A Model-Based Approach to Battle Execution Monitoring 

by Richard C. Kaste and Charles E. Hansen 

ARL-TR-3394 January 2005

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Army Research Laboratory 

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ARL-TR-3394 January 2005

A Model-Based Approach to Battle Execution Monitoring 

Richard C. Kaste and Charles E. Hansen 

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A Model-Based Approach to Battle Execution Monitoring

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6. AUTHOR(S) 

Richard C. Kaste and Charles E. Hansen

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13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

The work set forth in this report is intended to lead to portions of an integrated system to dynamically link automated plan

generation and analysis with execution monitoring. The report deals with algorithms for, and interfaces to, a prototype battletrajectory decision aid. We explored complexities of navigating parameter tradeoff spaces and performed detailed studies of

strength/attrition parameter estimation and calibration techniques. We investigated software for solving differential systems 

and for visualizing and interacting with decision aid data. We considered aspects of the analytical theory of dynamic systems:

mathematical chaos, stability, and control. We set the groundwork for formulation of models to produce systems exhibiting

inherently chaotic behavior and for studying qualitative aspects of phase plane trajectories. It is probable that nonlinear 

analytic techniques will soon be considered as invaluable to commanders. 

15. SUBJECT TERMS

combat modeling, military decision aiding, dynamical systems, cause of action analyses 

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Richard C. Kaste a. REPORT 

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ii
Contents 

Contents iii

List of Figures v

List of Tables v

Acknowledgments vi

1. Introduction 1

1.1 Project..............................................................................................................................1

1.2 Report..............................................................................................................................2

1.3 Background .....................................................................................................................3 

2. Basic Idea 4

2.1 Concepts..........................................................................................................................4 

2.2 Design..............................................................................................................................8

2.3 Decision Aid....................................................................................................................9 

3. Initial Formulation 10

3.1 Motivation .....................................................................................................................10 

3.2 Model.............................................................................................................................11

3.3 Sketch of Algorithm......................................................................................................11 

4. Theoretical Preliminaries 14

4.1 Fundamental Notions ....................................................................................................14 

4.2 Analytical Results..........................................................................................................16 

5. Aspects of the Decision Aid 20

5.1 Introduction ...................................................................................................................20 

5.2 Trajectory Tracking.......................................................................................................21 

5.3 Sensitivity Analyses......................................................................................................23 

5.4 Qualitative Behavior......................................................................................................26 

5.5 Other Desiderata............................................................................................................27 

6. Software 28

6.1 Consideration of Options...............................................................................................28 

iii
6.2 Dynamics Solver ...........................................................................................................30 

6.3 Contingency...................................................................................................................31 

7. Estimation of Parametric Values 33

7.1 Introduction ...................................................................................................................33 

7.2 Recurrence Relation Derivation ....................................................................................35 

7.3 Least Squares Approach................................................................................................36 

7.4 Modifications.................................................................................................................37 

7.5 Illustrations....................................................................................................................39 

8. Techniques for Analysis of Dynamic Systems 45

8.1 Introduction ...................................................................................................................45 

8.2 Theoretical Notions.......................................................................................................46 

8.3 Chaos.............................................................................................................................47

8.4 Stability .........................................................................................................................48 

8.5 Deviation From Plan .....................................................................................................51 

8.6 Control Theory ..............................................................................................................52 

9. Future Efforts 53

9.1 Extension of Initial Model.............................................................................................53 

9.2 Attrition .........................................................................................................................54 

9.3 Next Formulation ..........................................................................................................55 

9.4 Modeling Continuous Forces........................................................................................58 

9.5 Heuristics and Estimation..............................................................................................59 

9.6 Other Applications ........................................................................................................60 

10. Conclusion 62

10.1 Summary .......................................................................................................................62 

10.2 Impact............................................................................................................................63 

10.3 Issues.............................................................................................................................64

10.4 Final Observations.........................................................................................................64 

11. References 66

Distribution List 70 

iv
List of Figures 

Figure 1. An example of a battle trajectory. ...................................................................................5 

Figure 2. Blue and Red force levels over time..............................................................................13 

Figure 3. Planned trajectory with alerts as actual trajectory breaks envelope..............................23 

Figure 4. GUI input of the square law solution. ...........................................................................32 

Figure 5. Graph resulting from GUI input....................................................................................33 

Figure 6. Predicted strength. .........................................................................................................39 

Figure 7. Smoothed estimate.........................................................................................................40 

Figure 8. Y-force strength levels. .................................................................................................41 

Figure 9. Predicted strength levels................................................................................................42 

Figure 10. Modified situation. ......................................................................................................42 

Figure 11. Attrition rate quantities................................................................................................43 

Figure 12. Predicted Iraqi forces...................................................................................................44 

Figure 13. Delayed casualty reports..............................................................................................44 

List of Tables 

Table 1. Concepts and symbols. .....................................................................................................6 

Table 2. Stability of (0,0) for { ' x = + ax by y, ' = cx + dy} ............................................................49 

v
Acknowledgments 

This work was performed under the auspices of the U.S. Army Research Laboratory (ARL) 

Director’s Research Initiative FY03-CIS-04. The authors gratefully acknowledge the assistance 

of Janet O’May, Fred Brundick, Andy Neiderer, and Pat Jones in this project. Ms. O’May 

generated scenarios, implemented simulations, and wrote data collection/analysis programs. Mr. 

Brundick was instrumental in developing the improvements to the interface and plotting software 

discussed in subsection 6.3. Mr. Neiderer performed background research into interface 

software and facilitated the illustrations. Ms. Jones offered editorial suggestions that improved 

the readability of this report. 

vi
1. Introduction 

1.1 Project 

The project “A Model-Based Approach to Battle Execution Monitoring” involves a form of 

exploratory mathematics for military application that is intended to lead to more basic research 

with greater payoff. We compare an updated-parameter model trajectory to a planned-parameter 

model trajectory. There are interesting problems in doing such comparisons from a “leastsquares” type standpoint and basins of attraction. As a practical matter, various tactical 

considerations such as attrition levels in selected time make such comparisons more 

straightforward. We examine such in this project in the hopes of being able to continue the more 

difficult research based on the tools and foundations developed here. This is not the innovative 

basic research, but rather sets the stage and tool formats for subsequent exploration and novelty. 

Briefings to advisory bodies have been generally positively received, and management has 

offered good suggestions for improvement. 

This project was motivated somewhat by the changing nature of course of action (COA) 

analysis, in particular, incorporation of the notion of commander’s intent. This formed a 

justification for the gross modeling of battle in the abstract, which lends itself to qualitative 

analysis, use of rough estimates, and consideration of the derivatives of the trajectories of 

conflict. Here is a good place to mention a quote from the father of this sort of combat analysis, 

“There are many who will be inclined to object to any mathematical or semimathematical 

treatment of the present subject, on the ground that with so many unknown factors such as the 

morale or leadership of the men, the unaccounted merits or demerits of the weapons, and the still 

more unknown ‘chances of war,’ it is ridiculous to pretend to calculate anything. The answer to 

this is simple: the direct numerical comparison of the forces engaging in conflict or available in 

the event of war is almost universal. It is a factor always carefully reckoned with by the various 

military authorities; it is discussed ad nauseum in the press. Yet such direct counting of forces 

is, in itself, a tacit acceptance of the applicability of mathematical principles, but confined to a 

special case. To accept without reserve the mere ‘counting of the pieces’ as to value and to deny 

the more extended application of mathematical theory, is as illogical and unintelligent as to 

accept broadly and indiscriminately the balance and the weighing machine as instruments of 

precision, but to decline to permit in the latter case, any allowance for the known inequality of 

leverage (1).” 

The usefulness of mathematical modeling for combat analysis can then be considered wellaccepted. We can use it to understand somewhat the dynamics of conflict, and hence can invoke 

it, at least iteratively, as a tool for developing plans for the mission. But a more difficult 

problem, again one that helped motivate this project, is the question of what triggers the need for 

replanning; that is, how one realizes that the actual battle is deviating unacceptably from that 

1
envisioned. This is certainly part of the military art of experienced commanders, but whether it 

can be brought practically into the realm of science remains an open question. One reason to 

monitor a battle is to see if it goes too far off track. How do we know it is off track?

The basic intent was to use this project as a launcher for novel research into other types of 

execution monitoring and dynamic replanning. An ongoing area of research is the problem of 

evaluation of “goodness” of a COA. A COA is a feasible way to accomplish a mission that 

follows the commander’s guidance, will not result in undue risk, and is noticeably different from

other actions being considered (2). That is, what makes one plan “better” than another? Can it 

be better statically, as is often done by planners, or only dynamically? Can a plan be shown to be 

good only at the end of the battle?

We proposed an application, building on classic and newer techniques of mathematical 

modeling, to dynamic battlespace decision support. Technical challenges included investigations

into goodness of the underlying model, determination of input parameters and values, and 

transformation of results for use in the tactical domain. We intend to develop a set of algorithms

to be coded into a computerized prototype of a command staff decision aid. This work 

constitutes portions of a prototype system dynamically linking battle plan generation/analysis 

with execution monitoring. The technique should permit comparison of a plan, in terms of 

desired end state and other measures, with the trajectory of the conflict in phase space. During 

planning, and particularly during monitoring, if analysis shows the trajectory does not permit the 

end state, the user is to be helped with adjustments enabling the end state or be alerted to the best 

results achievable. 

We sought application of results in nonlinear dynamics to such problems. There has been 

considerable philosophical discussion concerning the sensitivity of combat to small variations in 

initial conditions (“for lack of a nail the kingdom was lost”), but we seek mathematical rigor in 

treating combat as a dynamic system. We are also interested in whether notions of optimal 

control theory can be applied to such analyses, perhaps ultimately leading to tools for monitoring 

divergence from a battle plan and making changes. Can combat be analyzed (at least in a 

qualitative sense) that allows reasonable measurement of the influences of various factors?

1.2 Report 

In this report, we concentrate on expositions concerning the main technical results with regard to

the design and implementation of the prototype, including the nature of decision aids based on 

these concepts and particular parameter estimation techniques. These discussions are followed 

by subsections dealing with ancillary aspects (such as the interface and control theory) that will 

be developed as part of the follow-on work. Note that some of this is like a tutorial. This is on 

purpose. The mathematical derivation is necessary to enable the construction of algorithms and 

hence the prototype. We entered into this project with the intent, as explicitly laid out in the 

Director’s Research Initiative program, to justify further developments along these lines as part 

of the division mission. 

2
Therefore, we set forth, in a kind of white paper, a summary of emerging results and areas we 

deem as having potential for fruitful further development. The format involves a series of 

discussions of the topics, each ranging from an introduction of concepts to more in-depth 

development and areas for further research. The reader will notice several aspects to developing 

a decision aid of this sort. We have made progress on several. We spell out various topics in the 

hope of coordinating efforts for future researchers, including interns and students. Various 

design desiderata are set forth as guidance for the coding of the software. Some additional 

discussion concerning the actual computer work is provided in later portions of the report. In 

particular, results on the parameter estimation routines and the curve plotting routines are 

examined in some detail.

We generally assume continuity, as opposed to discreteness, in the systems being discussed. 

However, in some instances discussion of discrete time intervals is more appropriate to the 

analysis, but this is clear in context and should cause the reader no particular confusion. A 

similar comment applies to nonlinearity, as opposed to linearity. Also, for the sake of brevity, in 

some cases the terminology is a bit loose, but this will be made stringent in future examinations 

of specific systems. 

1.3 Background 

A fundamental challenge of the 21st century battlefield is to reduce vast amounts of data into 

contextually relevant and actionable knowledge. Achieving the intent of the Chief of Staff’s 

Army Vision to improve operational jointness and ensure responsive dominance will require 

cognitive amplifiers for application to Future Combat Systems. By ARL’s treatment of combat 

modeling, we intend to facilitate provision of the right tools for evaluating and improving 

combat effectiveness. This research focuses on identifying and developing techniques for 

modeling and simulation to meet these fundamental challenges. The overall objective is to 

develop and evaluate COA tools for the mobile commander in the Future Force. Enhancement 

of the staff’s planning capability will allow increased options in an engagement. The technical 

focus is on research into techniques for automated COA evaluation incorporating “reasonabletime” battlefield information and development of a COA analysis testbed and associated decision 

tools. An essential part of this work is extension of the mathematics of combat modeling. 

Analysts use differential equations (DEs) to model aspects of combat. Parts of so-called 

Lanchester theory are fairly well developed. Such models, generally solved through 

computerized difference equations, can estimate the winner and duration of a conflict and track 

the history of force levels. More recent aspects include stochastic formulations and examination 

of optimal combat responses. Moreover, attrition of moving forces lends itself to use of partial 

DEs, which handle density and geometry, and these formulations may permit analysis of 

interpenetration and nonlinear battles in future scenarios. 

Various techniques might be applied to the analysis/assessment/evaluation of a COA, given that 

it has already been derived. One of particular interest to analysts is wargaming, which may be 

3
manual, mathematical, computerized, or a combination. Several approaches, mostly derived 

from wargaming, are traditionally somewhat feasible in examining changes to battlespace 

parameters (weapons and tactics). These include combat ratios, firepower potential, history, unit 

effectiveness, rate of advance, casualties, and vehicles lost. Innumerable ways exist to measure 

or calculate such things. Moreover, other methodologies utilize these battle statistics as inputs 

for subsequent numerical or symbolic analyses: decision trees, Bayesian belief nets, expert 

system advisors, etc. We would like to develop a procedure, automated if possible, for 

indicating the effectiveness of forces, COAs, and tactics in various scenarios. The intent is to 

utilize the killer-victim scoreboards (or other measures of effectiveness) extracted from

computerized wargame outputs as inputs to the procedure. A theoretical problem is whether 

there could be a way to evaluate the general utility of forces (or other aspects of the battlespace) 

without regard to specific situations. What we basically seek is an optimal technique or 

(possibly combined) set of measures, probably "situation dependent," useful for evaluating 

COAs in general (3). 

We considered many aspects of such a project before narrowing it down: the modularity of 

planning and gaming, ongoing work at ARL on genetic algorithms (4), improvement of combat 

simulations (5), commercial AI tools, military scenario-based games, display techniques, 

qualitative physics and model-based reasoning, training vice functional/operational applications, 

and so on. Some of the work to be discussed emerged from an attempt to address such notions. 

2. Basic Idea 

2.1 Concepts 

We consider as a fundamental concept that one can compare plans or COAs within the context of 

a given scenario. (Many terms will take on technical meanings in the subsequent development.) 

Such comparison can be done, as alluded to earlier, statically; for instance, whether certain 

aspects of the envisioned conflict are addressed. However, it makes more sense to do the 

comparison in terms of “trajectory” and realized “end state,” where trajectory and end state are 

related by notions to be discussed. 

We can define a COA in terms of the paths it precipitates in a space of parameters describing the 

conflict. For example, a (rather simplistic) kind of Blue COA can be defined as a given rate of 

fire and a given rate of movement at a given time. Such a COA can then be tied into Lanchester 

modeling. Let us consider that a COA is analogous to a vehicle being controlled through an 

environment. The environment here reflects the hostile activities of the opponent: the Red 

“wind” blows the Blue position and strength back. We can track the Blue and Red strengths (or 

whatever attributes we can meaningfully assess) in another phase plot for monitoring. The 

details of such tracking, with regard to human-factors-based navigation of a parameter space, 

4
will be discussed in section 5 on decision aid aspects. By way of introduction, figure 1 gives an 

example of a battle trajectory: the plot shows a schematic portrayal of the force ratio over time. 

3:1

1:3

Red Strength

Blu

e

Stre

n

gth

Figure 1. An example of a battle trajectory. 

We therefore take it as a given that we can consider Blue and Red as having abstract trajectories

in a “battlespace” with regard to missions, assets, and actions. The idea, then, is to consider a 

model of the conflict, say a system of DEs. The commander can control certain aspects of the 

battle. That is, he has reasonable power to modulate certain attributes. For instance, he might 

modify the number of his troops, via reinforcement or withdrawal, even in the midst of the 

conflict. To some extent, he can modify the rate at which he kills the enemy (say, increasing it 

by utilizing heavier firepower) and the rate at which the enemy inflicts attrition on his force (say, 

decreasing it by improving defensive posture). A desirable decision aid would constantly give 

the parametric set(s) that would enable attaining the goal. This could be demonstrated by having 

an (invisible) enemy changing his values. 

Note there is really nothing special about the form of the model. As long as we have parameters 

that influence the trajectory, be it discrete or continuous or symbolic or whatever, we can 

proceed with such methodology. Indeed, as computational power increases, the nature of the 

model can be more and more sophisticated; as the Future Force becomes operational, the data to 

populate such models should become readily available in near real time. However, even very 

high fidelity models and accurate information may not approximate reality sufficiently well for 

the commander’s purposes. Therefore, another novel concept set forth in our work is that of 

attempting to adapt (or eventually even actually develop, probably through powerful statistical 

techniques) the model on the fly from observed data. A fundamental question, then, to be 

considered in some depth later in this report, concerns the feasibility (technical and tactical) of

computing the parameters or the model itself from the observed values. 

What does it mean for combat to be considered a dynamic system? As a lead in to the 

discussions of analytical methodologies of section 8, we note that a dynamic system can be 

5
considered mathematically, in one sense, as a vector field. As a practical matter for initial 

development of the prototype decision aid, however, we will develop it in terms of set of 

differential equations describing aspects of behavior, basically as functions of time. Notions of 

space are, initially, fairly abstract. 

The concepts and symbols set forth in table 1 will be necessary for the development of the theory 

and implementation of the proposed decision aid. Some of these are relatively straightforward, 

but most require some discussion of the underlying philosophy. (Note that no letter convention 

is followed with regard to a symbol denoting a constant or variable.) 

Table 1. Concepts and symbols. 

Concept Symbol

Time (as variable) t

Friendly (Blue) force level x

Enemy (Red) force level y

Rate at which Red kills Blue a

Rate at which Blue kills Red b

Blue breakpoint x p

Red breakpoint y p

Initial (t = 0) 0 0 t , x (sub 0) 

Specific time of interest τ

Known/measured variable x (no mark) 

Desired by Blue xτ % (tilde over variable) 

Assumed by Blue yτ

( (cup over variable) 

Modeled (by Blue) xˆ

τ (hat over variable) 

(Blue) desired time to end t

ω

%

End state (desired by Blue) Ω%

“Breakpoint” connotes the percentage of its initial force level at which a force would attempt to 

disengage and break contact. The time to end desired by Blue represents the time that Red 

would be defeated, where this is of necessity in this scheme related to other Red parameter(s), in 

particular casualty level. “Known/measured” means that the Blue commander has sufficient 

ground truth or intelligence to take the value of the attribute as effectively true; “assumed,” on 

the other hand, means that the Blue commander is postulating this attribute to have a certain 

value for his purposes of assessing the mission; and “modeled” means that the value of this 

attribute has arisen from calculations of the active system of equations for the conflict. In a 

“degenerate” case, we can simplify the analysis somewhat by assuming the breakpoints are 0. 

That is, one side wins when the other is totally annihilated. However, this is not tactically 

realistic, and the algorithm is not overly complicated by dropping this assumption. 

6
Kill rates (or attrition rates) require a little discussion. Anderson and Miercort (6) note that 

comparisons of combat forces are generally calculated as follows. All resources on each side are 

grouped into categories. Each resource is assigned a (nonnegative) score. These scores, which 

can be functions of the numbers and effectiveness parameters of the resources, are constant 

within categories and can vary across categories. Force strengths are formed by summing, over 

categories, the product of the number of resources in each category times the score. 

Comparisons of forces are then computed using (real-valued) functions of these strengths and, 

perhaps, other parameters. Thus one must consider how to calculate scores for resources, and 

how to combine them to form force strengths. We note that over the years much analytical 

thought has gone into implementing such ideas, but we propose additional developments based 

on our current data mining work and extension of its methodologies (7). 

Thus, for our purposes, kill rate can be computed in terms of the probability of hit times the 

number of shots per individual per unit time. For example, is the rate at which a Blue combat 

element destroys a Red combat element, considered derived as a product of the rates of trials and 

probabilities of kill. That is, in this kind of abstract modeling it is best we consider the 

“effectiveness” of a unit’s action against other units, as opposed to individual weapon vs. 

individual target as in our Killer/Victim Scoreboard analyses (8). The unit of measurement for 

kill rate may be thought of in terms of (casualties of “target force”)/(strength of “firing force” 

times time). The rate at which fighting occurs may to a considerable degree be measured in 

terms of the trials with which probabilities of hit and kill are associated. At time t ,

b

( ) Bf t denotes 

the average rate at which trials are generated by a single Blue force-unit; ( ) Rf t is the 

corresponding rate for a Red unit. For an individual trial, kill probabilities are denoted by for 

Blue and for Red. The unit killing rates are 

B k

R k B B c k f = B , the number of Reds that a Blue unit 

can kill per unit time and , for the number of Blues that a Red unit can kill per unit of

time. Attention is confined mainly to homogeneous forces comprising for each side units 

identical in kill probability and in fighting–rate (9). Moreover, we tacitly consider an integrated 

“effective” kill rate if the contributions of effects such as reinforcements or disease are not 

known. 

R R c k = fR

A “plan” or “course of action” for our purposes is defined as: the initial Blue force allocation (a 

positive number), a Blue breakpoint, a desired kill rate (by Blue of Red), a desired time to win, 

assumed initial Red allocation (a positive number), an assumed kill rate (by Red of Blue), and an 

assumed Red breakpoint. We see that there are three types of parameter: known, desired, and 

assumed. “Actuality” comprises true force allocations, kill rates, and breakpoints. Note that in a 

more sophisticated formulation, several parametric values can change as part of the plan: e.g., 

anticipated Blue reinforcement or weakened Red kill rate. 

The phase space of the battle is a fundamental notion. Phase space, for the purposes of general 

discussion, is the n-dimensional space accessible to an object with degrees of freedom. A 

somewhat tighter notion important to subsequent technical discussions is that of manifold, a 

n

7
space locally like n-dimensional Euclidean space, but lacking a preferred system of coordinates 

and possibly having unusual global topological properties. In most of this decision aid 

development, we are concerned with the trajectory, the path of an object through phase space. In 

particular, we are concerned with characterizing the relative positions and movements of the 

Blue and Red forces in the following technical sense: each force may be thought of as a tuple of 

coordinates, which may include force level, geographical position, kill rate, ability to move, 

ability to communicate, etc. For the initial formulation leading to a prototype decision aid, we 

consider that the trajectory of the battle occurs in a two-dimensional space: the x-axis is the force 

level of Blue; the y-axis the force level of Red. As indicated elsewhere, this simplistic start 

results in a number of technical/computational and cognitive/display challenges. 

2.2 Design 

The technique we are considering would permit comparison of a plan, in terms of desired end 

state and other measures, with the trajectory of the conflict in phase space. During planning, and 

particularly during monitoring, if analysis shows the trajectory does not permit the end state, the 

user would be helped with adjustments enabling the end state or be alerted to the best results 

achievable. The algorithm involves the following sequence. Input the model form (based on 

postulated type of operation), parameter values (known, such as Blue force level, and assumed, 

such as Red break point), and desired end state (e.g., by a certain time cause a certain Red 

casualty level before Blue suffers a certain level). Compute the resulting trajectory; portray 

graphically. If the end state cannot be met, assess what changes to parameter values must be 

made. If a parameter is not established, compute what values will enable the end state. As the 

(simulated, for this project) operation is monitored, when a value change occurs, recompute the 

trajectory. Determine if the desired end is still achievable. If it is, reassure the user (perhaps 

with visual indications of changes); if not, compute the parametric region allowing achievement 

and enable user trade-offs. Further, during battle monitoring, when new data are obtained, the 

model itself can be improved: either explicitly, such as by utilizing direct observations of Blue 

casualties; or implicitly, as in calculating Red attrition rates from force level snapshots. The user 

is guided to parametric regions allowing end state achievement so that trade-offs can be made if 

necessary. Values are to be noted as actual or postulated in portrayal of current/projected battle. 

Initially, for concept demonstration, we utilized a simple model with the intent to then increase 

the sophistication, eventually monitoring a high-fidelity combat simulation to better demonstrate 

dynamic model updating. 

This is based in part on the idea of iterative wargaming. In this paradigm of battle planning, the 

analyst sets forth a COA for a relatively short period of time (or until some significant event 

occurs) and simulates the conflict during that span. Of course, a desired outcome for the conflict 

and the commander’s intent are guiding the choice. At the end of the span, the simulation is 

stopped, the results tallied, the situation assessed; and the process begins again with a new COA 

from that point. So, the notion of trajectory correction is similar in that, when a span is reached 

8
in the sense of an updated value for a battle parameter, the situation is assessed for COA 

changes. The intent of both processes is to bring the long-term goal to fruition if possible. 

If a trajectory does not permit the end state, it should be corrected. It is possible that in some

formulations and with some ranges of parametric values the end state is mathematically 

unattainable. For example, the Blue breakpoint might always occur at a time prior to the Red 

breakpoint with the given model and input ranges. The user should, of course, be alerted to this 

fact, and the battle would as a practical matter not be entered into. 

In the portion of the algorithm concerned with battle monitoring, we shift the intent slightly. 

Although the concept of checking for ability to reach the desired end state still applies, the nature 

of the analysis changes to comparison of trajectories. Whenever the commander’s staff obtains 

“intelligence,” the system should use this information, if possible, to update the model. In our 

initial prototype, the updates take the form of snapshots concerning one or the other force level 

or kill rate. The data can be explicitly ascertained, as in a Blue casualty report that updates the 

known force level. At this point, the trajectory of the conflict is recomputed, and hence ability to 

reach the desired end state. If the end state cannot be reached, the necessary changes to 

parametric values are developed, similarly to the technique in the planning phase. 

The new data may also lead implicitly to model updates. An example of this is a new Red kill 

rate, which could be based on an assumed Blue kill rate, assumed Red force level, and known 

Blue force level. Techniques for handling such implicit updating have been the focus of a fair 

amount of research in this project, and their development is discussed in section 7. 

Recomputation of trajectory and end state parametric analyses are essentially the same as for the 

explicit case, at least in our initial model formulation. There is a possibility that, with some

model formulations, the implicit case may be handled more efficiently through other methods. 

In any event, there is then a “loop” transition back to the earlier monitoring mode. Any update 

results in a reassessment of the situation as just noted. As discussed elsewhere, updates ideally 

are handled as they arise, essentially randomly with regard to type, confidence, and time 

distribution. We have considered in earlier work (10) methods for transforming, basically 

through interpolation, these into a uniform distribution for ease of computation. Whether this is 

the best approach from a tactical standpoint, particularly with regard to concerns of levels of 

indirection, remains to be seen. Notions of sensitivity and computational efficiency also enter 

into the development. That is, not all updates are of the same utility. Some, for instance, may in 

effect corroborate the existing model and some may not warrant much additional calculation. 

The rub, of course, is how to ascertain this in a reasonable manner, given the dynamics of actual 

combat. 

2.3 Decision Aid 

The system should develop and plot “envelopes” concerning the trajectory. We had many 

philosophical discussions among ourselves and with Battle Command Battle Lab personnel 

9
about what this really means. From a technical standpoint, it is not too difficult to calculate 

“parallel” curves (analogous in some ways to confidence intervals) that bound the trajectory by 

placing a range of one or more parameters into the mix. It is debatable as to whether this has 

tactical utility; for example, in enabling the commander to more easily perform risk or sensitivity 

analyses concerning the feasibility of his plan. Ultimately, we intend this to be the subject of

Advanced Decision Architectures Collaborative Technology Alliance (ADA CTA) 

experimentation. 

At any point, we can compare, at least conceptually, planned (desired) Blue and Red trajectories 

with actual values (or estimated/projected values). Further, we can modify any of these 

trajectories (separately, or more realistically, together) by changing certain defining parameters, 

and perform additional comparisons for sensitivity analysis. We would like the Blue commander 

to receive warnings like “unless the Blue kill rate is doubled by time τ , the goal will not be 

met.” 

The decision aid should also, in a real sense, answer questions the commander might have 

concerning the planned or unfolding battle. Again, this has been (and continues to be) the 

subject of much debate. As examples of the kind of “what-if” questions that appear universally 

desired, we set forth the following. If the enemy maintains this intensity, how much 

reinforcement do I need at time τ to reach my goal? How much do I need to increase my kill 

rate or decrease his in order to win? If I desire the battle to be over by some time interval, and 

know initial troop strengths, what relationship between attrition rates must hold? Such things 

seem intuitive, indeed even straightforward, yet couching them in forms that are mathematically 

tractable while militarily reasonable is proving to be a mixture of both art and science. 

3. Initial Formulation 

3.1 Motivation

As mentioned in subsection 1.3, analysts have used DEs for years to model aspects of combat, in 

particular attrition. Of particular interest is the possibility of parameterized studies to "optimize" 

tactics, force deployments, and weapon characteristics. For instance, suppose friendly forces 

face an enemy in two echelons. Is Red better served by committing the second echelon at some

particular time or by continuous reinforcement? How should Blue counter such tactics?

New types of DE models may permit other kinds of investigations. One aspect that might be 

pursued is application of (multi-variate) control theory to determine (even existence of) optimal 

responses in combat situations. Perhaps we can characterize through proper variables the state of 

the battlefield and what might be meant technically by an objective state. Then we can analyze 

whether the conflict system can be controlled (perhaps via feedback loops) and whether any state 

10
is optimal. Ultimately, such work might be applied to multistage optimization for monitoring 

and managing battle execution. 

This research could also yield reduced complexity for large simulations; sensitivity analyses may 

show the relative importance of parts of the model, even allowing for reduction of scope if 

certain items are found to be superfluous. In any event, it would be useful to investigate 

situations and parameters for which greater “fidelity” may not necessarily be better for the 

analysis. 

3.2 Model 

These observations having been offered as motivation, we now consider the initial model, a set 

of ordinary DEs: 

, dx dy ay bx

dt dt

⎧ ⎫ ⎨ = − = − ⎬ ⎩ ⎭ . (1)

For obvious reasons, we set forth the constraints . Subsection 4.1 shows why this 

model is known as the Lanchester “square law” formulation. 

x t( ), y(t) ≥ 0

The x initial force level and breakpoint are known, the y initial force level and breakpoint 

assumed, and the time to end desired. The “White” commander (that is, ground truth) inputs 

initial Red level and the kill rates. The friendly commander inputs initial Blue force level, 

desired rate at which Blue kills Red, and desired goal state. Examples of goal forms are 

{ | ( ) f y y t }ω ∃τ τ τ = ∩ < % , (2) 

where t

ω

% can be infinite and 

{ | ( ) ( ) } f f ∃ = τ y y τ ∩ x τ > x . (3)

The Blue commander faces the situation { . Given the starting levels, the kill rates,

and the breakpoints, strictly determined are the winner (if not a draw, which is unlikely), ending 

force levels, and battle duration. Our particular desired end state 

} 0 0 0 0 , , y a ,b ( ( % x

Ω% was defined to be “Red 

reaches his breakpoint before (condition 1); Blue reaches his breakpoint after (condition 

2).” In general, we plan based on the initial set , where the first four 

elements drive the trajectory. However, at any time we can change any such parameter and reevaluate. We now “start the clock,” and the system traces x(t) and y(t) . The system also 

calculates and displays in manners to be discussed later the required parameters when changes in 

force levels are observed or postulated. 

t t ω

% ω%

0 0 0 0

ˆ { , ˆ ˆ , ˆ , , ˆ , ˆ , } x y x y a b p p t

ω

%

3.3 Sketch of Algorithm 

We now build on the discussions of section 2. The prototype development is based on the 

following sketch of an algorithm, which is broken into four subaspects: inputs, end state 

determination, correction guidance, and battle monitoring. (Our main development is couched as 

11
the Lanchester square model, but, again, keep in mind there is nothing magic about that 

particular formulation. It just enables a convenient prototype and facilitates discussion of 

subsequent concepts.) The reader will see that development of even an apparently 

straightforward decision aid of this sort offers a number of significant challenges. 

In general, we will have a model formulated in terms of a set of equations. Now, in augmenting 

this set of equations to form a system, we can input “knowns” (e.g., Blue force level), 

assumptions (e.g., Red force level), and desiderata (e.g., time to end). Input the model, meaning 

the formulation, the representation of the battle, be it a system of differential equations, a 

complicated simulation, or whatever. Input the parameter values, meaning the numbers or 

reasoning symbols that drive the current manifestation of the model, which as indicated, has 

several components (actual, postulated, desired, modeled). Input the desired end state, meaning 

the situation in the conflict to which the Blue commander strives and is requesting assistance 

from the decision aid. We must determine that all parameters are established as constants 

(although see section 5 on notions such as bounding and what-iffing); initially, this is done 

through interaction with the user. 

Compute the trajectories (meaning the trace of the conflict in parameter space) based on the 

solution equations, through numerical integration, or whatever method is appropriate. Plot them, 

or at least the one selected as being of interest in the chosen display (for visualization). Values 

are noted as actual or postulated in portrayals of current/projected battle. The computation 

immediately establishes whether the end state can be met. If the end state is achievable, reassure 

the user; if not, compute the region (which may be complicated, as will be discussed later) 

permitting achievability in the overall parameter space. The aid must guide the user to determine 

what changes might be made to bring about the end state. If the end state is mathematically 

impossible, the user must be so informed; if the aid determines that it is reasonably impossible to 

complete a required change, then the user should be given a threshold at which to break off 

consideration. 

The next portion, a looping routine, deals with battle monitoring: when a value changes or an 

unknown becomes a known value, update (improve) the model and recompute the trajectories. 

The information may be implicit, such as Red kill rate based on assumed b , assumed , and 

known . Again, ability to reach the end state is recomputed. If the end state cannot be reached, 

the system recomputes the changes needed. The looping for new information then continues. 

y

x

Let us now discuss in a little more detail certain aspects of the algorithm. The basic initialization 

subalgorithm first merely asks the user to input what he knows, what he assumes, and what he 

desires. It then checks whether these inputs are consistent, that is, whether the desiderata are 

achievable with the given inputs. The user initiates the decision aid program, which identifies 

itself to the user and cites the time. The program then prompts the user for inputs in the planning 

stage. First, the knowns: initial Blue force level 0 x , Blue breakpoint x p (defined as the fraction 

of 0 x at which Blue stops the conflict), and time to start (which is used for 0t

12
planning/replanning purposes in the eventual decision aid; for most of our mathematical 

development at this point, we can take as 0 without loss of generality). Second, the 

assumptions: Red force level , rate at which Red kills Blue 

0t

0 y

( a

(

, rate at which Blue kills Red b

(

, 

and Red breakpoint (defined as the fraction of 0 y

( at which Red stops the conflict). Third, the 

desiderata: in this formulation, time to end . There are of course many “mechanical” aspects to 

this. For example, “sanity checking” of breakpoints must ensure these are percentages (and, 

perhaps, reasonable percentages). We assume that such aspects are handled by the program;

ideally, they are foolproof graphic inputs. We also assume for the prototype that all these basic 

inputs are established; however, dealing with uncertainty in the values must form an important 

part of an actual system. 

The program computes Blue and Red trajectories, based on the strictly-determined solution 

equations, and plots them starting at and ending at the highest time of interest. Figure 2 

illustrates this schematically. The plot shows time as the abscissa and force level as the ordinate. 

Moreover, the breakpoints are cited as both time and level for both x and y forces; is given as a 

vertical line.

0t

Figure 2. Blue and Red force levels over time. 

Let us call the modeled time at which Blue reaches his breakpoint ˆ : xt 0 ( ) ˆ

x x x t = x p , where we 

have taken advantage of the monotonicity of the trajectory. Similarly, we have $ 0 ( ) ˆ

y y y t = y p ( ( . 

Now

y 0 p y

( (

0t tω

%

x

y

x 0 p x

y p

(

t

ω

%

t

ω

%

13
The program checks whether is achievable: condition 1 of < and condition 2 of < . If 

these conditions are met, the program so informs the user. 

Ω% ˆ

x ˆ t

y t t t

ω

% ω

%

Now the plot thickens, so to speak. If condition 1 is not met, we want the decision aid to guide 

the user to correct the situation for planning. More-or-less obvious suggestions can be made 

involving the “raw variables” end time and Red breakpoint; for instance, relax the end time to 

accommodate the calculated value. (Such suggestions actually turn out to be somewhat less 

obvious than what might be thought, due to their dependency of these on the larger “mix” of 

variables. This will be considered later in more depth.) However, we want more sophistication 

for our prototype decision aid. 

In order for trajectories to have value in what-iffing and sensitivity analyses, we must have a way 

to compare them. Comparisons based on time would seem relatively straightforward, and 

indeed, as a practical matter, this is a tactically reasonable way to proceed. That is, are the 

desired parameter values plotted over time as the commander desires? (This will be examined in 

terms of force levels in the initial formulation. However, deviation from plan is difficult to 

assess even in this simple case, as will be seen.) We might consider comparing parameters 

without the notion of time. This at first seems more abstract as far as tactical applicability. 

However, meaningful comparisons can be made; again, as will be seen in the initial formulation 

when we examine relative force levels as determining the outcome. 

One question the friendly commander will ask is, “What can I do to move the enemy breakpoint 

time to where I want it?” Since he has no particular influence on or a 0 y ( (other than by 

changing his assumptions, which we consider in a subsequent discussion of sensitivity), we see 

we may want to solve for 0 x or in terms of desired enemy breakpoint time. At this point, we 

need to consider some mathematics. 

b

(

4. Theoretical Preliminaries 

4.1 Fundamental Notions 

We begin with some terminology. Let us consider a system 

{x& & = f t( , x, y), y = g(t, x, y)} , (4)

with 0 x(0) = x and , where both 0 y(0) = y f and have continuous first partial derivatives. The 

functions ,

g

t x = x t( ), and y y = (t) in an interval define a solution if 

x&( )t f = (t, x( )t , y(t)) (5)

and 

y t &( ) = g(t, x( )t , y t( )) (6)

14
 for all t in the interval. Now the (x, y)-plane is called the phase plane and the parametric curve 

{x = x t( ), y = y(t)} is the trajectory of the system. If the points of a trajectory are in one-to-one 

correspondence with the parametert , the direction in which t increases is called the positive 

direction; arrows are sometimes used in plotting to indicate this. A collection of trajectories is a 

portrait. 

Generalizing to higher dimensions, the vector traces the trajectory in the -dimensional 

phase space of the state vectors, where 

x(t) n

x x = (t)is a solution of x& = f (t,x) in an interval. The

point (t , x(t)) traces a time-state curve in the n +1-dimensional space of the time and statevariables. The projection of this curve onto the ( , )-plane is the i t x i x -component graph. Note 

that in subsequent discussions we may speak of component graphs or other functional traces of 

parameter values as “trajectories,” the meaning being clear in context. 

An autonomous system has the property that only initial position and elapsed time are important 

in phase space: if 

{ x x = (t),α < <t β } (7)

is a solution, then so is 

{ x x = + ( ) t k ,α − k < t < β − k , k constant}, (8) 

and the trajectories are seen to be identical. Whether combat in general is an autonomous system

is open to debate; but note that t does not explicitly appear in equation 1, so our initial model is 

autonomous. Anyway, it is apparent autonomous systems are simpler to analyze. 

Continuing to look at our model, since is not equal to zero in the first quadrant (the region 

of tactical interest), we find 

dx / dt

/

/

dy dt bx

dx dt ay

− = − (9)

or 

dy ay bx

dx = , (10)

where time has, in effect, been replaced by x as the independent variable. Put another way, the 

initial formulation yields the ratio of the two equations in the system: 

dx ay

dy bx = . Now writing 

this as 

0 0

bxdx aydy

τ τ

= ∫ ∫ (11)

yields 

15
2 2 2 2

0 0 b x( − = x (τ )) a( y − y (τ )) , (12)

or 

, (13) 2 2 2

0 ay ( ) τ τ − = bx ( ) ay − bx 20

hence the characterization of equation 1 as the “square law” model (11). 

4.2 Analytical Results 

The so-called “state equation” 

(14) 2 2 2 2

0 0 b x( − = x (t)) a( y − y (t))

(which does not explicitly consider time as a variable) is in itself useful for deriving results of

potential utility in our analyses. For instance, consider the force level ratio . It turns out 

this ratio satisfies 

r x = / y

dr 2 br a

dt = − . (15)

This equation provides qualitative information about the force ratio: x is winning if and only 

if x / y > a / b ; a necessary and sufficient condition that x wins a battle to annihilation in finite 

time is 

0 a b/ < x / y0

2

 (12). (16)

That is, battle outcome is determined by the initial force ratio and the relative attrition, even 

though there are four parameters in the model. 

Again, writing 

, (17) 2 2 2

0 0 ay ( )t − = bx ( )t ay − bx

and calling the right side constant c , note that if c < 0 , then x wins, since x does not vanish, but 

y = 0 when x = c / a . Blue wants conditions in which 

2

0

0

y a

x b

⎛ ⎞ ⎜ ⎟ <

⎝ ⎠

. Due to the quadratic nature 

of this inequality, note that a small increaseε in could change the predicted win to a loss: 0 y

2

0

0

y a

x b

⎛ ⎞ + ε ⎜ ⎟ >

⎝ ⎠

 (13). (18) 

Such calculations form the basis for the dynamic decision aid. Note also the notion of “basin” 

(in the sense of qualitatively different areas of the space) that surfaces in this last observation 

concerning ε . 

16
Similarly, the state equation can be manipulated to derive solutions for the force levels as 

functions of time: 

0 0 ( ) cosh sinh a

x t x abt y abt

b = − (19) 

and 

0 0 ( ) cosh sinh b

y t y abt x abt

a = − . (20)

Further, we have 

0

0 0

( ) cosh sinh x t y a abt abt

x x b = − , (21) 

showing the force level depends on three derivative parameters: , , and . It is seen 

that ab controls the length of the conflict. Such derivations have been pursued in great detail by 

several researchers, in particular, by Prof. Taylor (12). However, for decision aiding we must 

consider more realistic conditions such as breakpoints and desired time to finish. 

ab 0 y / x0 a / b

Consider the four main inputs to the model: x, y a , , and b . These all impact the modeled Red 

breakpoint time, which we shall denote . Let us solve for , and then consider the change in 

each variable, at first separately, that will result in condition 1 being met. Later, as an 

improvement to the decision aid, we will explore the tradeoff in parameter space that will enable 

this. Since c ands , some algebra and the reasonable 

assumption >> 0 yield 

osh ( )/ 2 x x x e e− = + inh ( )/ 2 x x x e e− = −

ˆ

y t

ˆ

y t

ˆ t

y

0

0 0

2 ˆ ln y

y

y p

t

b

y x

a

≈

−

( (

( ( (

. (22) 

We now have a relation between the modeled enemy breakpoint time and the four Lanchester 

parameters. We can now develop sliders involving multidimensional plots of the parameter 

space under these conditions. For example, we can fix = t

ω

% , , and 0 y a

(

; and then show a plot 

of 0 x vs. as an indicator of the tradeoff between initial force level and kill rate required to bring 

about the end of the conflict by the desired time. (This, of course, assumes condition 2 is met. 

We must at some point in the tradeoff analysis check whether the Blue force is diminished to too 

low a level at the desired end time, clearly an unacceptable situation.) Alternatively, we find 

b

( ˆ

y t

0 2

0

[ ( 2 )y

y b a e e p

x

τ τ ≈ − ]

( ( ( % , (23) 

where τ here indicates the desired time of enemy break, computed based on the model solution 

17
0 0 0 ( ) cosh sinh y

b

y p y y ab x ab

a

τ = = τ τ − (24) 

and again assumed >>0.

Let 

, (25) 2

0 0 m y = / , x a = k b

with all quantities positive. Based on equation 19, the Blue strength level at any time can be 

written as 

0 0 0 0

1 ( ) [( ) ( ) ] 2a a abt abt x t x y e x x e

b b

− = − + + , (26) 

which implies 

0 ( ) [(1 ) (1 ) ] 2

x a kbt a kbt x t m e m e

b b

− = − + + . (27)

Note that as , , 0 . Moreover, kbt kbt t e e → ∞ → ∞ − → 1 a

m

b

± are constants. If x loses 

(i.e., x → 0 ), then it is necessary that 

2

2 1 0 1 a a

m m a

b b

− ≤ ⇒ ≥ ⇒ ≥ b

m . (28) 

Similarly, if 2

b

a

m

< , then Blue wins (i.e., ). Also, note the further one deviates from this 

critical point where 

y → 0

2

b

a

m= , the more decisive the win. Critical values for a and b such that 

2

b

a

m= result in a battle where both forces essentially go to 0. Values of a greater than 2

b

m

result in Blue losing, and of a less than 2

b

m

, Blue winning. 

Let km > 1, i.e., Blue loses. Rewriting equation 27 using k, m, and 0 x , and letting E = ekbt, we 

get 

0 (1 ) ( ) [(1 ) ] 2

x km

x t km E

E

+ = − + . (29)

Blue may decide to end the battle when its strength reaches a level less than 0 x , say x 0 p x , 

where [0,1]. Then we have the equation x p ∈

18
0

0

(1 ) [(1 ) ] 2 x

x km

p x km E

E

+ = − + . (30)

If we multiply equation 30 by –E we get 

2 ( 1) 2 (1 ) x km − E + − p E + km = 0. (31)

The roots of this equation are 

2 2 2 1

1

x x p p k m

km

− ± + −

− . (32) 

Since > 0 and the root kbt E e =

2 2 2 1

1

x x p p k m

km

− − + −

− is less than 0 for km > 1, it cannot be a 

valid root. Thus, the only root of interest is 

2 2 2

0

1

1

x x p p k m

E

km

− + + − = − . (33)

In the event x loses, values for will give a real root of equation 33. In the event x

wins, then the x strength level does not fall below some critical value

[0,1] x p ∈

xc 0 p x , while the y strength

falls to 0, and a root exists if x xc p ≥ p

E = E0

; otherwise, no root exists. A similar argument holds for y. 

We can then solve the equation or 

0 exp(kbt%ω ) = E (34)

to obtain 

0 loge E

t

kb %

ω = . (35)

In general, equation 30 can be solved numerically for one of the variables { x p , ,b ,t }, given 

the other three. The decision aid might use such calculations to help the commander address 

such situations as the following. I am involved in a battle, but by time intervals I need to be 

elsewhere to support another battle. Having determined I can inflict casualties on the enemy at 

rateb , what casualties can I accept in order to win the battle within time intervals and retain at

least

k

et

et

e 0 p x of my forces? The question can be answered by using equation 31 to find the attrition 

rate a corresponding to the strength needed and equation 34 to determine whether the mission 

can be accomplished within time intervals. et

We see that the algorithm can be modified to give the maximum attrition rate that allows mission 

accomplishment: answering this question reduces to solving equation 31 for , where , 

yielding the maximum attrition that allows mission accomplishment. A related situation is: if 

Red is winning the battle, at some point in time intervals into the battle what could happen in 

k 2 a k = b

et

19
terms of force strength or attrition rates so that Blue wins? The critical inequality to consider is: 

Blue wins if , i.e., if . A Blue win occurs by an increase in strength of 2 a b < / m

2

ay < bx2

/ e e t t ∆x > − a by x . (36) 

Alternatively, if Blue can decrease Red’s attrition rate by 

2

2

e

e

t

t

ay b b

x

∆ > − , (37) 

or if Blue can increase its own rate by 

2

2

e

e

t

t

bx

a a

y

∆ > − , (38) 

then Blue will win. 

5. Aspects of the Decision Aid 

5.1 Introduction 

We now examine this work from a somewhat different perspective. While still technical, this 

section is less concerned with mathematical modeling and computational algorithms. It sets 

forth a variety of human-centered display considerations for research and development in pursuit 

of an actual prototype decision aid for model-based execution monitoring. We postulated 

various displays that show the past, present, future; real, assumed, desired, modeled factors; 

boundaries, confidence levels; and other information of interest to the commander. Almost any 

one of these could be a project in itself, and we intend that they be approached in an orderly 

manner by a design/programming team. 

At some point in monitoring the battle, we may receive new information. As sketched earlier, 

we can input the updated parameters and compute a new trajectory from that point (and check 

that the desired outcome is obtainable). We use the former model’s values at that time to form

the non-new information, such as Red strength level. A more complicated computation would 

involve fitting the form of the model to the observed data, and this sort of approach is discussed 

in some detail in section 7. A more sophisticated interface would be required to explore the 

effects of model formulation on conflict trajectory. Curve fitting or regression software might be 

leveraged for both these purposes. An abundance of information is available on curve fitting, 

and many of the software packages mentioned in section 6 utilize curve-fitting algorithms. 

Continuing in a bit more detail, let us again distinguish among model values; that is, consider 

separately the “best estimates” (e.g., $

x(t)) and the actual reported values (e.g., x(t)). Let us now 

20
consider the battle to be on; the system is in execution monitoring mode. At time τ , the Blue 

commander receives a report that x( ) xτ τ = . If $ x( ) x x( ) τ τ = = τ , he is on track with regard to the 

plan playing out properly. If $ x( ) τ > x(τ ) (and all other things being equal – which they are 

probably not, as discussed elsewhere), he is doing better than expected with regard to this 

parameter of interest. The system now modifies the working model (or more properly for the 

initial prototype, its set of parameter values) to reflect this latest strength information in the 

solution set 

$( ) cosh ( ) ˆ sinh ( )

ˆ ˆ ( ) cosh ( ) sinh ( )

a

x t x ab t y ab t

b

b

y t y ab t t x ab t

a

τ τ

τ τ

τ τ

τ τ

⎧

⎪ + = + − + ⎪

⎨

⎪ + = + − + ⎪

⎩

τ

. (39) 

A new estimate of the end state is also calculated, and if the Blue commander is not satisfied 

with the situation, he can estimate the consequences of changes in parameter values. 

Similarly, if $ x( ) τ < x(τ ), the commander is in trouble (all other things being equal). Again, the 

model is modified and a new estimate of the end state calculated (presumably Red will win or 

Blue will be too late). What can the commander do to “correct” the situation? We might 

assume x(τ ) must be increased to $

x(τ ) in order to get back on track; however, a problem arises 

due to the probability that Red is now better off, so computations involving y(τ ) as well now 

become germane. 

At any time, we should be able to display graphically for the commander various aspects of the 

trajectory spaces. Of fundamental importance is what region he is in, in order to assess whether 

he is meeting his goal with respect to his chosen criterion or criteria. Of course, auxiliary 

windows should indicate position in different parameter spaces. The commander would usually 

access such displays purposefully, but if some dangerous situation arises that requires alert 

(perhaps as determined by methods discussed in section 8), the appropriate auxiliary alert 

window should be displayed automatically. 

The commander can also be presented with options given a fixed parameter. That is, the 

commander would choose a value for “parameter 1” and be presented with a curve in the space 

defined by “parameter 2 (abscissa) and parameter 3 (ordinate)” yielding that value. Multiple 

such curves should be plottable on the same set of axes. 

5.2 Trajectory Tracking 

It is fundamentally important to be able to plot trajectories under various assumptions. These 

could generally be displayed in sets, perhaps analogously to iso-surfaces. The best way for the 

user to immediately grasp information being portrayed is an area of active research that is 

beyond the scope of this effort. However, the reader is referred to pioneering work being done at 

the Institute for Human and Machine Cognition (14). 

21
A related aspect is computation and display of rates of divergence of the actual trajectory from

that planned. For simple trajectories and concern with a single parameter, such divergence can 

be handled in a basically intuitive manner. However, determining what this really means in a 

stringent technical sense turns out to be somewhat difficult. It is hoped that subsequent research, 

probably utilizing Lyapunov analysis (discussed in subsection 8.4) will shed more light on this 

problem. 

Another aspect of the trajectory of concern to the planner and execution monitor is the rate of

change along the chosen trajectory. Auxiliary concerns arise analogously to those just discussed. 

An area of research addressable by cognitive scientists in the ADA CTA (15) is whether 

transformation to the first derivative should be done by the computer prior to displaying such 

information. That is, are points in a “derivative state space” easier for the user to interpret than 

following the actual motion along the trajectory? In any event, information such as time to reach 

the boundary of a critical region if no parametric values are changed must be developed and 

displayed to the user. 

An even further extension is consideration of “acceleration” in phase space. That is, analysis of 

second derivatives could help the commander with questions of criticality concerning model 

parameters, particularly with regard to time. For instance, if the approach to an undesirable 

region is increasingly rapid, determination of parameters and correction of values becomes of 

paramount importance. One issue that arises in such extensions is that of meaningfulness. That 

is, as the analysis of the conflict takes on more degrees of indirection, it perhaps becomes less 

relatable to the actual tactical situation. At the very least, the notion of sensitivity vis-à-vis 

numerical analysis is germane if only in a technical sense. 

One relatively easily developed plot is that of Blue (Red) trajectories as a function of Red (Blue) 

initial force level. With an appropriate graphical user interface (GUI), the user can explore times 

to reach certain levels as either “dependent” or “independent” variables. This technique also 

illustrates the notion of a simple possibility envelope, achieved by setting bounds on the assumed 

initial level.

Another approach to the overall system design shows some promise: develop an allowable 

envelope and track the actual trajectory in that context. Figure 3 illustrates this schematically; 

the actual trajectory is plotted, with alerts shown as the envelope is intersected. This seems 

particularly attractive in light of the desire expressed by the U.S. Army Battle Command Battle 

Laboratory that the user be provided not only an alert but also what to do about it. Of course, 

this could be made more sophisticated by incorporating simple devices such as having the past 

plot fade with age or by color-coding the breaks according to exit/enter top/bottom. Further, the 

user could be warned and guided based on derivative analysis of apparent approach to the 

envelope. Time to reach the boundary of a critical region if no parametric values are changed 

could also be displayed. 

22
Figure 3. Planned trajectory with alerts as actual trajectory breaks envelope. 

As mentioned earlier, we can plot various aspects of the battle. In particular, in our Lanchester 

formulation, we can look at the force ratio trajectory. This trajectory is of particular interest, 

since it is readily indicative of battle-state regions, and is a good starting point for further work 

on nonlinear dynamics in the next formulation. 

5.3 Sensitivity Analyses 

We now have, based on the given initial parameters, plots of the Red and Blue trajectories and an 

assessment of whether the enemy can be defeated. Another form of decision aiding can be 

sensitivity analyses in the form of overlaying on the “real” trajectories variations in which certain 

parameters are “dithered.” For example, the commander should be able to have sliders in which 

he changes 0 x ,b

(

, , or singularly or in multiple fashion. (Such notions will be discussed in 

more detail shortly.) The manifestation on the plot would be shifting trajectories in which the 

user can see the influence. Of course, the breakpoints can also be changed, resulting in moving 

of the “crosshairs.” All these are readily computable from the solution equations. 

0 y a

(

23
Various kinds of sensitivity analyses appear to be useful to the staff. Some of these are discussed 

in other subsections. One way to perform them is having the user choose a parameter(s) to vary, 

with the others defaulting to (pre)selected values. With a reasonable GUI, the user can easily 

and dramatically explore interactions among the model parameters, such as the influence of 

enemy initial strength on friendly loss. Derivative aspects of the trajectory would seem to be 

worthwhile, especially in an extremely time-critical environment. This notion includes rates of

change of aspects of the battle, and how they influence the outcome. Dithering certain values 

and rerunning (or running in parallel) the repercussions would seem to be useful, especially for 

logistical analyses. This notion includes consideration of possibilities of bad intelligence. As an 

example, the commander is probably immediately interested in analyses of what happens if the 

enemy force is 10% larger than he is estimating for the game. Again, the actual tactical utility of

such capabilities remains to be measured. 

A more sophisticated guidance mechanism would indicate to the user the qualitative nature of his 

proposed changes. For example, given that the user wishes to “raise” the blue curve, there 

should be a plot in n-dimensional parameter space showing the region(s) in which this occurs. 

We considered natural language type conversations and symbolic aspects of the system to be 

beyond the scope of this project. However, we will consider a simplified version of this in order 

to explore the navigational aspect of the decision aid. Let us have a tactically reasonable input 

for the user: desired end time, in the sense of Red break point manifestation. The trajectories are 

strictly determined, and the end time can be slid back and forth over the graphs, with a changing 

read-out of py, where all other parameters are fixed. This changing readout has nothing magic, 

though, about py; by fixing py and selecting, say, x0, the sliding end time can show the changing 

read-out in terms of Blue initial force level required to bring about the desired end time. This 

can be done for any of the parameters; there is a color-coded indicator to the user of 

given/assumed in the plots and numerical readouts. It is, of course, up to the user to exercise 

proper military judgment in assessing the options resulting from such “dithering” of the decision 

aid. 

We plot x(t) and y(t) from t0 to, say, twice the input py. (Again, a nice aspect of the graphical 

decision aid would be to have this window stretchable and automatically scaled.) The input 

delimiters (e.g., Blue breakpoint) are plotted with both vertical and horizontal lines dropped to 

the axes. The user inputs a time to end t

ω

% by clicking on the plot; a vertical line appears through 

that time, and horizontals as well from the force level plots. All abscissa/ordinate pairs are 

displayed on the plot, having been immediately calculated from the equations for x(t) and y(t). 

An auxiliary screen appears. In it is displayed the py associated with the time to endt

ω

% ; this is 

computed easily from . Now the user can drag this vertical line left or right, and the 

changing py is computed and displayed continuously, with the slider “stopped” when py reaches 0 

or 100%. 

y t p 0 ( ) y y %

ω =

Now, let us thicken the plot, so to speak. What is the display like if there are two variables being 

changed? Let us take a specific example again. We select py and x0 as the changeable variables. 

24
Now each position of sliding the desired end time results in a locus of pairs (py, x0) that produce 

the desired result. Since for any selected τ we have the model solution (assuming the original 

parametric values) as 

0 0 ( ) cosh sinh b

y y ab x ab

a

τ = − τ τ (40)

 (and 0 0 ( ) cosh sinh a

x x ab y a

b

τ = − τ τb ). (41) 

Note that these make tactical sense only until one side is annihilated. We can now compute from

the y-trajectory the function 

1

0 0 (cosh )( sinh ) y

b

x y ab p ab

a

τ τ − = − . (42) 

This curve can be shown in an auxiliary screen; the curve changes as the slider moves, 

changingτ . Note that there is a relationship between py and x0, rather than having them strictly 

determined. This may seem strange because, after all, we have two equations in two unknowns 

for any givenτ . However, the link between the trajectories for x and y is strict, due to the nature

of the initial model previously analyzed. Moreover, we had no problem considering py in the 

one-variable case. This gives a kind of reasonable assistance to the commander, in that he can 

see what his initial force level must be in relation to the Red breakpoint, given a desired end 

time. Of course, we must still check for a reasonable x trajectory. 

Continuing, let us consider three variables as changeable: we select , 0 x , and . Each position 

of sliding the desired end time results in a locus of triples ( ,

b

(

0 x ,b

(

) that produce the desired 

result. This surface can be shown in an auxiliary screen, with the surface changing as the slider 

moves. Now we enter the realm of information displays per se. For example, software can be

utilized/developed that enables the user to rotate the surface for better examination. Another 

possibility is to let the user fix one of the three variables and produce a curve display as 

previously shown. However, again, we considered this beyond the scope of the project. 

p

py

y

A desirable feature of a COA in general is that it be “robust.” This is usually interpreted to mean 

that the plan produces good results even in the face of widely differing enemy COAs, and is 

assessed by wargaming against a most likely and a most dangerous enemy plan. In this modelbased decision aid, we would like to continue this notion of robustness, but probably in the form

of risk assessment. It seems that sensitivity analytical techniques can be used to examine 

subtleties in the battle construction that could result in “squeakers” or the onset of chaos (a 

notion discussed in subsection 8.3). These techniques can be either closed-form analytical for 

relatively simple systems like the prototype uses or numerical analytical (or even simulationbased) for more complicated models. Development of what might be called confidence bands is 

25
a worthwhile extension of the notion of trajectory envelopes: a new idea is to use deltas in 

parameters to yield bounds and hence probabilities. 

5.4 Qualitative Behavior 

Let us return to the notion of examining the qualitative behavior. As a practical matter, the user 

may not be looking for exact values of the changeable variables (at least not at first); the user just 

wants to see whether increasing one of the parameters, say, or increasing one and decreasing 

another, brings about the desired result. We now enter the realm of partial derivatives. For 

example, go back to the initial example where we were sliding the end time and looking at the 

changes of py. If we differentiate the solution curve(s) with respect to that variable, we have 

another equation that can be examined similarly. But this one is in a sense more straightforward, 

in that we have natural regions of positive and negative (“slope”) behavior separated by zeroes. 

This can be transformed back automatically for the user onto the original plot showing where the 

variable produces the desired change. 

This idea is generalizable. Suppose the user wants to examine the effect on certain variables 

within the model of changes on other variables (where the “change” may involve fixing it as a 

constant). This requires a shift in thinking to considering the model as a relation or set of 

relations among the variables. 

The notion of “possibility envelopes” is a bit harder to describe. Perhaps it is best to think of it 

as examining a “tradeoff space.” It is basically a kind of inverse problem, where the conditions 

that would bring about a desired situation must be derived. An example might be to calculate 

and display a function of several parameters (a curve or surface in three-space, for example) such 

that any point in this set yields the desired result or range of results, given a selected model 

formulation. 

If any of the enabling conditions is violated, a yellow region can be visualized; if violated 

beyond someε , a red region. Of course, it is problematic how to assess ε scientifically, as 

opposed to just having the user make educated guesses. It is important that the system assess at 

every time (or “phase” of the battle, in the sense of agreed-upon tactically-significant events) 

what conditions would result in not meeting the goal. For example, from the perspective of 

( , x ) τ τ what region of -space causes: (1) defeat of Blue (a red situation), (2) longer time

to win than desired (yellow situation), and (3) more Blue casualties than desired (yellow 

situation). Such assessments, if properly visualized (perhaps overlaid with probability curves, 

another challenge), can help the commander assess risk/sensitivity and needs for intelligence or

change of tactics. If a slight increase in a would cause many more Blue casualties, we may want 

to reassess our COA. Visualization may be based on a fixed 

( , y a,b)

( , x ) τ τ , for example, so that the 

desired region is a kind of “floating rectangular solid.” A more sophisticated development, but 

still based on the mathematics of the model, would enable the user to click on any point in this 

space to get the projected time to defeat or lose, Blue casualties, and so forth. Another type of 

26
visualization is exemplified by fixing a at a chosen value and having the system plot curves of 

constant time to end. 

So we see how we can visualize red and yellow situations for a given model. How do we 

compare plan and reality? For a desired end state and a given set of fixed parametric values, we

can assess our ability to reach the end state. We can click on a point in red/yellow regions to 

check on values of the orthogonal space that might enable the desired end. We can play the plan 

model in real time and note if snap-shotted actual-combat values fall into red/yellow regions. 

But it would be most valuable to track the motion of these values, warn the user if any is headed 

into a danger region, and offer suggestions about possible corrections. Moreover, these concepts 

are intimately related to the techniques for analyzing dynamic systems. The philosophy and 

mathematics behind such a system is discussed in more detail in section 8. 

5.5 Other Desiderata 

Obviously, the ability to display information about all the various parameters is essential. This 

includes human-centered displays of values (whether actual, assumed, or modeled), relationships 

among them, allowable ranges, and so forth. A related notion, previously alluded to, is that of 

implementing mouse sensitivity of various objects; moving the cursor over certain regions or 

clicking points on a given curve/axis would automatically display more information about that 

object. For example, choosing the projected portion of a trajectory might display the actual and 

assumed parametric values that yield it. Moreover, many aspects can be animated, with colors 

indicating the nature of the portrayed information, such as modeled based on real data, assumed 

data, or both. 

At a given time, there are generally a set of conditions that guarantees the Blue goal, a set 

guaranteeing the Red goal, and a set guaranteeing a “draw.” Note that we say “generally.” 

Exhaustive characterization/determination of such sets for an arbitrary model is important 

follow-on work. As mentioned, determining reasonable techniques to display these conditions is 

ongoing research in several arenas. 

It is considered desirable in some circles to show the user the battles. That is, a graphical 

portrayal of the predetermined simulation and/or the unfolding conflict is displayed on a 

computer screen filled with tactical overlays. This aspect is somewhat tangential to the intent of

this initial work. The model-based results of the prototype decision aid could, however, be tied 

into such a display by future engineering. It is hoped that some emerging results of the ADA 

CTA can be leveraged in this regard. 

A related notion is to tie in our work on developing and visualizing “terrain ownership.” Again, 

the idea would be to augment the model-based decision aid results concerning the simulated and 

actual battles by providing the user with a distilled portrayal of realizable combat power in the 

area of operations (7). 

27
Many of the sorts of engineering details we have been discussing may seem to be trivial 

mechanical implementations. Once the mathematical underpinnings are developed for the 

model-based execution monitoring, however, such display aspects are vital to enable a 

reasonable decision aid, and should be developed in conjunction with the other analytical 

aspects. Indeed, we see that many of them are complex mathematical and cognitive challenges 

in themselves. 

6. Software 

6.1 Consideration of Options 

Based on the developments previously mentioned, there naturally arose several aspects of the 

project to which we thought to apply commercial software packages: differential equation 

solvers, curve fitters, and data plotters for experimental mathematics. We performed a fair 

amount of investigation into the availability of software both for solution of systems of 

differential equations and for display and evaluation of data in a GUI. We considered the 

particular design applications arising, and looked in this context at a variety of factors such as 

ease of use, maturity of development, reputation, and expense. 

DE solvers can be based in some senses on algorithms set forth in the Numerical Recipes series 

(16). Ordinary DE problems can always be reduced to sets of first-order DEs. New variables 

can be derivatives of each other and the original variable. Sometimes, one can incorporate some

power of the independent variable or other factors; different auxiliary variables can often be 

chosen to mitigate computational difficulties. Solutions can be obtained in finite form for 

equations of all orders having linear constant coefficients. Infinite series solutions can be 

obtained for many linear equations with variable coefficients. Good numerical approximations 

can be obtained for equations whose complexity makes them analytically intractable. It is 

probably true that all solutions of any (solvable) system can be found with reasonable accuracy. 

However, as alluded to earlier, many questions concerning a system can be addressed by 

considering only descriptive properties. One such notion, stability, will play a vital role in future

investigations. 

We considered many systems, including: MATLAB, Maple 9, LabVIEW 7 Express, 

OriginPro 7, Gauss 5.0, Scientific WorkPlace, FlexPDE 3, Nmath Core, Interactive Data 

Language, IRIS Explorer 5.0, PV-WAVE, JMP 5, STATISTICA, Mathcad, Mathematica, SPLUS, DPGraph, ActivityBase 5.1, JMSL, Diffpack, Tecplot, and EnSight. We even considered 

certain imaging software, but found it generally too specialized to image analysis per se. (An 

exposition of the plusses and minuses of these packages might be a useful derivative effort to this 

project.) As the reader can see, much such software is available as commercial packages. Of 

28
course, many existing Java tools and applets lend themselves to interactive plotting, as may be 

seen by a few Google searches. 

Discussions with various graphics and scientific visualization experts were useful in narrowing 

down the possibilities. Some of these discussions produced somewhat unexpected results. For 

instance, MATLAB certainly appeared to be a good possibility. It includes DE solvers,

multidimensional data support, and interactive capabilities for 2-D plots and 3-D surfaces. It 

enables customization of functional source code and linking to external software and data. 

However, discussions with the U.S. Army Research Laboratory (ARL) scientific visualization 

group were discouraging with regard to ease of use of the GUI and lack of availability of a 

current MATLAB “guru.” We similarly eliminated S-PLUS, although useful for canned 

statistical evaluations. We participated in an ARL seminar on Analysis and Visualization of 

Large Data Sets that helped us with visualization techniques, and a review of Major Shared 

Resource Center projects resulted in additional knowledge concerning high-performance 

computing software that might be leveraged. 

One possibility for enabling a visualization environment template for the prototype, and one that 

is a natural follow-on to collaborative work already being performed in the ADA CTA, is to

utilize the Seeker-Filter-Viewer technology of the Ohio State University Laboratory for Artificial 

Intelligence Research. This system, described extensively in various references (17, 18), utilizes 

a Java-based simple GUI by which the commander can specify what data to see in what kind of 

display. Another opportunity for collaboration is via work being done at the Center for the 

Representation of Multi-Dimensional Information (19). We have been interacting with Prof. 

Foresti at the University of Utah and, given a proper state of maturity, intend to utilize the 

sophisticated software developed by his multidisciplinary team. 

Various software packages exist for exploring nonlinear dynamics. All require a graphics 

display (such as video graphics array, enhanced graphics adapter, or color graphics adapter), but 

that should not be a real problem. Chaos, the Software, by Rucker, Autodesk Inc., illustrates 

various ideas of chaos such as the Mandelbrot set, attractors, magnetic pendulum, and fractals 

(20). INSITE is a software package of graphical interactive programs useful for examining 

nonlinear systems. Written in C and running in DOS or UNIX, it is a possibility for future 

algorithmic development (21). Phaser, by Kocak, is companion software to one of our 

references (22). A fairly complex program that covers a range of systems and iterated maps, 

Phaser permits the user to analyze specific difference/differential equations of theoretical or

practical importance with regard to their dynamic behavior. It has been used for undergraduate 

courses at Brown University, and has sophisticated interactive graphical capabilities that make it 

a useful exploratory research tool. However, it is somewhat dated. We have found another tool, 

discussed in the following subsection, that appears well-suited to our work. 

Physics Academic Software makes available several dynamics packages. Mapper, by Harold, 

computes trajectories, Lyapunov exponents, and Poincaré maps. It allows entering one’s own 

29
equations and handles 2-D iterated maps and DE systems. Chaotic Dynamics Workbench, by 

Rollins, generates state space diagrams, Lyapunov exponents, and Poincaré maps. It handles 

systems described by a few ordinary Des and permits changing parameters and examination of 

transients. Chaos Demonstrations, by Sprott, permits adjustment of certain parameters and 

demonstrates the Mandelbrot set, fractals, DE systems, and iterated maps (20). Moreover, other 

specific programs are in the literature. For example, Peitgen et al. provide a BASIC program

that computes a particular strange attractor (23). 

6.2 Dynamics Solver 

As just mentioned, the downloadable freeware Dynamics Solver 1.01 appeared to be an excellent 

fit for this project (24). However, we could not install it on laboratory hardware, due to 

organizational network security restrictions. However, we describe it here in the hope that 

software restrictions may be lifted for the next phase of this project. (We opted not to pursue 

funding for other packages until the issues were resolved.) 

Aguirregabiria developed the 32-bit edition of Dynamics Solver for Windows over several years. 

It handles initial- and boundary-value problems for continuous and discrete dynamic systems: 

single ordinary differential equations of arbitrary order, systems of any number of first-order 

ordinary differential equations, a large class of functional-differential equations and systems, and 

iterated maps and recurrences in arbitrary dimensions. 

Generally, no programming is needed. User-friendly dialog boxes are used for entry. Numerical 

results and complex graphics may be easily obtained. A built-in compiler translates a large class 

of mathematical expressions written in a standard format for rapid execution. One can also 

compute quantities involving the solution and its derivatives, parameters, and initial conditions. 

It is possible to draw phase-space portraits, Lyapunov exponents, histograms, bifurcation 

diagrams, etc. Results may be projected and subspaces of the phase space or space of initial 

conditions easily analyzed. Any problem may be saved to and retrieved from an editable disk 

file. The program is configurable and extensible, and a complete help system is available. 

Different kinds of results in graphics and text formats may be displayed in one or more windows. 

They may be sent to Windows-compatible devices or collected in a file for processing by other 

programs. Dynamics Solver may be used to construct many geometric figures, including 

parametric curves in two and three dimensions and a large class of fractals. The goal is to have 

completely correct figures in a device-independent format that can be translated for combination 

with output files from text processors. 

Dynamics Solver is a powerful tool for engineers and mathematicians as well as a teaching 

device concerning dynamic systems. It is a reasonable numerical laboratory in which problems 

may be more readily analyzed and comprehended and explanatory graphics developed, in many 

cases without programming. Borrelli and Coleman is a useful companion volume to such efforts, 

30
and we hope to utilize the program in such context in follow-on work (24). However, for this 

project, we were led to consider a contingency route. 

6.3 Contingency 

We have been developing an initial GUI by extending a Java-based plotting routine called JPlot 

(25). As an indication of the kind of work being done, we cite briefly in this subsection a few 

Java details; more extensive documentation will be released when the code is further along. For 

the first pass, we constructed the manner in which functions work as generation of an array of (t, 

f(t)) points. JPlot was written to plot data, not functions, and this appeared to be the quickest 

way to implement functions. Besides, a delta t should not be smaller than a pixel. There were 

many design considerations even in leveraging this existing software. For instance, when (if) the 

plot window is resized, should more points be computed, or should we always compute the same

number of points based on the screen width? As for entering the functions, initially we preferred 

a list of canned functions for which the user provides the coefficients: ax + b, ax^2 + bx + c, etc. 

Eventually, there should be a mechanism by which a programmer could provide a function 

written in Java to avoid a clumsy (and complicated) translator. 

We developed a working parser and evaluator for arbitrary expressions. However, for this initial 

version, the expression must be entered in postfix, or reverse Polish, notation with white space 

between the tokens. In other words, 2.3+sin(t) becomes “2.3 t sin +.” The infix form may be 

used for the description that the user sees. The initial version recognizes six binary operators 

(^*/%+–), 10 unary functions (six trigonometric functions, exp, log, sqrt, abs), the constants pi 

and e, and the independent variable t. These have all been successfully tested and hooked into 

JPlot. After some experience with the software, it proved trivial to add new operators, such as 

sinh and cosh. We are considering the inclusion of neg (as in unary minus). We wanted to use 

Knuth's parser to convert infix to postfix, but volume 5 of The Art of Computer Programming

will not be ready until 2010. However, given that the function can be parsed into a tree, 

volume 1 describes how to traverse the tree and build the postfix equation 25. 

We have also made progress on when and how to build the DataArray, which in turn is plotted. 

We believe the function code has reached a beta level, despite current use of the JTable defaults. 

At this point, a plot of an array of data and a plot of a parsed function may be portrayed on the 

same graph. The user may define Reverse Polish Notation functions in a function file. The 24 

letters of the alphabet, besides e and t, may be used as user-defined variables (named constants). 

The parametric range and increment may be supplied in the function file, as may the values of 

the named constants. When the user attempts to open a graph, the parametric values are checked 

to ensure they are defined and valid (e.g., tmin < tmax and dt > 0). All variables referenced in 

the function(s) are also checked. At this point, it is possible a function that is not selected uses 

an undefined variable, which will prevent the graph from being generated, but generally there is 

a small chance of this actually happening. 

31
We have added (conditionally) to PlotPanel a button labeled Vars to invoke an editVariables 

method in FunctionFile. It opens a JDialog containing two buttons and a JTable listing the 

names and values (if any) of all variables used in the current function file. The system ignores 

variables to which the user may have assigned values but did not use in a function. The Cancel 

button causes the changes to be ignored, while the OK button copies all of the values into a 

UserVariables array. 

As an example of the GUI, the reader is referred to figure 4, which shows one prototype of the 

Jplot modification being used to input the square law model solution, and figure 5, which shows 

the resulting graph. This work is leading to a set of software that can link seamlessly with the 

main decision aid algorithms and enable interactive manipulation of the data, both in terms of 

input to the modeling and examination of the output. More JPlot updates are forthcoming, as 

well as a new application that permits use of sliders to adjust the variables. Sliders are easier (if 

more restrictive) than a relatively static Jtable for enabling the important requirement of “what 

if” analyses.

Figure 4. GUI input of the square law solution. 

32
Figure 5. Graph resulting from GUI input. 

In concluding this section, we note that since we could not gain authority to use the package of 

choice, we pursued different avenues. We did not delve into the experimental aspects of chaos 

and control, but rather looked more into design issues and setting the stage for follow-on 

mathematical and programming developments. In particular, we developed the parameter 

estimation aspects for the current model and Java plotting routines for the proposed GUI. We

decided to go with the Java development, and to pursue use of DPGraph/Dynamics Solver and 

possibly Mathcad for future efforts. We have not given up completely on utilizing EnSight or 

PC MATLAB, although either of these may be overkill as total packages. In any event, we note 

that this software issue resulted in a change of plans in this project, and must be resolved to 

allow substantial effort along the lines originally considered. However, this was considered in a 

sense part of the research; moreover, the unexpected graphics development spin-off may lead to 

improved software for the Java community and a possible GUI patent. We also plan to pursue 

use of Dynamics Solver, DPGraph, and possibly Mathcad for future efforts. 

7. Estimation of Parametric Values 

7.1 Introduction 

We now resume discussion of mathematical aspects of this project. As mentioned in the 

conceptual exposition, an important aspect of the prototype is that of improving the model on the 

fly. In order to do this, we assume that the battle may be monitored for estimated kill rate or, 

33
more directly, estimated forces; an updated model of the battle may then be derived from the 

observations. As a particular application we chose to develop the parameters and from 

observations of troop strengths at several times. This led to interesting research into techniques, 

based initially on least squares curve fitting and difference equation approximations, with wider 

potential application. 

a b

If we assume force attrition on the battlefield obeys the Lanchester square law, then the solution 

is known and involves two parameters. Our original intent was to estimate these two parameters, 

given a set of force levels at various times during a battle. A recursive relation was found 

between the force level at a time and the forces levels at the previous two times. This relation 

allowed us to use a conventional least squares technique to determine the two parameters based 

on the available data. 

The accuracy of future force levels is dependent on reported force strengths. Casualty reporting 

time intervals may depend on a number of factors, including commander’s instructions, strength 

levels of the forces, and perhaps the importance of having timely strength data during the heat of 

battle. A battle expected to last only hours may require frequent casualty updates, while a battle 

expected to last days may not require as frequent an update cycle. Although accurate estimates 

of enemy force levels may be difficult to acquire and although attrition will, in general, not be as

predicted by the calculated solution, smoothing techniques may be used on force levels to 

moderate the lack of precise knowledge, resulting in smaller changes in our estimates of the 

parameters. Smoothing techniques are used to better approximate the model solution, and an 

algorithm based on weighted moving averages and smoothed “derivative” estimates addresses a 

solution to distant estimates based on limited points. 

Recall the discussion of kill (or attrition) rates in subsection 2.1. An extension to how

commanders use their military judgment in assessing kill rates is beyond the scope of this report. 

However, many mathematical approaches are possible and are applied more or less consciously. 

For example, linearization is often used as a first approximation for estimating aspects of the 

battle. As a crude example, consider the following two observations: at time 0, and 

; at time 10, and

x(0) = 10

y(0) = 100 x(10) = 5 y(10) = 90 . Now, the Blue loss rate over the interval is 

(10–5)/(10–0) = 0.5. Since these losses were inflicted by an average of (100+90)/2 = 95 Red, 

one argument is that the rate at which Red kills Blue is 0.5/95 = 0.005. Similarly, the Blue kill 

rate can be computed as ([100–90]/[10–0])/([10+5]/2) = 0.13. A more sophisticated technique 

we considered, and would like to develop further in future work, is based on the eigenvalue 

method of Anderson known as antipotential potential (27). 

In any event, it would be desirable to present visual estimates of predicted troop strength levels 

to the commander along with the probable outcome of the battle. Toward this end we developed 

an algorithm to estimate force attrition levels based on reported casualty data. We made two 

important assumptions in this analysis: attrition obeys the Lanchester square law model as set 

forth by equation 1, and casualties are reported uniformly in time. Neither of these assumptions 

34
will, in reality, be strictly true. However, to the degree they are true, some valuable insights into

the progress of a battle may be gained. 

The discussion that follows deals mainly with the strength levels of the Blue, or x, force; an 

analogous approach applies for the Red, or y, force. The general approach to predicting troop 

strength and thereby battle outcome will be the following: 

1. Determine future troop strength recurrence relation based on previous troop strength as a 

function of a single variable c . 

2. Estimate the value of this constant c based on the available casualty data. 

3. Estimate the individual attrition rates a and b for forces x and y, respectively, based on the 

constant c and the available casualty data. 

4. Based on initial troop strengths 0 x and and attrition rates ,b > 0, predict battle 

outcome and the effects of changes in the attrition rates or the probable outcome if

additional or fewer troops are available. 

0 y a

7.2 Recurrence Relation Derivation 

Given initial troop strengths 0 0 x , y for forces x and y, respectively, and attrition rates a andb , 

and assuming the force algorithm obeys the square law, then the solutions for x and y strengths at 

time t are, as noted in subsection 4.2, 

0 0 0 0

1 ( ) [( ) ( ) ] 2a a abt abt x t x y e x y e

b b

− = − + + (43) 

and 

0 0 0 0

1 ( ) [( ) ( ) ] 2b b abt abt y t y x e y x e

a a

− = − + + . (44) 

For this derivation, we are interested in integral numbers of time intervals and so replace by an 

integer , where it is understood that a time of means n intervals.

t

n n

A simple recurrence relation for x(n) can be derived. Let 

abt abt c e e− = + . (45) 

Strength at time 

n −1 is ( 1) ( 1)

1 2

1 ( 1) (

2

ab n ab n x n c e c e − − − − = + ), (46) 

where 

35
1 0

a

c x y

b = − 0 (47)

and 

2 0

a

c x y

b

= + 0 . (48) 

Multiplication by c yields 

( ) ( 2) ( 2)

1 2 1 2

1 ( 1) ( ) ( 2

2

abn abn ab n ab n cx n c e c e c e c e x n x n − − − − − = + + + = + − ) . (49) 

This finally implies 

x n( ) = cx(n − − 1) x(n − 2) . (50) 

A similar derivation holds for y. 

If we knew attrition obeys the square law, and if we knew x(0), x(1), , and exactly, 

then it would be an easy matter to determine the value of above and therefore predict all future 

strength levels for both forces exactly. These conditions are generally not satisfied, however, 

and so we must estimate the value of given imprecise strength levels. Having a value for 

does not require knowledge of individual attrition rates. In order to gain knowledge about the 

battle beyond simply predicting force levels, we estimate the individual attrition rates as

constants based on the actual casualty figures. 

y(0) y(1)

c

c c

7.3 Least Squares Approach 

Given N + 2 reported strength levels for x, { | 0,1,..., 1} i x i = N + , we seek to determine a c value 

that minimizes the sum of the squared differences between the actual strength levels, as reported, 

and the predicted strength levels, as calculated. Let 

. (51) 

1

2

1 2

2

( )

N

i i i

i

S cx x x

+

− −

=

= − ∑ −

Minimizing using the least squares approach will yield a particular value for c , say . Our 

interest in is that it will allow us to calculate

S 0 c

0 c ab , since by definition in this case 

ab ab S e e− = + . (52)

Generally, in more complicated formulations (e.g., considering varying attrition rates as 

functions of time) we would have to solve analogous expressions numerically; however, in this 

case we can generate a closed form solution. If we multiply equation 34 by ab

e and let abz e = , 

then we have the following quadratic equation: 

2

0 z c − z + =1 0 . (53) 

36
We solve this to obtain 

2

0 0 4

2

c c

z

± − = ⇒

2

0 0 4

log ( ) 2 e

c c

ab

± − = . (54) 

By the nature of equation 34, we are interested only in the root 

2

0 0 4

log ( ) 2 e

c c

ab

+ − = . (55)

Note that the least squares process may give us a <2. Since the range of the 

function

0 c

2 ab abe e− + ≥ , this procedure will fail for <2. In this case, a default value slightly 

>2 is used. 

0 c

Given a value for ab , sayα0 , we seek to determine the value for the quantity . Using α0

and rewriting equation 43, we obtain the equation 

a

b

0 0 0 0 0 0 ( ) ( ) ( ) 2 2

x y t t t t a

x t e e e e

b

α α− − α α = + + − + . (56) 

Let 

0 0 0 0

2

1

0 0

2

( ) ( ) 2 2

N t t t t

i

i

x y a S e e e e

b

α α α α + − −

=

⎛ ⎞

= + ⎜ ⎟ + − + −

⎝ ⎠ ∑ x . (57) 

We seek to minimize S , which will give us the value for a

b

 that minimizes the sum of the 

squared differences between the actual and predicted x strength levels. Suppose the value of 

a

b

 is α1 . 

We now have the following two equations: ab =α0 and a

b =α1 . Solving this system yields 

a =α0α1 andb = 0 / α α1 . These values for a and b are then, in the least squares sense, the best 

estimates we can make given the realization that reported casualty figures will not strictly adhere

to the square law. 

7.4 Modifications 

The force strength curve in a real battle would more likely resemble a step function having 

constant strength over a number of time intervals. In an effort to make the reported strength 

curve more closely resemble the curve one would get if attrition obeyed the square law, we 

modify the originally reported data and generate two additional sets of data. 

37
The first modification uses a weighted moving three-point average. Currently, we use 

, . The superscript denotes that these values have been 

smoothed. 

1 1 ( 2 )/ 4 s

i i i i x x x x = + − + + i = 1,2,..., N

The second modification is more complicated. A force whose attrition obeys the square law has, 

in addition to a “smooth” strength curve, a “smooth” rate of loss (i.e., first derivative) curve. In a 

real battle, the loss differences would likely vary wildly as a function of time. Again, in an effort 

to force the loss difference curve to more closely resemble the curve one would get if attrition 

did obey the square law, we fit a quadratic polynomial to the loss differences. In this scheme, we 

replace the actual loss differences with ones calculated based on quadratics B and fitted to the 

backward and forward differences, respectively. To modify the smoothed point 

F

s

n x , we set 

, (58) 1 1 ( 2 D s B s s F

i i i i i i x x x x x x = + − + + + − )/ 4

where B

i x and Fi x are the quadratic-generated derivatives at the i th time interval using the 

backward and forward differences, respectively. The intent is to take into account the fact that 

nearby points are changing smoothly and predictably in developing a process that will result in 

reasonable strength values. We place two restrictions on the derivatives generated: all 

derivatives are less than zero and all derivatives are strictly decreasing. In the event either of 

these conditions is violated, the terms B

i x and Fi x are ignored in equation 9, which then reduces 

to a weighted moving average of smoothed points. 

For each of the three data sets, we generate values minimizing the squared errors between 

reported and predicted x levels, similarly for the y levels, and similarly for both x and y

simultaneously. For each of the three 

c

ab , values then calculated for each set, we calculate 

the a / b value minimizing the squared errors for the x force and similarly for y. For each of the 

18 combinations of ab and a / b we calculate and . We rank by size the sum of the 

squared errors, subtract the smallest value from each, and eliminate those beyond a selected 

maximum allowable value

a b

EMAX . (As an alternative, we ranked errors by the sum of the absolute 

differences, but found such ranking did not change significantly; its effect on the calculation of 

an average a and b was negligible.) The rest are weighted by 

( , ) ( ) MAX i K

i i i

E E

w a b − = ∆ , (59) 

where is the actual sum of squared errors, Ei ∆ is the partition size, and is an exponent used to 

change the distribution of the weighting scheme. The attrition coefficients used for 

characterizing the reported strength data are then

K

1

1

ˆ

w

w

N

i i

i

N

i

i

a w

a

w

=

=

=

∑

∑

 and 1

1

ˆ

w

w

N

i i

i

N

i

i

b w

b

w

=

=

=

∑

∑

, where is the 

actual number of combinations used in the final determination. 

Nw

a,b

38
A little more discussion on EMAX is warranted. This value is basically a function of the number 

of points used in the calculation of the sum of squared errors, and increases with the number of

points. For example, if we believe all errors >1000 would indicate a “bad fit” and we chose 

=100 and (i.e., linear weighting), then we generate weights in [0, 10]. A value of 

increases the range of values to [0,100] and has the effect of giving increasing weight to 

those ’s andb ’s having smaller errors. 

∆ K = 1

K = 2

a

The final values used to characterize the reported strength data are then 1

1

ˆ

w

w

N

i

i

N

i

i

a

a

w

=

=

=

∑

∑

 and 

1

1

ˆ

w

w

N

i

i

N

i

i

b

b

w

=

=

=

∑

∑

, where is the actual number of combinations used in the final determination. Nw

7.5 Illustrations 

The following paragraphs describe plots illustrating some of the results of this work. The plot in 

figure 6 shows the predicted strength using 10 and 20 reported strength levels when force 

strength obeys the square law. The strength prediction was good and tells us that if casualty 

figures are reported on time and are accurate, then a good estimate of future strength will be 

derived. Attrition rates are a = 0.01, b = 0.02. 

Figure 6. Predicted strength. 

39
We calculate a smoothed estimate of the difference in force levels for each time interval 

(figure 7). XD_EXACT represents the differences one would expect if x strength data were 

generated exactly as a square law model. XD_ORIG represents the difference after the exact 

strength data was modified to reflect some delay in reporting strength data resulting in constant 

strength over several time intervals. We replace each calculated difference by the value of 

, for some constant nominally set to a weighted average of the first few values and for Bt A0e A0

B which minimizes the errors between XD_ORIG and predicted values generated by the 

function in either the least squares sense or the sum of absolute errors sense. In this 

instance, the absolute value approach more closely approximated XD_EXACT, although as 

discussed this was generally not the case. The upper and lower sets of curves used 10 and 20 

points, respectively. 

Bt A0e

Figure 7. Smoothed estimate. 

The plot in figure 8 shows an example of y force strength levels used by the algorithm. The 

exact trace of a square law generation is shown as Y_EXACT. Y_ORIG is derived from

Y_EXACT by assuming that several time intervals may elapse before casualties are reported. 

Y_DERIV is generated by averaging a smoothed value with its previous value and next 

value projected one time unit forward and backward, respectively, by utilizing our estimates 

of the smoothed derivatives at those points. 

i y i 1 y −

i 1 y +

40
Given x and y force strength data X_ORIG and Y_ORIG, we plot the predicted force levels, 

using the algorithm, based on 10-point and 20-point reported strength levels (figure 9). We

believe these are good predictions. Attrition rates are a =.01, b =.02. 

The plot in figure 10 shows a situation in which x initially has 600 troops and y has 500 troops, 

but attrition rates are such that if the battle proceeds without change, then x will lose in 88 time 

intervals. At the fiftieth interval, x and y strengths are 210 and 300, respectively. At this point, x

receives an additional 240 troops, which allows x to withstand enough losses and inflict enough 

casualties to win. 

The plot in figure 11 illustrates the quantities used in determining the attrition rates for the two 

forces. The 18 and b values are plotted, along with their weights, for each of the ranked 

combinations, as well as the final a _average and _average. Conditions were: x = 2000, 

y = 4000; = 0.01,b = 0.02; casualty reporting delayed by a random number of intervals. 

a

b

a

Figure 8. Y-force strength levels. 

41
Figure 9. Predicted strength levels. 

Figure 10. Modified situation. 

42
Figure 11. Attrition rate quantities. 

Figure 12 shows predicted levels of Iraqi forces in Desert Storm based on 5, 10, and 15-day 

assumed casualties. Daily reports were not available, and the following assumptions were made: 

coalition forces, 360,000; Iraqi forces, 540,000; coalition casualties, 358; Iraqi casualties, 

100,000; duration, 43 days (28). Attrition rates were chosen so that after 43 days roughly 

100,000 Iraqi troops had been killed. We calculated force levels assuming a square law and then 

added a random number between –5000 and 5000 to reflect uncertainty of coalition estimates of 

Iraqi casualties. Agreement between predicted and assumed force levels is quite good. Note that 

the actual daily casualty figures may be vastly different than what is shown and that these force 

levels reflect a constant attrition rate over the duration of the war. The agreement between 

predicted force level and the assumed force levels was quite good. 

Figure 13, based on the same scenarios as in figure 12, reflects the situation where Iraqi casualty 

reports were delayed by a randomly selected number of days (uniform distribution between 0 and 

5). 

43
Figure 12. Predicted Iraqi forces. 

Figure 13. Delayed casualty reports. 

44
8. Techniques for Analysis of Dynamic Systems 

8.1 Introduction 

We now shift gears a bit to discuss some germane concepts that are being used in ongoing work 

to develop the decision aid with regard to qualitative analyses of the system model. As 

mentioned earlier, we are seeking application of results in nonlinear dynamics (a portion of 

which is sometimes known as mathematical “chaos” theory) to COA problems. Philosophical 

expositions concerning sensitivity to small variations in initial conditions have been valuable for 

raising the consciousness of the networks and information integration (NII) community. 

However, we seek rigor in treating combat as a dynamic system. For instance, we are concerned 

about defining the senses in which combat can be treated in terms of iterative function theory. 

We are also interested in applications of optimal control theory (observability, controllability, 

and stability), particularly since such investigations may ultimately lead to tools for monitoring 

and even correcting divergence from a plan. 

Any analyst, when confronted with understanding or designing a system, will consider certain 

questions. What are the parameters? Can they be measured and controlled? Does the system

exhibit nonlinearities or feedback? Is it repetitive? Is the phase space bounded? Do the phase 

variables mix? Particular technical challenges may be expressed in terms of two “simple” 

questions. Does so much variation exist in combat, given even nominally similar plans and 

scenarios, that there is no way to group them into “bins” of similar trajectory in phase space? Do 

natural ways exist to develop envelopes, forbidden regions, or attractors that permit assessment 

of the influences of various combat factors? A related consideration is that of assessing 

confidence in the outcome and, by extension, actual sensitivity analyses of chaotic combat 

situations. 

Difficult problems abound in qualitative investigations of nonlinear systems. For some, no 

general method has been developed (29); for instance, the existence of integral curves of certain 

kinds not restricted to lie in the neighborhood of a certain singular point, and limit cycles 

characterizing the disposition of integral curves in the large. Thus, these are worthwhile 

investigations, even in the realm of pure mathematics, and they should be somewhat more 

readily related to when applied to specific systems as we are doing. 

Several commercial or academic software packages designed for “chaos exploration” or “fractal 

exploration” exist and are described briefly in section 6. We intend to utilize such software in 

conjunction with the other commercial off-the-shelf packages and experimental software for 

further research and development of both the theoretical underpinnings and the actual decision 

aid prototype. Along these lines, we note another germane use for the large OneSAF/DISAF 

databases mentioned elsewhere in this report. The results of executing the stochastic simulation 

45
based on the same COA inputs can be plotted in different kinds of phase space and analyzed for 

trajectory “bins” and the existence of attractors. This sort of work is of interest to the U.S. 

Army Battle Command Battle Laboratory and holds much promise for valuable insights into the 

nature of at least certain kinds of combat. 

We are interested in graphically portraying the behavior of continuous-time dynamic systems; in 

particular, limit sets, basins of attraction, and trajectories. Initially, we consider only secondorder systems, because all possible types of behavior can be displayed in the plane. Moreover, 

we intend to examine only structurally stable systems due to numerical analytical considerations. 

Such notions will be discussed shortly. 

8.2 Theoretical Notions 

Czerwinski has noted, “By post-Newtonian, we mean the arrangement of nature—life and its 

complications, such as warfare—to be nonlinear, where inputs and outputs are not proportional; 

where phenomena are unpredictable, but within bounds are self-organizing; where 

unpredictability frustrates conventional planning; where solution as reorganization defeats 

control as we think of it; and where a premium is placed on nonlinear reductions. And where 

rewards go to those who excel in coping with the bounds in order to command and manage—not 

on prediction and control” (30). Also, Weaver suggested three classifications of complexity: 

“organized simplicity,” described by a small number of parameters deterministically linked, but 

some of which can be neglected (and which we generally analyze through mathematical 

modeling); “disorganized complexity,” in which a large number of distinct nondeterministic 

parameters play the main role (and which we tend to describe statistically); and “organized 

complexity,” in which many interrelated but deterministic parameters all contribute to the 

system (31). We hope further examinations of chaos theory will help us understand the third 

type, of which combat is an example. With these observations as motivation, we sketch out 

some concepts that will prove helpful in subsequent decision aid development. 

A dynamic system is a way to describe how one state in phase space evolves into another over 

time. A dynamic system is a smooth action of the real numbers (a continuous system) or the 

integers (a discrete system) on another object. If f is a continuous function, the evolution of a 

variable x can be given by the formula 1 ( ) n n x f x + = .This equation can be transformed into a 

difference equation that forms the discrete analog of the differential equation x& = f x( ) , and the 

difference equation formulation provides some incentive for the solution to and set forth in 

section 7. 

aˆ ˆb

A set of points in phase space is said to be an attractor if it is invariant under the dynamics and 

certain neighboring points asymptotically approach the set. In particular, an attractor upon 

which trajectories are periodic is said to be a limit cycle; trajectories circle around a limiting 

trajectory which they asymptotically approach. A set of points in the space of system variables 

such that initial conditions chosen in this set evolve to a particular attractor is said to be a basin 

46
of attraction. An attractor is the smallest unit which cannot be decomposed into more attractors 

with distinct basins, a distinction needed because a system may have several attractors, each with 

its own basin. 

Moreover, in a system with multiple basins of attraction, the boundaries may be in a fractal 

pattern in which nearby initial conditions lead to radically different trajectories. Decision 

making in such environments will probably be difficult without a detailed aid. James makes this 

point concerning a decision space formed by the number of reinforcement troops available and 

the time intervals between reinforcements (32). An interesting question is whether in the tactical 

domain we see fractal boundaries of the basins. If some trajectories within some epsilon of an 

initial point converge to one attractor and some to another, this point is said to be “unsafe” (23); 

one cannot safely predict the final state associated with the initial point. Moreover, fractal 

boundaries with a large dimension further hinder predictability in such nonlinear systems. It is 

obviously important to understand these aspects of combat, if only to try to “know what one does 

not know” or place limits on the reasonableness of attempting to observe or control such 

systems. 

Nicolis and Prigogine point out that in systems having dynamics reducible to a 2-D phase space 

the only attractors are limit cycles and fixed points (33). With additional dimensions we find 

more complicated topologies of trajectories around these types of attractors, as well as new types 

of behaviors. One way to explore these is through studying the succession of points in a plane 

that cuts the trajectories; by using the so-called “Poincaré map,” we derive information about the 

attractor. 

8.3 Chaos 

A system described by a function f is said to be chaotic if it (1) is sensitive to initial conditions 

(that is, initially neighboring states can evolve to distant states), (2) has a dense set of states with 

periodic trajectories, and (3) is such that, given any two open sets U and V , a positive integer 

exists such that the intersection of 

n

( ) n f U and V is not empty (that is, neighborhoods of points 

get flung out eventually to “big” sets) (34). The first item is the one most popularly associated 

with chaotic systems, “unpredictable” behavior. Simple physical examples of chaotic systems 

are a pendulum with another attached to the free end and a magnetic pendulum moving over a 

plane containing two attractive magnets. Motion and end state of both systems are highly 

dependent on initial position and velocity. 

A chaotic system must be bounded, nonlinear, nonperiodic, sensitive to small disturbances, and 

mixing (in the sense just discussed). James believes these criteria are necessary and sufficient 

(32). Also, such a system usually has these observable features: transient and limit dynamics, 

parameters, definite transitions to and from chaotic behavior, and attractors. In 1976, Rössler 

discovered what is probably the most elementary geometric chaotic system: 

{ ( x& & = − y z + ), y = x + ay,z& = b + xz − cz} (23). This is an artificial formulation to create a 

47
strange attractor using the simple stretch and fold chaos generator. We are attempting to develop 

an analogously simple system with rationale in the tactical realm. 

Several techniques, such as Greene’s method and resonance overlap, exist for predicting the 

onset of chaos in a system as parameters are adjusted. However, these are beyond the scope of 

this initial exposition. Again, follow-on efforts should address the analysis of orbital transition. 

Bradley has developed a (computationally intensive) software approach to controlling chaos in 

electrical circuits, by using information about dynamics on the attractor to control individual 

trajectories to a target point in phase space (35). A few years ago, this was also being done at 

ARL in the context of communications (36). 

Some detailed mathematical expositions have been done in the combat modeling community 

with regard to chaotic behavior. For example, Oak Ridge National Laboratory (ORNL) has done 

work on DE-based models (37) exhibiting chaos, and RAND has shown chaos arising from

certain models, even very simple ones (38). In particular, Dockery and Woodcock have 

analyzed several models, including Lanchester-based and ones incorporating reinforcement, 

utilizing Lyapunov exponents (discussed in the next subsection) and fractal mathematics (39). 

8.4 Stability 

In studying systems (linear or nonlinear), one generally tries to characterize them by looking for 

equilibria, seeing if they are stable, estimating growth rates, considering behavior resulting from

perturbation, considering limiting behavior, and so forth as we have been discussing. We are 

concerned about stability in the face of changing parameters, asymptotic behavior of solutions, 

and periodic or chaotic behavior. 

Qualitative properties can be developed without explicit solutions, which are generally difficult. 

Portraits, usually via numerical solution, provide information about the general behavior. Timespace curves are another tool. Of course, in high dimensions we have the usual visualization 

problems, and may turn to component graphs or other projections of the full state space onto a 

subspace. 

Observe that dy / dx gives the slope of the trajectory that passes through the point( , x y). If the 

numerator and denominator are both zero, the point is a critical or equilibrium point of the 

system; if there is a circle around ( , x y) that contains no other critical point, it is said to be

isolated. The isolated critical point is stable if and only if for every distance ε there is a distance 

δ such that any trajectory that comes within δ of the point remains within ε of the point 

subsequently. The point is said to be asymptotically stable if and only if it is stable and every 

trajectory that comes sufficiently close to the point actually approaches the point. 

Wiggins notes that it would be useful if structural stability (where, roughly speaking, this means 

“nearby” systems have “qualitatively the same” dynamics) of a system could be characterized, if 

only since it might be presumed models of natural phenomena should be structurally stable (40). 

Unfortunately, such determination does not exist (although partial results are known). However, 

48
Wiggins discusses at length one approach to such characterization, namely identification of 

generic properties of dynamic systems. It would be interesting to know whether certain models 

are structurally stable since, by definition, perturbations of unstable dynamic systems can 

produce systems having radically different behavior. Again, such knowledge would be valuable 

to the commander; however, the mathematical development lies in the future. 

Understanding the nature of possible configurations of trajectories of simple linear systems is 

essential for the descriptive study of general systems. If an entry in the system matrix of a linear 

system is modified, so are its trajectories, and (as just noted) structures vary in sensitivity to 

parameter changes. The stability of (0,0) for the system{x& & = ax + = by, y cx + dy} (where 

are constants) can be examined via the characteristic equation (the determinant of

the operational coefficients equated to zero), 

abc , , , and d

 (41). (60) 2m − + ( ) a d m + (ad − bc) = 0

Borrelli and Coleman show how to use eigenanalysis to construct the solution of a homogeneous 

linear system with constant coefficients. They also classify all portraits for a real 2 × 2 matrix, 

based on characteristics of the eigenvalues: improper node, deficient improper, star, saddle, 

vortex, and spiral (13). Table 2 gives stability properties of the critical point (0,0) of the linear 

system just mentioned. 

Table 2. Stability of (0,0) for { ' x = + ax by, y ' = cx + dy}. 

Nature of the 

Characteristic Roots Nature of the Critical Point Stability of the Critical Point 

Real, unequal, of like sign Node (improper) Asymptotically stable if roots –;

unstable if + 

Real, unequal, of unlike sign Saddle point Unstable 

Real and equal Node (proper or improper) Asymptotically stable if roots -; 

unstable if + 

Pure imaginary Center Stable, but not asymptotically 

stable 

Complex but not pure 

imaginary Spiral point Asymptotically stable if real part 

of roots -; unstable if real part + 

In particular, expressing the square law equations in the form x A &( )t = ( )t x(t) results in the 

system matrix . Hence, the characteristic equation for our initial model 

is , with roots 

0

0

a

b

⎡ − = ⎢

⎣ ⎦ − A ⎤

⎥

2

m a − b = 0 ± ab , and therefore the origin is an unstable saddle point. This is 

also observable in earlier illustrations of the tactically significant quadrant (that is, where x and 

y are both non-negative) of the larger (x, y)-plane plot of the system. 

As just alluded, if the system is linear, the signs of the real parts of the eigenvalues generally 

determine the stability properties. Different tools exist for nonlinear systems and one might also 

49
look for threshold effects or other aspects. The most useful tool for nonlinear systems is the 

summarizing function of the system state vector, and it is often possible to describe its behavior 

by an approximate first-order DE. Analyzing that equation then summarizes the system. The 

summarizing functions used for examining the stability of nonlinear systems are called 

Lyapunov functions. 

In the method of Lyapunov, stability of complex systems is generally measured in terms of 

“distances” of several evolving states from an equilibrium state and determining whether these 

distances decrease, or by measuring the energies of evolving states and determining whether they 

decline (13). Lyapunov exponents generalize the notion of eigenvalues at points of equilibrium, 

and can be used to investigate such points, periodic, quasiperiodic and chaotic behavior. We can 

assess the motion’s instability through the use of the mean exponential rate at which two initially 

close trajectories diverge. The Lyapunov exponents are the projections along the n independent 

directions in phase space of this mean rate: 

0

0 0

0

1 ( , ) ( , ) lim log l e

t

t

λ = ∆x X

X ∆x

∆x , (61) 

where we are considering two trajectories having initial conditions and and the 

limit is taken and 

X0 X0 + ∆x0

t → ∞ 0 ∆x → 0 (33).

By measuring the sensitivity of trajectories to perturbation, Lyapunov exponents provide 

information that can help the commander find battle states in which a plan is less sensitive to 

disturbance, for instance, of regions of parameter space in which the combat variables may 

produce less predictable outcomes. We will be examining such calculations in follow-on work, 

possibly utilizing Raglin’s expertise garnered in the image processing realm (42). 

Elementary catastrophe theory deals with equilibria and families of potentials and how these 

equilibria change when external control parameters change. However, it cannot describe the 

dynamic processes in transitioning between static equilibria. Moreover, it is difficult to extend 

its results to dynamic systems, which have no analogous canonical forms, although certain 

methods of phase portrait analysis can be used to facilitate analysis of some dynamic systems. 

Qualitative topological methods can provide much information with relatively little effort. Also, 

catastrophe theory may be applied to problems (such as combat, arguably) in which neither the 

variables nor equations governing the system are known. Given limited modality, a small 

number of functions of state variables actually drive the system, and qualitative behavior in the 

neighborhood of certain critical points may be reasonably inferred (43). 

Understanding the nature of the manifold upon which the battle trajectories play out would shed 

more light on possibilities for predicting outcomes. Parker and Chua set forth algorithms useful 

for finding stable and unstable (one-dimensional) manifolds of a second-order system, and note 

that the algorithms can also be applied to higher-order systems (21). They also note that 

50
techniques for finding higher-dimensional manifolds are not known. Mathematical analysis of 

the phase space itself is a ripe area of research. Understanding the nature of the manifold upon 

which the battle trajectories play out would shed more light on possibilities for predicting battle 

outcomes. Notions of conformal mapping and area/volume preservation in phase space may 

seem far removed from actual battle, but investigation of such topics may have its place in 

development of future military decision aids. 

8.5 Deviation From Plan 

As we have discussed, it is important for the commander to appreciate when the events of the 

actual battle are causing, or probably will cause, significant departure from the projected

trajectory of the conflict as modeled based on his input plans. Of course, there are many 

simplistic approaches such as just tracking the Blue casualties and alerting the commander 

whenever the number exceeds the postulated number. However, it would be much more 

valuable to develop meaningful measures of departure, in a spirit of understanding the larger 

dynamics of the conflict. The fact that we have vectors of parameters characterizing the battle 

makes the mathematics interesting and challenging. 

Certain approaches have been discussed in section 5. Lyaponov analysis is another possibility. 

Divergence measures could be based on schematics like 

. (62) 

0

0

0

( ) (0) ( )

( ) (0) ( )

( ) ( )

t

t

t

P t P v t dt

P t P v t dt

t v v dt

⎧

⎪ = +

⎪

⎪

⎪

⎨ = +

⎪

⎪

⎪∆ = − ⎪⎩

∫

∫

∫

% %

%

In such a formulation, we could consider the actuality to be critically deviant from the plan if∆

exceeds some threshold. But here again the rub is how to choose the threshold in a meaningful, 

dynamic, tractable, and “situation-dependent” manner. One concept worth considering is that 

the divergence is too great if the model shows the commander will be unable to bring the 

trajectory back to the desired path within some desired time. This sort of “derivative” modeling 

has been discussed earlier, and forms to a large extent the motivation for the control theoretical 

aspects being pursued. 

We cannot (just) consider time in the analysis, since even relatively simple “phase shifts,” as 

opposed to the “amplitude differences” generally analyzed, could cause apparently dramatic 

departures; we must consider the plan vis-à-vis enemy strength, friendly position, and other 

parameters that are in some senses independent of the temporal flow of the battle. Moreover, 

both the time-stamped and the event-based indicators are subject to all the uncertainty and error 

51
associated with the fog of war. As indicated in section 7 on parameter estimation, an approach 

dealing with assumed certain updates is challenging enough initially. 

We will be exploring divergence criteria in more depth in follow-on work. Since we have in 

effect -dimensional pulsating clusters as a function of time, we hope to be able to apply some

of Liao’s methodology (10, 44). Further, since time can serve as a “dependent” variable as well, 

we must consider notions of critical deltas and combined deltas. 

n

8.6 Control Theory 

We remind the reader that the idea that some aspects of the conflict are unknown to the 

commander is an important aspect of this project. With this in mind, we sketch some notions 

that should be quite important in future developments. If we want to consider the existence of a 

solution to designing an optimal system, we must investigate whether its state is “observable” 

and whether it is “controllable.” Usually, control theoreticians attempt an optimal solution to a 

well-defined, modifiable system. However, the types of model we have been considering are in 

certain senses unmodifiable. Moreover, we have found in other related investigations that it is 

difficult to develop an objective function for combat (3). These aspects of the system probably 

make calculation of observability and controllability more worthwhile than that of an optimal 

solution, which may not even exist (45). It is hoped that analytical investigations into 

observability, controllability, and stability will yield improved understanding of the dynamics of 

the conflict. 

As noted earlier, our continuous time system can be expressed as x A &(t) = (t)x(t) . The so-called 

state transition matrix, by which multiplication of an initial state vector yields the state at a future 

time, can be calculated from (t ) e

A −τ , for our time-invariant case. The details of this development 

are beyond the scope of this exposition; however, we note certain parallels in the recurrence 

relation derivation of subsection 7.2 (46). 

Now, the general form of the state space equations of a lumped-parameter, linear, continuoustime system is 

{ x A & = + (t t )x B( )u(t) , y(t t ) = C x ( ) + D(t)u(t) }. (63) 

Typically, the control problem comprises determining what the input should be so that the 

state or the output

u(t)

x(t) y(t), or both, behave acceptably as the goal of the system is achieved. 

Controllability, a property of the coupling between input and state, involves and B ; 

observability, a property of the coupling between state and output, involves and C (47). 

A

A

A system is said to be completely observable if by observing the outputs we can deduce the 

initial state in finite time. Many control designs are based on the assumption that the state vector

can be measured completely all the time; the current input is specified as a function of the 

current value of the state vector. Intelligent control should be based on the current state. 

However, in many systems (in particular our combat model) the entire state vector is not 

52
available for measurement (for a variety of reasons), and so control of such systems must be 

based on a subset of the state variables. One could deal with this by looking for techniques that 

require fewer measurements or by constructing an approximation to the full state vector based on 

measurements that are available. It can be shown that an approximation to the state can be 

computed by another linear dynamic system whose state vector generates missing information 

about the state of the system being studied (46). 

The system is said to be completely controllable if for and any 

given there is a finite and a piecewise continuous input {

x A &( )t t = + x( ) Bu(t) x(0) = 0

1 x 1

t 1 u( )t t, ∈[0,t ] } such that 

. It can be shown that a continuous-time system is completely controllable if and only 

if the controllability matrix is of rank (46). Again, the details of 

this exposition are beyond the scope of this report. However, these sorts of analyses will play a 

significant role in development of sophisticated decision aids that incorporate the notion of 

controlling the battlespace based on observations of the unfolding conflict, particularly if 

appropriate models turn out to be highly nonlinear, as appears likely. 

1 x( ) t = 1 x

2 1 [ , , ,..., ] n− M B = AB A B A B n

Analytical solutions to systems of nonlinear equations are generally not possible. However, 

there are techniques that can obtain approximate solutions, yielding understanding of the 

behavior of a system when perturbed from a known solution and developing iterative corrections

often required when controlling such solutions (47). 

9. Future Efforts 

9.1 Extension of Initial Model 

Many extensions are possible, to varying levels of difficulty, to make our simple model less

abstract and bring it more into line with tactical reality. For instance, it is well known from

historical battles that unit “posture” affects kill rates on both sides. Incorporating this “simple” 

change into the Lanchester formulation results in another degree of internal self-reference and 

complexity. Similarly, attrition factors are probably dependent on time, as manifested in terms 

of fatigue and destruction of equipment if nothing else. Change in killing power could be due to 

Blue improved/worsened weapons, Red worsened/improved defense, or changing distance. 

Other changes to the system of differential equations modeling the conflict would be relatively 

straightforward, at least in a theoretical sense if not a tactical. “Reinforcement” of one or both 

force levels could be brought about by some function; it could be negative, like disease or 

desertion. Change in breakpoint could be due to morale or commander’s guidance. Any or all of 

these factors could be considered to be changing in an essentially continuous manner. 

Explicit consideration of troop movements complicates the solution immediately (even if 

handled at a high level of abstraction), as will be discussed in subsection 9.3. At the least, 

however, we must consider reachability of phase lines and achievement of intermediate goals. 

53
More explicit treatments of subconflicts involving split forces and of phases of battle are 

desirable extensions with great analytic power. Work was initiated in this project along these 

lines, as we considered reinforcement at timeτ , optimal allocation, Blue engaging two Red 

forces, and effects of delay on the assault. However, much remains to be done; we plan to report 

on such developments in future reports. A more complicated improvement would be inclusion 

of disparate groups of forces in a multisided game, analogously to that being pursued by other 

ARL researchers (4). Again, initial mathematical results are setting the groundwork for 

programming algorithms to improve the prototype. 

9.2 Attrition 

Taylor notes several functional forms for attrition rates. For instance, Lanchester himself

considered the effect of firer and target strength on both sides in the formulation 

, dx dy axy bxy dt dt

⎧ ⎫ ⎨ = − = − ⎬ ⎩ ⎭ (64) 

with state equation 

0 0 b( ) x − x a = ( y − y) , (65) 

hence the so-called “linear law.” Considering the effect of target, vice firer strength, on both 

sides yields 

, dx dy ax by dt dt

⎧ ⎫ ⎨ = − = − ⎬ ⎩ ⎭ (66) 

with state equation 

0 0 log log e e

x y b a

x y = , (67) 

hence the “logarithmic law.” 

Considering firer strength on one side and both firer and target strength on the other, 

, dx dy ay bxy dt dt

⎧ ⎫ ⎨ = − = − ⎬ ⎩ ⎭ (68) 

produces the mixed law 

2 2

0 0 ( ) ( 2

b

x − x a = y − y). (69) 

As alluded to earlier, such extensions can go on to the limits of the analyst’s abilities to justify 

the formulation and deal with the generally complicated and intractable solutions. As another 

illustration of this, the system

54
, dx dy ay x bx y dt dt β α

⎧

⎨ = − − = − − ⎩ ⎭

⎫

⎬ (70) 

could arise by combining square law interactions among forces with fire support that is not 

attrited, or alternatively by considering square law interactions among forces having operational 

losses (48). 

Hartley claims the Lanchester square law and linear laws do not provide good models of combat 

attrition as evidenced by historical data, but that a particular homogeneous linear-logarithmic law 

provides a good approximation to historical data (6). Another possibility we considered, one 

having some additional theoretical basis, is that of using a Helmbold-type system; for instance, 

exp(1 ) , exp(1 ) dx x dy y a y b

dt y dt x

ω

⎧ ⎫ ⎛ ⎞ ⎛ ⎞ ⎨ ⎬ = − ⎜ ⎟ − = − ⎜ ⎟ −

⎩ ⎭ ⎝ ⎠ ⎝ ⎠ ω x (49). (71) 

9.3 Next Formulation 

By way of introduction to this subsection, we note the extensions to the initial model cited earlier 

(even significant changes to the attrition representation), although improvements in an abstract 

sense, are obviously not sufficient for combat situations having any basis in reality. For instance, 

consider positional effect on attrition. Let us say as a first approximation that lethality follows 

an inverse-square scheme with distance, based on presented area considerations of sensing and 

weapon-delivery error if nothing else. Again, as a tactically realistic improvement, there should 

be a minimum and maximum range. Notions such as these are more or less readily incorporated 

into system equations. 

Ancker finds combat theory and modeling basically deficient (50). However, he asserts that two 

statements can be considered as axioms: a firefight is a terminating stochastic target attrition 

process based on a discrete state space with a continuous time parameter, and all combat is a 

hierarchical network of firefights. He also notes that stochastic duel theory considers 

“microscopic” features of combat (e.g., time between firings, cover) as opposed to Lanchester’s 

aggregated effects. He also notes, however, that it has been shown possible to proceed from

relatively easily measured microscopic parameters (51). Invoking theoretical relationships then 

helps lend veracity to macroscopic developments. We hope to investigate such linking of 

granularities in further mathematical analyses by follow-on collaboration with another ARL 

researcher who is extending such work into the realm of survivability analysis (52). 

Battle command is harder to model than attrition, partly because of the immensely complicated 

nature of the interactions among entities. We realize this sort of work has been debated for years 

in military operations research circles. For example, there are Spradlin-type extensions

(discussed in the next subsection) to more realistically model the behavior of an intelligent army. 

Such an entity has radios to relay information instantly, can react to the presence of opposing 

forces before actual contact, learn from mistakes, and not become stuck in undesirable equilibria. 

55
Such items can be added incrementally as improvements to the modeling process. Moreover, our 

basic decision aiding methodology lends itself to essentially any of the formulations considered 

by analysts, given properly tailored mathematical algorithms. 

However, we considered how to modify our initial formulation to reflect the fundamental 

functions of a military entity: shoot, move, and communicate. The intent was not to definitize 

any particular formulation, but rather to set forth a system that lends itself to incremental 

development in portraying more dimensions of combat beyond simple attrition. This serves two 

purposes: (1) enable an extension of the analytical/display aspects of the evolving decision aid 

prototype and, more importantly, (2) produce a dynamic system that can exhibit inherently 

chaotic behavior. 

One approach was to add onto the set of “shoot” attrition equations 

, s s

s s

dx dy

ay bx

dt dt

⎧

⎨ = − = − ⎩ ⎭

⎫

⎬ (72)

a “move (or position)” set, say of the form

( ), ( p p

s s

dx dy k y k x

dt dt α β ⎧ ⎫ ⎨ ⎬ = − = − −

⎩ ⎭

) , (73) 

and a “communicate” set, say of the form 

, c c

s s

dx dy Ax By dt dt

⎧ ⎫ ⎨ = = ⎬ ⎩ ⎭ . (74) 

Assume further that Blue is attacking on the move and Red is defending from a stationary 

position. We make this particular assumption basically in order to leverage the simulation 

results of the DISAF experimentation being done for our datamining research. However, there is 

some rationale in keeping the formulation relatively simple at first, in a spirit of iteratively 

increasing the sophistication as problems are encountered and solved. As noted in 

subsection 8.3, even apparently simple systems of equations can produce unexpectedly 

complicated behavior. 

In this context, we considered qualitative descriptions: at fixed distance and communication 

ability, rate of destruction is proportional to strength; at fixed strengths and communication, rate 

of destruction ∝ distance ; at fixed strength and distance, rate of destruction communication 

of opponent; at fixed strength and distance, rate of destruction 

−2 ∝

∝ (own communication) −1; at 

fixed strengths, movement own communication. These sorts of criteria can be sharpened with 

experience and generalized into other tactical considerations. Such systems can be much more 

interdependent as attempts to increase realism are made. For example, we need to develop 

another system of linked equations in which sense/perceive and decide are modeled as well. 

∝

56
Modification of attrition to include effects of communications is problematic. It could be argued 

that in the context of our modeling perfect communication defaults to the basic square law. If no 

communication among entities exists, one could say that no effective fire, and hence no attrition 

of the enemy, is possible; but that seems unreasonable, even as a first cut, and one should 

probably consider a minimum effect even in the absence of communications. However, perhaps 

it would be better to consider with a multiplicative effect on attrition of the form , 

where the parabolic effect is based on an network centric warfare (NCW) argument 

concerning the number of nodes in a network (53). 

C ∈[0,1] 2 C

2

n

By considering such arguments, we developed systems of the following form that we intend to 

analyze using the software previously described. 

Shoot: 

( )

( )

B

R B

R

B B

dS aS C P C

dt

dS bS C P C

dt

= − −

= − −

R

B

 (75) 

Move: 

( )

( )

B B

R

R R

B

dP S d

dt S

dP S f dt S

α

β

= −

= − −

(76)

Communicate: 

2

2

B

B

R

R

dC AS

dt

dC DS

dt

=

=

(77)

These are again notional formulations, where represents Blue strength, Blue position, 

Blue communication ability; and analogously for Red. In any event, the thrust of the 

investigation is to describe mutual qualitative influences, using given extremal input values 

yielding reasonable outputs. 

B S PB CB

Another approach was precipitated by considering the positive-coefficient system 

{ x& = −( ) α + by x = −α x + bxy , y c & = ( ) β − = x y β y − cxy }, (78) 

a model developed by Lotka and Volterra for interaction among a predator population x and a 

prey population y. The linear terms –αx and β y model decay of predator and growth of prey in 

the absence of the other population. The quadratics bxy and −cxy model the effects of 

57
interaction, where the number of interactions is assumed jointly proportional to the population of 

each species: the “mass-action” coefficients and measure the predator’s efficiency in

converting prey into reproduction and the probability that an interaction kills one prey (13). This 

is an interesting model to consider as a partial basis for the next formulation, not only because of 

the analogies between the natural situation and combat, but because it is a system of nonlinear 

DEs with closed form solution. Other such models can be developed for situations such as 

competition for resources, cooperation, harvesting, overcrowding, and predator satiation, again 

with parallels in the realm of military conflict. 

b c

Further, by setting x y & & = = 0 in the Lotka-Volterra system we find equilibria (0,0) and 

( / β c,α / b). Borrelli and Coleman show that the orbits are closed and the solutions periodic; the 

component curves show predator peaks lag prey peaks, the period of an orbit increases with 

distance from the equilibrium point, and the orbits go clockwise (13). It is readily apparent that 

these kinds of analytical results, based on combat information available in the modern 

battlespace, could be quite useful to the commander. 

9.4 Modeling Continuous Forces 

We would like to consider methods for dealing with battles in which the combatants are 

distributed in space according to density functions. For example, in a general formulation each 

differential element of force would engage the enemy according to a kill rate function (e.g., of 

distance to points within the enemy mass) and would be killed at a rate comprising some kind of 

summation over the enemy elements. Advancement of one or both of the forces would be a 

function of ongoing combat as well as of mission/strategy, so computation of the vector field 

would generally be difficult. 

In a sense, the traditional Lanchester approach is analogous to Lagrangian fluid dynamics. 

However, for modeling the distribution and movement of forces, Eulerian equations are arguably 

more appropriate. In particular, attrition of forces interacting as a result of movement of the 

forward edge of the battle area lends itself to use of partial DEs, which can naturally handle 

densities and geometry. In particular, such formulations may permit analysis of interpenetration 

of forces and "unusual" battles that may occur in future scenarios. However, they are often 

mathematically intractable. 

Spradlin and Fields have augmented the discrete modeling of the individual soldier with larger 

forces represented as continuous density distributions (54). Then movements and attrition can be 

modeled via variations on reaction-diffusion equations in two spatial variables and time. We

need to examine this notion further, building on earlier work by ORNL researchers, in particular 

Azmy and Protopopescu (37). In guiding troop movement, many factors must be included 

(attrition, terrain, mission, visibility, obstacles, etc.), so computation of the vector field is 

generally difficult. This work has been pursued for some time, but for greater realism 

improvements to the method are needed, such as reaction prior to contact, instantaneous 

communications, and learning from mistakes. Certain aspects of controlling the battle, with or 

58
without such formulations, were set forth in subsection 8.6. Perhaps we can eventually 

investigate application of Cauchy analysis to battle execution by means of controlling its 

boundary. 

9.5 Heuristics and Estimation 

A related effort would be investigation of other methodologies for developing numerical values 

for the “Lanchester” coefficients. Of course, the best situation obtains when the systems can be 

validated by actual combat figures. Traditionally, this is quite difficult, but we are confident that 

the not-too-distant future will bring the ability to obtain reasonable near-real-time values for 

many of the attributes, at least for Blue. Such empirical population of the abstract formulation 

might be based on historical data or records of field experiments. Also, problems of perception 

and measurement noise (associated with the “fog of war”) arise both in the formulation of 

equations and determination of coefficient values. 

Dupuy made valiant attempts to use historical data in modeling combat (55). He lists 73 

“variable effects factors:” weapons effects, terrain factors, weather factors, seasonal factors, air 

superiority, posture, mobility effects, tactical air effects, other combat factors, and intangible 

aspects. He also considers whether they are calculable, sometimes calculable, probably 

calculable but not yet, and probably individually incalculable. In the last category are morale, 

leadership, as well as (somewhat surprisingly, in our opinion) intelligence and technology. We

further note that his work did not involve aspects of NCW or information operations, at least in 

any meaningful sense. 

One issue that emerged during the investigations into parameter estimation was that of

applicability. This is not the same as the notion discussed elsewhere as to the practicality of 

applying these sorts of mathematical analyses in general to combat. Rather, it involves 

considerations of, for example, the numbers of troops required for reasonableness in the 

estimated solutions. We found that although the basic approach to estimating attrition rates for 

opposing forces seems to work in analyzing cases of conventional combat, there seems to be a 

problem with low attrition values, as in the case of the Iraq war for allied forces. We are looking 

at combinations of smoothing techniques and schemes for picking the attrition estimates of 

attrition to determine the best method to be used in the majority of cases. It may prove necessary 

that certain situations require special treatment, a more refined approach. We hope to develop 

this further in the follow-on efforts. 

Other plans for work in parameter estimation include a mixture of types of improvement. For 

instance, it is vital that non-constant reporting intervals be addressed, as well as the fact that the 

commander generally gets Blue and Red updates at different times. Although such items can be 

handled in a sense via interpolation schemes, it is important to consider the tactical reality of

sporadic updates and estimates of various degrees of certainty. Dealing with bands of 

uncertainty and their effects on parameter estimation and replanning is a ripe research area. 

More sophisticated approaches to dealing with parameter combinations are being considered; for 

59
instance, “running averages” of Blue and Red kill rates and force levels. Code optimization 

techniques to reduce calculation time are always a factor in our work. 

Further model formulations can be inspired by Dunnigan’s twelve rules of thumb, “historical 

outcomes that consistently repeat themselves (56).” These are exemplified by (1) combat causes 

losses of 1–5% casualties per day per division, (2) combat vehicles are lost at 5–10 times the rate 

of personnel, and (3) an attacker needs at least three times (varying with size of forces) the 

combat strength to overcome a defender. 

A qualitatively different result of continuation of the project could be a “library” of scenario runs 

categorized in many dimensions. Utilizing National Training Center exercise data as preserved 

in the Automated Historical Archives System has been cited as a possibility in this regard, as has 

the work of Dupuy (55) and various case-based reasoning researchers such as Thompson (57). 

However, the internal ARL work with OneSAF/DISAF lends itself more directly to this sort of 

ontology due to the control and consistency of the data collection. We were concerned early on 

about possible difficulties in handling asymmetric warfare. We knew that guerrilla formulations 

might have to be used for the model if the technique is to be accepted for Future Force missions. 

However, this surfaced somewhat unexpectedly in the form of special requirements needed for 

parameter estimation in casualty situations like those encountered in the Gulf Wars. Future 

research will extend the techniques used for estimation; and the fact that modern solvers can 

handle most systems of equations should alleviate certain technical problems with the conflict 

formulation. 

9.6 Other Applications 

We intend to use two versions of a single combat simulation program to develop techniques for 

transferring “real-world” data and simulation results back and forth among executing systems. 

For example, process A (using as input a certain COA) would be “monitored” to provide “actual 

execution” battlefield information. This information would be used to simulate portions of the 

battle in process B, the results of which would be fed back into the “real” battle. We can thereby 

explore algorithms for determining divergence of combat from the fight and for expeditiously 

developing modifications, including fragmentary orders, to the plan. 

We intend to tie execution monitoring into OneSAF initially and then other simulations such as 

the Joint Military Art of Command Environment. We have developed sets of files recording the 

execution of a number of runs of two types of scenarios: a conventional tank battle and an urban 

assault. These files contain extremely detailed time-stamped information about the unfolding 

battles (5, 58). We have developed programming techniques for converting such information into 

force level updates, and it will be relatively straightforward to utilize these files as drivers to the 

prototype. As precursors to such an exercise we intend to try simulation-based approaches of a 

different nature; in particular, we will monitor a OneSAF run. If it is determined that the typical 

trajectory of the scenario is not like the square law, we can develop a simpler simulation where 

60
the “real” values extracted for input to the decision aid are in fact developed from a square law 

situation, albeit one unknown to the user of the aid. 

An interesting project would be to consider an historical battle such as Iwo Jima or 73 Easting 

and attempt to improve the outcome via analysis of the dynamics, perhaps by monitoring 

casualties and shifting allocation of firepower. Another eventual effort could be to use the 

prototype decision aid during an exercise such as Prairie Warrior and see whether the command 

staff found the monitoring and trajectory change suggestions of any real use in modifying their 

tactics or force composition during the play. 

We note that there is nothing special about the orientation of this decision aid work toward the 

Blue commander, other than the fact that he is the intended user and that the information 

available to the friendly side is generally more accurate. We could use a tool similar to the 

prototype for analyzing the Red situation as well; that is, model-based planning and replanning 

lends itself to applications in intelligence and wargaming as well, due to certain symmetries in 

threat assessment. A related extension involves consideration of the postulated Red 

commander’s desires and his ability to reach that end state in analyzing the Blue situation. It is 

hoped we can leverage pioneering work in multisided game theoretical applications to combat 

that is being done at ARL (59). 

In a spirit of sketching out future work on other sorts of decision aiding using the paradigm of 

model-based execution monitoring, we have developed vignettes of different kinds of planning 

and replanning. Again, these utilize Lanchester modeling as a basis, but the ideas are extensible 

to other formulations. The intent is to inspire researchers to “think outside the box” about how 

the commander may be helped by computer-based aids. It may turn out that these sorts of 

mathematical abstractions are in fact too far removed from tactical (especially, small unit) reality 

to be immediately useful to the commander in a particular mission. However, it is still a 

challenge to prove this rigorously; and the work may still find application to different sorts of

analysis, including drivers for artificially intelligent assistants.

One aspect (examined briefly in section 7) along these lines is that of assessing the value of Blue 

troop reinforcement at a given time. This certainly is not a novel consideration, although the 

idea here would be more one of optimizing the time and amount. At the least the commander 

should be able to readily ascertain the value of reinforcement contingencies under consideration 

with regard to changing the flow of battle. 

Another larger area of study is that of expanded mathematical analysis of the value of NII. Put 

somewhat facetiously, what good is battle command? What does accurate (in the sense of valid 

intelligence, good communications, logical planning, etc.) command and contol buy the planner 

and executor with regard to achieving desired outcome and facilitating contingency planning? As

the DoD C4ISR Cooperative Research Program (CCRP) has pointed out (60), these notions have 

been studied for years, yet the answer is somewhat elusive. On the other hand, the CCRP notes, 

“NCW provides opportunities to improve C2 and execution ... because (1) decision entities or

61
C2 elements will be more knowledgeable; (2) actor entities will be more knowledgeable; (3) 

actor and decision entities will be better connected; (4) sensor entities will be more responsible; 

and (5) the footprint of all entities will be much smaller” (53). 

Coe and Dockery set forth some sketches of experimentation in partial DEs, stochastic DEs, Qanalysis, chaotic dynamics, catastrophe theory, cellular automata, fuzzy sets, fractals, and 

measure of effectiveness definition (61). Woodcock and Dockery also note the development of a 

program of advanced military analysis based on catastrophe theory, singularity theory, 

dissipative structures, reaction-diffusion systems, etc., which is leading to a combination of 

modeling and decision aiding in an “electronic workbench” (62). Such work must be followed 

up. 

We hope to be able to tie some of these investigations into the work Liao has begun on time 

series analysis of our combat data in understanding battle states (10). In particular, we hope to 

investigate the possibility of identifying the existence and fractal dimensionality of an attractor 

for a given time series or showing that the system has an irreducible stochastic nature. 

Moreover, it would be desirable to determine the fractal dimension of the attractor (which 

provides information about the system’s predictability and sensitivity to initial conditions) and 

the minimal dimensionality of its phase space (which yields the number of variables that must be 

used in describing the dynamics). Nicolis and Prigogine suggest an algorithmic approach to such 

investigations that we hope to pursue (33). In addition to the other types of conflict parameters, 

we will consider the notion of the energy or potential of the battlespace, perhaps building on our 

earlier work on calculating intensity (5). Again, all of these are potentially of use to the 

commander given a meaningful link between model and reality; with the proper software, we 

look forward to analyzing these aspects of models of tactical systems. 

10. Conclusion 

10.1 Summary 

By way of introduction to this conclusion, we note that, although we have presented several 

sections dealing with apparently disparate topics, they do in fact comprise an integrated “course 

of action” for continuing the work presented up to this point. The expositions dealt primarily 

with sketches of algorithms for, and interfaces to, a prototype decision aid. We considered 

several aspects of the analytical theory of dynamic systems, in particular mathematical chaos, 

stability, and control. Miscellaneous discussions involved “mechanical” follow-on work, such as 

simulated execution monitoring, parameter estimation, and other application-specific items. 

The intent was to build in reasonable stages toward both theory and prototype: define approaches 

to major technical challenges, develop initial algorithms, begin graphical prototype, prototype 

improvements for monitoring of simulated conflict, and begin considerations of control theory 

62
applications based on emerging results. We accomplished some, though not yet all, of what we 

set out to do in this project. We performed background research into approaches to various 

technical challenges. We designed an algorithmic solution to a prototype trajectory decision 

aiding system and explored some complexities of navigating parameter trade-off spaces. We did 

some detailed studies of strength/attrition parameter estimation and calibration techniques. We

investigated software packages for solving differential systems and for visualizing and 

interacting with decision aid data. We examined methods for utilizing the results of 

OneSAF/DISAF executions. We set the groundwork for formulation of models to produce 

systems exhibiting inherently chaotic behavior and for studying qualitative aspects of phase 

plane trajectories. 

As a result of the research, design, and software engineering begun in this project, we have the 

intellectual material for several in-depth expositions. These include the details of the parameter 

estimation investigations, applications of chaos theory to the study of military operations, and 

several spin-offs concerning COA theory, in particular the notion of plan divergence. We have 

more work to do with regard to iterative function theory and control theory applied to combat 

modeling, graphical prototyping, improving combat modeling via consideration of nonlinear 

dynamics, and approaches to applying stability analyses in combat monitoring. 

10.2 Impact 

It is probable that nonlinear analytic techniques will soon be considered as invaluable to 

commanders. We have seen that time series analysis can permit near-term prediction, attractors 

can indicate probabilities of outcomes, and Lyapunov analysis can help assess relative 

predictability of option results. 

This work, although obviously still in a basic research stage, does have transition potential. The 

work is intended to lead to portions of an integrated system to dynamically link automated plan 

generation and analysis with execution monitoring. Careful tuning of software to specific 

missions could improve the quality and timeliness of current planning programs. This research 

could also yield improved complexity for large simulations. Perhaps sensitivity analyses may 

show the relative importance of parts of the model, even allowing for reduction of scope. In any 

event, it would be useful to investigate situations and parameters for which greater detail may not 

necessarily be better for the analysis. 

Extensions of this effort can augment and improve ARL’s ongoing successful research program

toward methodologies to develop and evaluate COAs and to improve wargaming. Lessons 

learned may lead to extended COA measures of effectiveness. This work has potential for 

improving military command and control and analogous civilian systems. Scientific evaluations 

of plan attributes are of research value to the U.S. Army Communications—Electronics 

Research, Development, and Engineering Center and the Army Battle Command Battle 

Laboratory, and will benefit the soldier by revealing more numerous operational options during 

63
planning and replanning. Other beneficiaries could include any activity requiring rapid 

evaluation of operational plan status, in particular emergency responders. 

10.3 Issues 

Several programmatic challenges arose over the course of this project. As mentioned in 

section 6, apparent software security issues resulted in a deviation from the plan; they must be 

addressed to allow substantial future research and development. Personnel issues and other 

duties negatively impacted concentrated effort throughout the period of performance. These 

detours did, however, produce an unexpected graphics development spin-off, with potential for a 

possible GUI patent. 

We share the concern expressed in some circles over the ability of abstract systems to model 

combat, particularly military operations on urban terrain, in sufficient detail as to be generally 

tactically useful. Moreover, we believe discrete dynamics will play a larger part in analyses of 

small unit actions. However, we maintain that the utility of such techniques will be proven to the 

soldier in the forms of bounding parameters, portraying worst cases, and assessing possibilities. 

It is our vision that when the future force is operational, the Army will be able to effectively and 

efficiently utilize the enormous amounts of data provided by Future Combat Systems. 

10.4 Final Observations 

We trust the reader has been given a flavor of what work in the realm of “A Model-Based 

Approach to Battle Execution Monitoring” might involve over the long run. This report is 

merely an introduction to the possibilities for such decision aiding science and technology. 

Although we have just scratched the surface in developing a simple prototype, we have touched 

on disparate important aspects: the model, the mathematics, the display, and the dynamic 

analysis. This project precipitated more work on aspects of the theory of COA analysis and 

battle execution monitoring for replanning. More discussion of the development of these notions 

is planned for subsequent reports. We trust this work will be found worthy of continuance, 

because the true research sketched out by this framework is just beginning. 

Collaborations in the form of exchanges and discussions are being pursued with several 

individuals in government and academia. For example, Dr. Mary Anne Fields of the ARL 

Weapons and Materials Research Directorate and Dr. Greg Spradlin of the U.S. Military 

Academy Mathematical Sciences Department are developing a battlefield model using reactiondiffusion equations (54). Jeffrey Smith of the ARL Survivability/Lethality Analysis Directorate 

is developing a method for analyzing survivability in the context of a one-on-one engagement 

(52). James Thompson of Rice University is exploring applications of Lanchester laws to 

modular wargaming (63). We plan to share progress with: the Naval Postgraduate School 

Operations Research Department, a center for Lanchester research; the Ohio State University on

Multicriteria Decision Making; and the National Simulation Center on Command Decision 

64
Modeling. We intend to collaborate further with several of these groups as our work gels into a 

vehicle for more in-depth research. 

This project allowed us the opportunity to examine some novel issues, and we look forward to 

contributing further by extending our research into realms of deception analysis and 

understanding the value of information. Analyses of nonlinear dynamics, combined with the 

powerful techniques being developed by ARL researchers for combat simulation data mining, 

should result in better understanding of battle “predictors” and much improved battlespace 

decision support.

65
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