally move from manifest behavior to Equation 1, to Equation 4, fully evaluate them empirically, and move back to predict future behavior (_h,ij)

So far, conceptually, an actor has general dispositions to behave toward another in a certain way due largely to their relative interests, capabilities, and wills. In a particular situation, however, the actor's expectations of the outcome of manifesting these different dispositions will encourage some of them, discourage others, the result being one overall situational disposition to behave in a certain way. Therefore, for each situation involving the two actors each will have a situational disposition toward the other. Finally, a specific situational behavior of one to the other is then the outcome of this disposition weighted by his expected consequence of such behavior.

Now, to simplify the model for this initial effort, restrict h in Equation 4 to conflict behavior, such that the total conflict behavior in situation g of ij is some scale,

Equation 5:

S_g,ij = _ccgc,ij,

where

S_g,ij = some linear scale measuring the total conflict behavior of ij in situation g;

_c = weights that fit each type of conflict behavior k to some conception of total conflict;

_gc,ij = the common conflict behavior c in situation g of ij;

c = a conflict behavior from the subset of conflict behaviors 1, 2, . . . , c, . . . of behaviors 1, 2, . . . , h, . . . .

Moreover, consider only those conflict behavior dispositions w in Equation 4 that underlie the overall disposition ij to conflict:

Equation 6:

w^*_g,ij = _fgifw_f,ij,

where

w^*_g,ij = the disposition of i to behave conflictfully toward j in situation g;

w_f,ij = i's general conflict disposition to behave in f-way toward j;

_gif = i's expectation of the outcome of behaving in f-way toward j in situation g;

f = a conflict disposition from the subset of conflict dispositions 1, 2,. . . , f, . . . of dispositions 1, 2, . . . , k, . . . .

Then

Equation 7:

minimum f(S, w^*),

= S_g,ij - w^*_g,ij if S _g w^*_g

= w^*_g,ij - S_g,ij if S _g *_g

This is a fundamental assumption in the helix and for the catastrophe model. Within any helix, there is a strong psychological pressure for an actor to reduce the disparity between behavior and disposition: to increase conflict behavior when the disposition is more aggressive, to reduce conflict behavior when it is more pacifistic. A disposition is a psychological, not social or political, variable. Its power comes ultimately from the sources of energy in the leader's psychological field, the needs that underlie his active interests.

S_g defines total conflict, but the concern here is with the dimension of total conflict, namely intensity, especially that ranging from nonviolent conflict behavior like accusations and threats, through low-level violence, to total war. Denote this intensity by X and assume, not unreasonably, that each conflict behavior can be reduced to some function of X and modifying conditions of the specific situation (conflict helix):

Equation 8:

_1,ij = ₁₁X + ₁₂X² + ₁₃X³ + . . . + _1mX^m + . . .

_2,ij = ₂₁X + ₂₂X² + ₂₃X³ + . . . + _2mX^m + . . .

. . . .

. . . .

. . . .

_c,ij = _c1X + _c2X² + _c3X³ + . . . + _cmX^m + . . .

. . . .

. . . .

. . . .

where

X = an intensity dimension of conflict behavior;

_cm = m conditions of the situation g for i to j modifying the conflict intensity of i's conflict behavior c.

Then, by Equation 5

Equation 9:

S_g,ij = _mmXm

where

_m absorbs the _c and _cm coefficients of Equation 5 and Equation 8; and defines the conditions 1, 2, . . . , m, . . . of the i - j conflict helix that modify the intensity of i's overall conflict behavior toward j.

That is, the total conflict of ij within a conflict helix is a polynomial function of the conflict's intensity and helix's conditions.

For simplicity, assume that S_g w^*_g. Then from Equations 6, 7, and 9,

Equation 10:

f(X, w₁, w₂, . . . , w_f, . . .)

= ₁X + ₂X² + ₃X³ + . . . + _mX^m + . . . - _fgifw_f,ij

This gives us the bridge between the conflict helix and catastrophe theory. It says that within a helix the course of conflict of one actor toward another will depend on its intensity, conditions of the relationship, i's underlying disposition to behave conflictfully toward j, and the inhibiting or aggravating effect of i's expectations of the outcome. Most important, this behavior will tend to match dispositions.

Of course, there are other dispositions, such as cooperative ones, that behavior also tends toward, so that the overall behavior ij is like the balance between rubber bands stretched between four poles, each pole at the corner of a square, and the bands tied together near the center. Each band pulls back toward its own pole, while their interconnections form stable equilibrium between them. Slowly move one pole back away from the center, however, and its band will stretch further and pull the equilibrium among the bands in the pole's direction. Eventually, that or the opposite pole's band should break, causing the whole equilibrium to jump in one or the other pole's direction and suddenly to reform. The poles are like the dispositions; the placement of poles, size of the bands, and their interconnections are the conditions; behavior is that equilibrium among the bands; and the energy of the stretched band is like the psychological force pulling behavior toward dispositions.

Within the helix there are two such equilibria of forces: that constituting the social contract--the structure of expectations manifest in cooperative--peaceful behavior--; and the other being the balancing of powers resulting from a disruption of this structure and displaying conflict behavior. In both cases Equation 10 holds. As conditions, expectations, dispositions, and behavior shift within a structure of expectations (one equilibrium) or conflict (the other equilibrium), the minima of Equation 10 may change and the difference between behavior and disposition greatly increase, until some trigger event causes the whole equilibrium to jump to a new balance, suddenly bringing behavior much closer to dispositions.

The problem now is to reduce Equation 10, a function of X and many dispositions w, to an equivalent function of X alone. Doing this requires that the conditions be clarified. To wit, what are the conditions of a conflict helix affecting the intensity of conflict behavior?

Considerable work has been done on this, and the results can only be summarized here.¹² Theoretically and empirically, the intensity of conflict between i and i is dependent upon four primary conditions: the transition in power between them, the gap between their structure of expectations and balance of powers, their Joint freedom,¹³ and their ability to project power across the geographic distance between them. Specifically, then, the four conditions and their definitions are the following.

Power transition: the degree to which capabilities are shifting toward the actor who is dissatisfied with the status quo (the core of the structure of expectations).

Gap: the incongruity between the structure of expectations and balance of interest, capabilities, and wills.

Joint freedom: the degree to which i and j both have internal civil rights, political liberties, and economic freedom; the less such joint freedom, the more i's expectation of violence if the status quo is disrupted; if i and j are both liberal democracies, there is no such expectation of violence.

Joint power projection: the capability of i and j to project power across the geographic distance between them, as defined, for example, by the inverse "social gravity" function G²_ij/P_iP_j, where P is some measure of power to transcend distance and G is the geographic distance.

Now, we are concerned with the minima of Equation 10, the critical points of the conflict helix. Then, to study the behavior of Equation 10 locally, in the neighborhood of such critical points, we can use Thom's theorem (the core of catastrophe theory).¹⁴ For the four conditions and our purposes here, without loss of generality Equation 10 reduces to the structurally stable butterfly catastrophe,¹⁵

Equation 11:

f(X) = ₁X + ₂X² + ₃X³ + ₄X⁴ + X⁶

Equation 12:

0 = df(X)/dx = ₁ + 2₂X + 3₃X² + 4₄X³ + 6X⁵

For given conditions ₁, . . . , ₄, Equation 12 defines the values of the conflict intensity X that reflect an equilibrium of social forces in the field. There may be one value of X defining one stable minimum, or as many as five values defining three stable minima (and two unstable maxima). Of most interest here is when more than one minima exists, for this provides a way of modeling the jump from a peaceful equilibrium to a conflict one and vice versa--as will be made clear below.

For the moment assume that solving for the roots of Equation 12 gives one minima. This can be interpreted as a point (X, ₁, ₂, ₃, ₄) in a five-dimensional Euclidean subspace of the social field. This is critical, for it incorporates the butterfly model of the helix within the Euclidean vector space of the social field, wherein by theory the helix exists. Now, within the helix the conditions ₁, . . . , ₄, will vary in time, and as they vary they will change the minima (roots of Equation 12) defining peaceful or conflictful equilibria, creating a smooth surface, a manifold, in this space. Any curve drawn on this surface will consist of each of those points that describe the stable equilibrium values of the intensity of conflict for certain conditions ₁, . . . , ₄, expectations, and dispositions. The curve itself will describe the change in the intensity of conflict with the change in the equilibrium of social forces underlying the conflict helix.

To illustrate this, and especially the manifold when more than one minimum exists, we can use the simple three-dimensional cusp catastrophe. To reduce the five-dimensional butterfly catastrophe to a three-dimensional cusp, let ₃ = ₄ = 0. In Figure 3 we can plot the manifold M for intensity X and conditions ₁ and ₂. The projection of this manifold on the (₁, ₂) plane is shown below M (the interpretation and direction of ₁ and ₂ will be justified subsequently); and conflict intensity X is the vertical dimension.

Assume now that the change in conditions ₁ and ₂ for a conflict helix, ij, is shown in Figure 3 by the points P₁, . . . , P₈, each a minima of X, each a field equilibrium. The corresponding shifts and jumps in conflict intensity are then a projection on X. Note that the jumps from nonviolent conflict to violence, and from war to peace, constitute a sudden move through the fold of the manifold as its lip is approached, a move from one equilibrium surface to the other. In this way the jumps in the conflict helix are modeled.

A Maxwell rather than delay convention is used here. The delay convention would assume that jumps occur only at the lip of the fold, whereas for the Maxwell convention the jump may occur before the lip is reached. In the helix, the move toward the lip of the fold is largely due to an increasing gap between expectations and power, a move increasingly manifesting strain and tension between i and j. The closer to the lip, the more likely some trigger event will precipitate the breakdown of the structure of expectations and catalyze conflict behavior. Similarly, once war has succeeded in displaying a new balance of power, the disposition to conflict decreases, conditions move toward peace (from the underside of the fold) until some trigger event (e.g., a significant battle lost, as for the French defeat in 1954 by the Vietminh army at Dien Bien Phu) induces the final negotiation, and peace. Keep in mind that what holds the course of behavior ij to the surface of this manifold is Equation 7--the strong pressure for behavior to match underlying dispositions.

The region of the fold, as shown projected onto the (₁, ₂) plane, is where more than one stable minima exists (the surface between the top and bottom sheets of the fold is where X has unstable maxima); it is the bifurcation set in which X can have two stable minima (top and bottom sheets), and thus for some specific ₁ and ₂ X can have either of two values. That is, X can be bimodal. This is not only consistent with the conflict helix, it is a property of the model recommending it as a representation of the helix.

As the bifurcation set is oriented in Figure 3, ₁ would be interpreted as the power gap condition underlying the equilibria; ₂ as the power transition condition. And, as mentioned, an increasing gap between the structure of expectations and balance of powers increases the actor's disposition to conflict and thus moves the helix toward a disruption in the structure of expectations and a jump in behavior; that is, toward the lip of the fold in M. While ₁ is the disequilibrating condition, ₂ increases the intensity of conflict for a given gap. That is, the more the actor who is dissatisfied with the status quo approaches and exceeds the capability of the status quo supporting actor, the more intense the disposition toward conflict for a given gap. This is shown by the two-way tilt of M, one downward toward an increasingly intense conflict as ₁ enlarges; the other the dip toward the high ₁, high ₂ corner, the region of the most severe violence and war.

This interpretation of ₁ and ₂ is provisional, however, and depends on the empirical location and orientation of the cusp-lips for a specific dyad. If in reality the bifurcation set is rotated in Figure 3 such that increasing ₂, not ₁, moves behavior more directly toward the lips, ₂ should then be defined as the gap and ₁ as power transition.

Now to bring in the other two conditions, joint freedom and joint power projection. The degree of joint freedom between i and j will limit the range of conflict behavior, and for two liberal democracies exclude war and make official violence most unlikely. This influence is consistent with that of ₃ on the cusp as illustrated in Figure 4. Increasing ₃ (i.e., decreasing joint freedom) tilts the cusp toward more intense conflict, lowers the whole surface so that even peace is less secure, and reshapes the fold so that the jump to greater conflict or violence is easier, and that to peace is harder.

Finally, ₄ is interpreted as the joint power projection condition. Consistent with this interpretation, as the geographic distance between i and j decreases (relative to their power projection capabilities) the whole manifold M is lowered toward more intense conflict, the region of peace is decreased, and the jump in behavior is greater. That is, geographic distance is pacifying, especially between small powers.

With this background, Figure 5 charts the full five-dimensional butterfly model for varying conditions ₁, . . . , ₄. The (₁, ₂) plane of the manifold M is shown for certain points (₂ = 0, ₃ = -10), (₄ = 10, ₃ = -10), (₄ = 0, ₃ = 0), etc. The projection of M on X at (₁, ₂, ₄, ₃) is given by the numbers on the (₁, ₂) plane. These are the rotated roots of the simpler, but topologically equivalent equation to Equation 12,

0 = ₁ + ₂X +₃X² + ₄X³ + X⁵

where:

- 50 ₁ 50

- 50 ₂ 0

- 10 ₃ 10

0 ₄ 10

with the angles of rotation being 1º, -7º, and 7º in the (X, ₂), (X, ₃), and (X, ₄) planes, respectively. The purpose of the rotations is to tilt M in Figure 5 best to fit the theoretical relationship between the critical values of conflict intensity X and the conditions ₁, . . . , ₄ of the conflict helix.

The so-called pocket, a third sheet of stable minima that intrudes between the upper and lower sheets of the cusp, appears for certain negative values of ₄. The shape of this pocket is like a butterfly and gives the whole catastrophe its name. After some investigation of its qualities, however, I do not see this pocket of use in modeling the conflict helix.

It is possible to conceive of the three sheets as representing peace or cooperation, nonviolent conflict, and violence. But as the values on the four conditions are varied, the changes in the manifold around the pocket do not correspond to what should theoretically occur for the conflict helix. However, this pocket is only one region of the butterfly manifold. Elsewhere, the manifold also forms a cusp and it is this region of M that theoretically best fits the conflict helix for

Equation 13:

±₁

-₂

±₃

+₄

This region defined by Equation 13 contains a cusp like Figure 3, that changes its shape and width with varying ₃ and ₄, as plotted in Figure 4 and Figure 5. The bifurcation set for this cusp is pictured for each (₁, ₂) plane in Figure 5, with the two (rotated) minima of X (points on the top and bottom sheets of the fold) given for specific values of ₁ and ₂. Thus, for the top left plane in Figure 5, where ₁ = 0, ₂ = -50, the two (rotated) minima are 17 and -5. The former number is the projection of the top sheet of the cusp onto the conflict intensity dimension X; the latter is a projection of the bottom sheet. The edge of the bifurcation set corresponding to the upper lip of the fold of the cusp is shown as a thick line.

It should be clear that Figure 5 is a (to borrow the label often applied to topology) rubber sheet model, and that the actual position and rotation of a cusp for the history of a specific dyad may differ considerably from the picture. Figure 5 only fixes the cusp for a specific scaling of the conditions and X, and for a specific Equation 11. However, in application the scaling is arbitrary and a matter of convenience or design (such as to place the cusp point at the origin); and the equation represents a stable family of functions, any one of which would define a manifold topologically equivalent to that in Figure 5, but each with a somewhat different cusp.

If this all seems too imprecise a model, then give thought to the often employed regression model. The regression equation

y = b₀ + b₁x + b₂x² + b₃x³ . . . + e

also represents a family of functions, including

f(x) = b₀ + b₁x + b₂x² + b₃x³ . . . + e

And the location and orientation of the regression plane is dependent upon both the covariances of y with the dependent variables and the scaling of the data--itself usually arbitrary in the social sciences.

Recall that the conditions ₁ and ₂ are left ambiguous in Figure 3. Which of these conditions in Figure 5 is the gap and which the power transition is thus undefined. Were the reality of a dyad in line with the general east-west tilt of the cusp lips shown in Figure 5, then ₁ would be labeled the gap, since increasing or decreasing ₁ would most directly move behavior toward the cusp's lips; ₂ would be the power transition.

Finally, to illustrate how this manifold models conflict intensity, a possible region of war, non-war violence, nonviolent conflict, and peace is delimited on each of the (₁, ₂) planes in Figure 5. As we move from far to close (₄), the region of peace decreases and that of war increases.

In sum, then, the catastrophe model of the conflict helix is defined by Equation 11 and Equation 12, with restrictions of Equation 13. The geometry and substantive interpretations of the model are shown in Figure 3, Figure 4, and Figure 5.

To determine the testing model, we can utilize the relationship between Equation 12 and catastrophe dynamics (Cobb, 1981b, p. 59, Eq. 17),

Equation 14:

dx/dt = -df(X)/dx

Thus,

Equation 15:

0 = dx = -(₁ + 2₂X + 3₃X² + 4₄X³ + 6X⁵)dt

Let dt = 1, the interval for our time data, transform and X appropriately, and insert regression coefficients b to get the statistical regression model,

Equation 16:

Z_t+1 - Z_t = Z

= b₀ + b₁^*_1t+ b₂^*_2tZ_t + b₃^*_3tZ ²_t + b₄^*_4tZ³_t + b₅Z⁵_t + e_t

where

Z = (lnX - lnX_min)/s_x and lnX_min is the smallest value of conflict intensity X, and s _x is the standard deviation of In X for i - j over the time period's range;

^*₁ = (₁ - _{1, min})/s₁, where _{1, min} is the smallest value of ₁, and s₁ is the standard deviation for i j over the time period's range;

^*₂ . . . ^*₄ = same as for ^*₁;

b₀ = intercept;

b₁ . . . b₅ = regression coefficients;

e = residuals.

The In transformation is applied to the intensity scores on X to reduce the effect on the distribution of very extreme scores. Particularly, the intensity scores for the Indian-Pakistan wars of 1965 and 1971 would have virtually washed out the variance in intensity for other years. The transformation of Z and ^* helps better to delineate the cusp by putting the origin of the space at the data's minima and standardizing the data around it. And the multiple correlation coefficient squared will measure how well the data fit the butterfly model.

The intensity of conflict (InX) and In of cooperation intensity for the two dyads are shown in Figure 6 (6a and 6b) for 1948 to 1973 (for Pakistan India, the conflict intensity was zero in 1960; to get In = 0, a 1 was inserted for this year).

The data are the DI-conf and DI-coop scales from Azar and Sloan (1975), which constitute summations of annual event data weighted for intensity. I am not entirely happy with their weighting scheme, which for example accords a weight of 16 to an accusation and 102 to a war. This means that seven accusations together exceed any one war in intensity--questionable, indeed.

Since this weighting scheme thereby compresses the underlying intensity continuum and mixes up at any given scale level of intensity events that are really at different levels of intensity, some random noise is created in the measurement. When this is conjoined with the substantial random invalidity/unreliability variance of such event data collections, then whatever results of our tests are significant on these data should in reality be even more significant. In any case, these data should be good enough for our initial tests here, and they have the virtue of being defined and collected independent of these tests, and available.

To turn now to the conditions , the power transition is the difference in capability between the anti-status quo and status quo actors. This is measured here as the annual difference in defense expenditures (in $U.S.) between India and Pakistan. India, by virtue of its actions in Kashmir and regarding East Pakistan before the 1971 Indian-Pakistan war, is assumed the anti-status quo actor. Thus, Pakistan's defense expenditures are subtracted from those of India to get the power transition. The defense expenditure data are given in Table 1.

The gap between the structure of expectations and the balance of powers is the most difficult to measure. An approach is suggested, however, by recalling that the structure of expectations underlies cooperation on the one hand, and on the other the balance of powers consists of capability, interests, and wills--as these three elements change, so does the balance. Initially, the structure of expectations fits this balance of powers, but as this balance alters in time, the fit between expectations, and thus cooperation, becomes less. Therefore, this gap will be operationalized and measured as the residuals of the regression fit of cooperation to some measures of this balance:

Equation 17:

Â_t,ij = b₀ + b₁P_t,ij + b₂V_t,ij + b₃D_t,ij

A_t,ij - Â_t,ij = e_t,ij = _2,t

where

A_t,ij = the In intensity of i's cooperative behavior to j at time t;

Â_t,ij = the regression estimate (best fit) of A to P, V, and D;

P_t,ij = political distance: an accumulative measure of internal political changes and events in i and j that may effect the interests and will of i and j at time t;

V_t,ij = the foreign interests/policy distance between i and j, as measured by the correlation between their UN General Assembly votes at time t (on the correlation coefficient, see Understanding Correlation);

D_t,ij = the difference in defense expenditures i - j at time t;

e_t,ij = regression residuals = the gap at time t.

The data on cooperation A are given in Table 1, along with that for the independent variables. The political distance P is the same for both dyads, and its measurement is presented in Appendix 1. The regression statistics for Equation 17 are given in Table 2, and the residuals (gap) e in Table 1. Figure 7 (7a and 7b) plot the gap for each dyad against its In conflict intensity, where each time series is smoothed by a two-year moving average and for comparison transformed to a 0-15 scale.

For the joint freedom condition, ₃, a 1 to 7 rating was done separately for India and Pakistan, and then combined into a rank-scale of joint freedom. The measurements are described in Appendix 2 and the result is given in Table 1.

Finally, for ₄, joint power projection, India and Pakistan had a very long border along West and East Pakistan (until the 1971 war). We can therefore set ₄ to zero and drop b₄^*₄Z³ from Equation 16; which is equivalent to restricting the test to the left side (X, ₁, ₂, ₃, 0) of Figure 5.

Since the tests are to be conducted on time series event data, subject to both year-by-year specific effects and annual data source/collector variation, all the data have been smoothed by a two-year moving average. Although a statistician might argue for more smoothing than this, for my taste applying a three or five-year moving average reduces too much of the annual contrast between nonviolent conflict, violence, and war.

The next question, then, is whether the direction of relationship between ₁, ₂, ₃, and X accord with theory. To be clear about this, Table 3 presents the direction of scaling in the data; and Table 4 gives the theoretical versus actual direction of relationship between X and the helix's conditions.

Except in the case of ₃ and X for Pakistan India, the signs match the theory, with a p .11 against the random null model (binomial test with a probability of "success" at .5). However, the one contrary result is hardly meaningful, since at .03, a slight random perturbation of the data could bring the sign in line with theory (then giving a binomial test of p .016). If this unstable case is eliminated, the results are binomially significant at p .03.

On this score it is also helpful to look at the plots of ₂ against In conflict in Figure 7 (7a and 7b). Although the overall correlations are low, the movement of ₃ in the region of the two wars (1965 and 1971) is one would theoretically expect: before war the gap sharply increases, reaching its peak at about when war occurs, and steeply declining thereafter. As also should be the case, the gap is smallest immediately after war. Moreover, in evaluating the results for ₃ here and subsequently, one should note that joint freedom theoretically ranges from 2 to 14, while that for India and Pakistan dyads varied from 5 to 8 for 1948 to 1973, or within 25% of the range. With such a limitation of the range for this condition, it is easy to understand the low correlations of -.12 and .03. However, we can go beyond these results to check for the assumed relationship between ₃ and X. Elsewhere ("Libertarianism and International Violence"), I tested for this theoretical relationship of ₃ and X for cross-dyad data on all cases of war (1816-1980), threat and use of force (1945-1965), and conflict and violence (1976-1980). For all cases, the correlations were in line with the butterfly model and highly significant.

Considering, then, these results for ₃ and those for ₁ and ₂ in Table 4, the empirical data support the theoretical directions of relationship.

With all this as background, we can now test the butterfly model itself. Table 5 represents the relevant statistics. The model accounts for 42 to 47% of the variance in the change in conflict intensity, 1948 to 1973. This is a good fit by itself, but even more substantial when the previously mentioned nature of the underlying conflict and cooperation data is considered.

Using the F-test, the results would also be highly significant could we assume an appropriate random model. Here, the significance test is used only as a benchmark to gauge the size of the R against the degrees of freedom in the data. This is because the random sample (from a population) model is inappropriate, as is the combinatorial model (where the results are conceived of as one possible random combination among all the possible combinations of the given data). Clearly, the data do not constitute a random sample, but also even for the given data, the separate data combinations are hardly independent. They are of time series, where sequential values are dependent on previous ones and are not free to assume any value in the data's range. Moreover, by theory this history of the process of change is crucial, which contradicts the random selection assumption of the combinatorial model.

Finally, the results enable the likely shape of the bifurcation set in Figure 5 to be drawn for the two dyads. Figure 8 shows this for India Pakistan; Figure 9 for Pakistan India. These are drawn using as a guide Cardan's Discriminant (Cobb, 1981a, P. 76) based on (a) the fit of a cusp model to the data for X, ₁, and ₂; and (2) the full and reduced coefficients for ₁ and ₂ in the butterfly regressions of Table 5. The plots are for real time data, while the bifurcation sets are based on the results in Table 5 using the smoothed (two-year moving average) data.

Moreover, for simplicity the plots in Figure 8 and Figure 9 were made for all (X, ₁, ₂) values scaled positively. To get the plot appropriate to the model of Figure 5 and empirical results shown in Table 5, rotate the X and ₁ dimensions 180º around ₂, which is equivalent to multiplying X and ₁ by -1.

To return to the question of the proper interpretation of ₁ and ₂, note that in Figure 8 and Figure 9 ₂ is appropriately defined as the gap. It is the condition most clearly moving behavior toward the lips of the cusp, either in the more conflictful or more peaceful directions. As should be by theory, the ₁ condition (as the power transition) is shown to be the intensifier, not disequilibrator.

Besides these figures, we can also test this interpretation by rerunning the regression with the alternative interpretation of ₁ as the gap, ₂ as the power transition. The results are listed in Table 6, and as can be seen by comparison with Table 5, give a poorer fit. For both dyads, treating ₂ as the gap instead of ₁ accounts for about 10 to 17% more of the variance.

Figure 10 now gives a side view of the quantitative history of Indo-Pakistan conflict and the cusps of Figure 8 and Figure 9, displaying the jumps between the upper and lower sheets. In terms of the conflict helix, each movement up from the fold bottom to the fold top (or as for India-Pakistan in 1951-1952, up to the fold top and out to the top sheet) constitute the creation of a new structure of expectations (status quo); each movement down from the fold top (or from the top sheet to fold top and down) is the disruption of this status quo. The up-down movements of both dyads are positively correlated r = .70.¹⁸ This measures the expected symmetry in the conflict behavior of India and Pakistan toward each other. More important, that the correlation is not larger evidences what has been as assumption throughout: the subjectivity of each conflict helix. As seen in Figure 10, the formation and disruption of a structure of expectations depends on the situational expectations and interests of the actor.

Aside from the data's quality, however, there is one problem in the data sample that should be remedied in future tests. Pakistan and India became independent in 1947 and from the very start were engaged in conflict. The separation of the colony of India into these separate Hindu and Moslem nations caused a mass migration of millions of Hindus and Moslem and communal rioting and massacres that killed hundreds of thousands on each side and left a legacy of hostility. And this separation created territorial disputes over their common border, especially focused on Kashmir, that would be the source of aggravation, incidents, and military confrontations for decades. As a result, what structures of expectations developed were more like truces in an ongoing hostile conflict than the initiation of a period of peace and harmony based on a settlement of outstanding issues.

Therefore, the conflict data manifest only the bifurcation set and deep conflict surface of the catastrophe manifold, as clear from Figure 8 and Figure 9. Except for one point (in Figure 8) out of the 52 for both dyads, there is no movement across the bifurcation set to the side of stable peace. Future tests should select dyads and periods that have not only the extremes of conflict, but also times of peace and solidarity as well, as for the U.S.-Japan dyads, 1920 to 1980.

Alternatively, in line with Model II of social field theory,¹⁹ data could be used for the same actor. While different actors i will have different expectations, perceptions, dispositions, and interest, and thus be differently sensitive to a gap or differences in power, the same actor i should similarly weight these conditions for different others

In other words, a range of cooperative/conflictful data could be employed for the behavior of, say, the United States to each of all other nations for 1968 (a year in which there would also be a case of war). However, the conflict helix and its butterfly model is meant to describe a dyad's life path; to test the model across dyads is to test a longitudinal theory by cross-sectional data, not a most recommended procedure.

In any case, future work should concentrate on refining the measurements of conflict and cooperation and the helix conditions, and on quantitatively testing the model against the history of dyads important to world peace. For it promises some help in both the forecasting of conflict and in the fostering of peace.²⁰

* Scanned from Behavioral Science 32 (1987): 241-266. Typographical errors have been corrected, clarifications added, and style updated.

1. See Part VII in Vol. 2: The Conflict Helix; and Part IV in Vol. 4: War, Power, Peace.

2. See Part II of Vol. 4: War, Power, Peace.

3. See Part IX of Vol. 2: The Conflict Helix; Part V of Vol. 4: War, Power, Peace; Power Kills.

4. See Rummel, 1977; Parts II and III of Vol. 4: War, Power, Peace. On field theories in general, see Part I of Vol. 1: The Dynamic Psychological Field.

5. By "common" is meant that variance (communality) that a specific behavior shares with other behaviors among all actors. Technically, the communality (h²) is the degree to which a behavior is within the common-factor space of dyadic behavior. This is because factor and canonical analyses comprise the models through which these equations are operationalized.

6. For the empirical application of canonical analysis within this theoretical context, see Appendix 9A of Vol. 4: War, Power, Peace.

7. Rummel, 1977, Chapter 8.

8. Rummel, 1977.

9. Chapter 2 and Chapter 3 of Vol. 2: The Conflict Helix.

10. Although only recently developed, catastrophe theory has been widely described and applied. For an elementary, largely conceptual introduction with social science applications, see Woodcock and Davis (1978). For a more mathematical treatment that assumes only a year of calculus, see Saunders (1980). An excellent advanced (for the scientists, not the mathematician) and full development of the theory, with many social science applications, is Poston and Stewart (1978). Zeeman has been the most prolific in publicizing and applying catastrophe theory, and his Scientific American (1976) article has done much to make the scientific community aware of this development. See especially his collected papers (1977), which also includes a longer and more helpful version of his Scientific American piece. For social scientists, Behavioral Science is the best source of articles on relevant catastrophe theory developments and applications. See especially the full issue devoted to it (Vol. 23, 1978). Catastrophe theory has already been applied to model conflict, specifically prison riots (Smith, 1980; Zeeman, 1977, Chapters 13-14), war (Holt, Job, and Markus, 1978), and crisis (Phillips and Rimkunas, 1978). For a full critical examination and some telling and helpful points about catastrophe theory, see Sussman and Zahler (1978a). They have also published a much shorter version (1978b). A response is given by Oliva and Capdevielle (1980). A less substantial critique is by Kolata (1977). See also the consequent communications in Science by Senechal, Lewis, Rosen, and Deakin (Vol. 196, June 17, 1977 ); and Sussman's response (Vol 197, August 26, 1977 ).

11. See Appendix 9A and Figure 9A.1 in Vol. 4: War, Power, Peace.

12. See Vol. 2: The Conflict Helix, Vol. 4: War, Power, Peace, and Power Kills

13. See the Joint Freedom Proposition and Chapter 2 of Power Kills.

14. I have found Poston and Stewart's (1978, especially Chapter 7) development and treatment of this theorem most helpful.

15. The details of this transformation are beyond our scope here, but is like the reduction of many variables to one eigenvector (as in principal components analysis).

16. Jiobu and Lundgren, 1978; Adelman and Hihn, 1982.

17. For other systematic tests of catastrophe theory, see Oliva, Peters, and Murthy (1981); Smith (1980); Zeeman (1977, Chapter 13).

18. To get this correlation, scale the location of the overtime point for the dyads as follows: 1 = bottom sheet, 2 = fold bottom, 3 = fold top, 4 = top sheet.

19. This model argues that the parameters of dyadic relationships are constant for the same actor only; Model I assumes that they are constant across actors, as in doing a regression analysis of conflict and socio-political distances for all dyads; see Rummel, (1977).

20. Written in 1998: I had intended to do this retesting shortly after this article was finished. The data collection for this likely would have taken a year or so and in the meantime I became totally involved in work that I felt was far more important: collecting data on democide that eventually was published in Statistics of Democide; doing the scholarly historical study of the megamurdering regimes that resulted in Death By Government; and retesting and evaluating the democratic peace, which eventuated in my writing Power Kills.

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APPENDIX 1
POLITICAL DISTANCE

Here political distance refers to those accumulated politically relevant events in i and j that could cause a shift in the balance of powers for i. A balance of powers is an equilibrium in the interests, capabilities, and wills of both sides. Anything that could alter these elements could thereby change the balance. Such might be due to changes in leadership, coups, constitutional or revolutionary changes in government, changes in alliances with significant third parties, loss of domestic support, economic crisis, domestic turmoil, major disasters, etc.

Moreover, this balance is basically subjective, partially dependent on i and j's perceptions within a situation. Not only are events in i relevant to the balance i j, but events in j that will be perceived by i can also effect the balance for i. For example, if i perceives that a coup in j has put into power a dedicated foe of i, this may well affect i's interests in maintaining close ties with j. Thus, for j's new leader and from i's perception, interests (and perhaps, will) have changed, and thus will have shifted the balance of powers maintaining their structure of expectations.

All such events accumulate for a particular structure of expectations and help create an increasing gap between this structure and the changing balance of powers. Once, however, this gap causes a disruption of this structure and a more fitting one is formed through conflict, the accumulation of events begins anew--political distance begins at zero. As a first approximation here, however, political distance is accumulated over the whole period.

To measure political distance, types of events are scaled by their possible effect on the interests, capability, or wills of India and Pakistan, as listed in Table 1-1. Then actual events and relevant changes for India and Pakistan were rated, and the ratings were aggregated into an accumulative political distance score. This is tabulated in Table 1-2.

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APPENDIX 2
JOINT FREEDOM

As has been defined and measured elsewhere for dyads 1976-1980 ("Libertarianism and International Violence"), there are three components to freedom: civil liberties, political rights, and economic freedom (a free market). Throughout the period 1948-1973, India and Pakistan have had mixed capitalist-statist economies, with the government playing a strong regulative-interventionist role in the market. We can therefore treat this as a constant for both (which means that it will have no effect on the tests) and focus on the other two components, or what together constitute political freedom.

I could not find an annual rating of political freedom for India and Pakistan over the whole period, and therefore had to develop one out of (a) political freedom related ratings of these nations for certain years, and (b) political histories of Pakistan. Table 2-1 provides ratings from various sources that helped to form the overall rating for India and Pakistan, also shown in the table. The last column gives the joint freedom rankings that comprise the test data.

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