Abstract. In recent years terrorist acts have developed into a wide variety of actions based on new levels of organization and approaches that present characteristics of nonlinear behavior. This paper is an attempt to create a basic framework for a deeper understanding of the process based on nonlinear models of the said behavior. The potential occurrence of chaotic (deterministic chaos) regimes together with the description of discontinuities and the parameters that generate or avoid them are presented along with the basic equations of the model. Basic conclusions are drawn and a possible research program is proposed.
Whenever talking about emergency response and preparedness one is inclined to consider emergency as a state created by events occurring with a short time constant and having significant impacts. Response and preparedness relate to our capacity to avoid the emergency state or to confine its impacts once they occur. Like any other organizations the terrorist ones have several parameters that classify them. For instance, one may consider the level of influence such as individual, group and organization; or the level of geographic coverage of the impacts such as regional, national or global. Moreover, we may distinguish economic effects targeting from a single objective (e.g. blowing a military ship) to the whole economy (e.g. internet attack). There is also a social impact to consider that may enhance the consequences of such acts.
Such organizations are bonded by: belief (e.g. jihad), interest (e.g. drugs, separatism) and violence (usually masked under fighting for or against something). Also, one may consider the scale of response that depends of the structure of the society e.g. full control of the persons or free will, as well as the level of awareness and education that influences the reaction of persons. We may already mention here a correlation: more control requires less self-awareness while more freedom asks for increased awareness at the level of the people. Educated persons are less prone to radicalization and enrolment into terrorist organizations. Collective behavior that generates organization stems from the interplay of the parameters listed above. For example, opposing the terrorist organization should be done on various scales. The saying that mosquitoes should not be fired at with cannons underlines best the need for a scale-based approach. Not to confound scale with intensity i.e. at each scale the intensity should be such as to ensure results. From a nonlinear model point of view the fractal approach may provide a comprehensive view that will not neglect or diminish the importance of various scales. Thus, it becomes important also to know what you live aside (what you do not know) along with what you concentrate on (what you know).
We make a brief comment on the time scale: emergency involves sudden events (low frequency) having high consequence (severity). Does that mean that we should neglect the high frequency low consequence, but cumulative events? Just an example: imagine the slow build-up of the material for a dirty bomb where terrorist are transporting small (under detection levels) quantities of Uranium ore that may be processed in a long enough time (i.e. with small enough installations) to avoid detection. As wars nowadays have a different time constant (consequently a different use of resources and means) if a terrorist is patient and he has the right knowledge, he may make a dirty bomb in half a lifetime without being detected, instead of a state making it in half a decade and being detected at once. Just for the sake of an example let’s indulge in a simple calculation. Assume that one need 17.5 kg of U-235 weapons grade for a bomb, U-235 content in Uranium is 0.7% and Uranium content in ore is 3%. Multiplying one would need about 83 t of ore. Consider transports of 8.3 kg done by 10 people every week, we need 1,000 weeks (or about 20 years) to gather the right material. Obviously a 20 years processing would not require large installations but a few that would work continuously.
CHAOTIC BEHAVIOR AND ORGANIZATIONS
We have mentioned above that considering the parameters of a terrorist orga-nization one may identify types of actions that may range beyond present detection capabilities. Moreover, we mentioned that in order to plan and make their actions terrorists need knowledge. That raises two problems: (i) how does a terrorist IQ interact with his terrorist organization one or how do the members of a group enter an organi-zation and (ii) what is the productivity of the terrorist organization in terms of passing from planning various acts to doing them. Understanding the two processes above – that, by the way, are correlated – may result in conclusions on how to neutralize them.
Let us first describe the interplay between individual and organization IQ. When we choose IQ, it is because this parameter is measurable and second because we need to describe a specific behavior that oscillates in between planning and doing – in this case terrorist acts. We make a separation between’ group’ as a gathering of individuals and ‘organization’.
Putting Ig as the IQ of the group and Ii as the individual IQ of its members we may say that Ig is proportional to Ii. The proportionality depends on the rate (r) of individuals entering the group and with the degree of perception (p) of the advantage (in terms of protection, recognition, etc.) of being a member. We may write:
Ig(t) = p(r Ii) (1)
Going at the level of an organization we may notice that the effect of an organi-zation is to raise the low Ii and to diminish the high Ii of its members at least in terms of initiatives of behavior. As E. Hunt 2006, puts it ‘in economic terms it appears that the IQ score measures something with decreasing marginal value. It is important to have enough of it, but having lots of it does not buy you that much’. Thus, to describe such effect a second order expression is more appropriate, putting Io the organization IQ we have:
Io(t) = mIi(1-Ii) (2)
Finally we may infer that attracting private individuals into a group is proportional to the level of organization of that group at the previous moment:
Ig(t+1) = qIo(t) (3)
After some mathematics with formulae (1-3) above we get the following expression:
Ig(t+1) = qmIg(t)/(pr)(1-Ig(t)/(pr)) (4)
Let us consider that the rate of entering the group is inversely proportional with the perception of advantage of being a member i.e. pr = 1. Expression (4) becomes
Ig(t+1) = qmIg(t)(1-Ig(t))
One may recognize the second order map that describes a system whose behavior may be chaotic or convergent depending on the coefficient qm. Thus, as mentioned above q represents the degree of external perception of the organization: very small q may describe a secret organization while a large q means an organization well perceived that obviously is not a secret one any longer. In what regards m it represents the intrinsic correlation of rules and values inside the organization of the individual Ii. If m is change de.g. from outside information then the organization dynamic changes. So, several regimes may be described:
qm < 1 – the group is totally organized and individual initiatives are discouraged
1 < qm < 3 – an dynamical equilibrium is obtained where group and organi-zation behavior coexist and initiatives are recognized and used
qm > 3 – a chaotic regime sets in where the degree of organization of the group varies with apparently no rule; this regime may lead to the total dissolution of the organization and also makes it act inefficiently.
Let us take the first case qm < 1 and consider extremes: (i) q small and m large represents an organization with little external perception i.e. secret and with strong internal rules and values; while (ii) q large and m small is the organization known by all and having free rules and values.
The three regimes described above may be represented in the figure below. The fact that the organization may enhance or inhibit initiatives is not giving us the insight into its behavior in passing from planning to doing terrorist acts. Let’s describe now the’ productivity’ of the terrorist organization mentioned above.
Figure 1. Organization behavior in terms of individual initiatives
FROM PLANNING TO DOING – A DISCONTINUOUS DECISION
Let me start with the story of the five birds staying on a branch near a fence, three of them plan to move on the fence, how many birds remain on the branch; answer: still five, because from planning to doing there is a long way. Sometimes the long way may be suddenly short and unexpected. Consider now that the distribution of terrorist persons according to their level of intelligence (I) is a Gaussian:
N(I) = Kiexp(-(I-Im)^2/(2σ^2)) (5)
N(I) is the number of people belonging to an organization and having an IQ of level I, (Im is the mean I). Terrorist people may range from single or small isolated groups to large complex organizations, but they are all using their intelligent members to plan various actions. We are, thus, interested in the high I tail of the distribution that is described as a power law:
N(I) = Ki/(I-Im)^μ
where μ = 3 to 5 depending on the domain i.e. closer or further away from the mean.
Further on we put N(n) the number of terrorists producing n acts:
N(n) = Kn/n^Φ say Φ = 2 to 3 and Kn = constant (6)
Kn is proportional to the means (M) available at the disposal of terrorists (i.e. n is greater if more means are available:
No initiative / Smooth initiative / Chaotic initiative
Kn = kn.M (7)
The question we are trying to answer concerns the correlation between the number of terrorists whose IQ is I and N(n). Also, how many possible terrorist acts are done, that are planned in a given interval based on a flux of information η, resulting from previous acts, i.e.n = f(η).
Elementary terrorist acts are planned and done, but history shows that a terrorist organization can only do a limited number of acts n max in a given period of time. Thus, we may consider that the number of acts n and the flux of information η has a logistic shape. Putting the condition that
n(η0) = 1 and n(η)n max when ηoo and taking as a variable the reduced information flux (η/η0)=ξ, we may describe n by the smooth function:
n = ξ^γ/(1+ξ^γ/nmax) where γ is a number we shall determine. (8)
Such a function has an inflexion point S that is characterized by the zero of the second order derivative in ξ, which marks the creative terrorist act. We know that for n << nmax we may neglect ξ^γ/nmax compared to 1 and we have:
n(ξ) = ξ^γ (9)
We may determine the value of γ by noting that the relation n(ξ) allows us to determine the number of terrorists which have done acts generating a reduced information between ξ and ξ+dξ, which we note N(ξ)dξ. This is equal to the number of those who did N(n)dn acts with the condition that n = n(_) where from we have:
N(ξ) = N(n)dn/dξ (10)
But out of (9) we have (dn/dξ) = γξ^(γ-1) and using for N(n) the expressions (6 and 7) we have:
N(ξ) = knM/(ξ^((Φ-1)γ)+1) (11)
We are now going to assume that the distribution of terrorists with the reduced flux of information is similar to the distribution with the I, namely:
N(ξ)dξ = N(I-Im)dI (12)
From (12) with some algebra we have:
N(ξ) = N(I)(dI/dχ) = N(I)(ξ/(I-Im))^(-1).(dI/(I-Im))/(dξ/ξ) (13)
We define the coefficient of creative (qualitatively new) terrorist actions:
Α = (dξ/ξ)/(dI/(I-Im) (14)
Replacing in (13) N(I) and N(ξ) with their expressions (5) and (11), inserting from (14) we have:
Ξ = (knγα/KI)^(1/(γ.(Φ-1)).(I-Im)^((μ-1)/(γ.(Φ-1))).M^(1/(γ.(Φ-1)) (15)
If we set:
Ke = (knγα/KI)^(1/(γ.(Φ-1)), a = ((μ-1)/(γ.(Φ-1))) b = (1/(γ.(Φ-1)) (16)
We obtain a relation among the information flux ξ, the level of intelligence (I-Im) used and the means M of the terrorist organization that is similar to the production functions from economic systems (Cobb-Douglas):
ξ = Ke(I-Im)^a.M^b (17)
Because the dimensional homogeneity implies a + b = 1 or μ / (γ(Φ – 1)) = 1 for the minimal values μ = 3 and Φ = 2 we have γ = 3 which leads to a = 2/3 and b = 1/3.
Thus, (17) becomes:
ξ^3 = Ke(I-Im)^2.M = ε (18)
and we consider ε as a measure of the planning effort.
THE CREATIVE TERRORIST ACT AS A CUSP CATASTROPHE
We must underline that the above is valid for the condition ξ^γ << nmaxi.e. when we are far from an inflexion point. As we said above the creative (new type) of terrorist act occurs near the inflexion point of the curve n(ξ), when the accumulation of planning activity is enough to trigger action. To describe such a situation it is sufficient to consider that near the inflexion point the variation of ξ’ = ξ-dξ, with ε’ = ε-ε0 is the same as (18). ε0 is the planning effort necessary for the occurrence of a terrorist act. This way the relation among the information flux, the planning and the acting, which we will measure as τξ’, is:
ξ’3 = ε’+τξ’ with τ>0 (19)
We call τ an “action factor”.
In Figure 2 one may see the folded space corresponding to the behavior described by expression (19). Having introduced the action effort as a new dimension along with the planning effort we obtain a horizontal plane called the control surface. The result of terrorist activity is measured by the flux of information generated by the said action. The surface of this variable shows a fold. The evolution of the terrorist actions is represented as a trajectory on the surface. The crossing of the fold by the trajectory with the pass from one surface to another model the sudden pass from planning to action either in a smooth continuous way or in a sudden discontinuous one.
Figure 2. Folded plane for the planning-action behavior
We have thus, three regions in the space above:
- The region where the planning effort leads to usual action for which there are already emergency responses prepared and enough awareness level achieved in the population and the specific protection agencies.
- The region where planning effort does not lead to action; this region consumes the effort of the terrorist organization but generates potentially dangerous plans of action.
- The region of terrorist action is reached by the increase of the planning effort and of the action effort and is reached in a discontinuous way that triggers large and different impacts for which the response and preparedness are either poor or totally lacking.
One interesting thing to notice in the dynamics that may be described with the model above, is, for instance, the fact that a reduction of ‘usual’ terrorist activity means that the organization is concentrating its planning effort to generate creative terrorist acts and prepares its means for an action effort that may generate creative new actions totally unexpected by the protection structures. A response to this situation would be to incapacitate the planning capability of the terrorist organization meaning not only to neutralize the leaders (that has an influence on the action effort parameter above) but also the key members of the high IQ tail of the organization (that has an impact on the planning effort parameter).
Various other conclusions may be drawn related to how to monitor terrorist activity with a view to prepare and to increase awareness of not only standard responses but also non-standard ones. Moreover, the scaled approached proposed here contri-butes to getting an extended view to what we may start thinking to protect from in the future. Also, it gives hints on how to dismantle potential harmful dynamics of terrorist organizations by pushing them into unstable behavior regimes.
We finish this paper with a proposal for research that may on one side introduce measured values for the coefficients taken into consideration, in the specific case of terrorist activity and, on the other side, extend the model to more complex folded spaces that we have not described here.
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* Executive Director, FCCEA and c.m. AOŞR