Dyadic Processes, Tempestuous Relationships, and System Dynamics
by
Dr. Michael J. Radzicki
Department of Social Science & Policy Studies
Worcester Polytechnic Institute
Worcester, Massachusetts 01609-2280
...the term ‘many-body problem’ takes on new meaning in this context.
Steven H. Strogatz (1988)
Abstract
This paper describes two exercises that are useful in an introductory course in system dynamics. They
are centered around two models of a couple engaged in a tempestuous relationship. Although the
models are quite simple, the exercises can be used to introduce and practice a Surprisingly large number
of system dynamics skills.
Introduction
One of the great appeals of the system dynamics paradigm is the emphasis it places on the intuitive
understanding of the mathematics underlying dynamical systems. This emphasis stems primarily from a
modeler’s need to identify system structure and relate it to observed behavior. It is important to note
however, that it is also responsible for attracting a significant number of people with limited mathematical
backgrounds (e.g., no courses in calculus, differential equations, or control theory) to the field. These are
people that normally would never consider using, say, differential equations to address a problem, yet find
modeling with bathtubs, faucets, and pipes to be an understandable and useful endeavor. While people
such as these have always been welcome in the field and, indeed, have been encouraged to join, their
presence creates the need for a catalog of simple exercises that can generate insight into some of the
well-known relationships between dynamical behavior and mathematical feedback structure. This need
becomes even more acute when one considers that most people get exposed to only one semester's
worth or, in many cases, a few days worth of formal system dynamics training.
The purpose of this paper is to present two exercises that can be used to teach a large number of
system dynamics skills -- including those involving traditional mathematics -- in a short amoun’of time.
They are based on Clarence Peterson's (1988) newspaper account of Steven Strogatz’s (1988) article
“Love Affairs and Differential Equations,” and on Strogatz’s original piece itself. Peterson’s article
describes how Strogatz teaches undergraduates Giferenttal equations by, in part, relating them to
romantic relationships. Figure 1 presents a copy of this article.1
Strogatz’s Simple Model
As can be seen in Figure 1, Peterson is both intrigued and perplexed by Strogatz’s account of how
dyadic relationships can be modeled with differential equations. He reproduces one of Strogatz’s models
-- a second order, linear, harmonic oscillator [1] that Strogatz relates to a hypothetical relationship between
Romeo and Juliet -- and describes its dynamic behavior.
dridt = -atj
di/dt = b'r
where: __r(t) = Romeo's love/hate for Juliet at time t
j(t) = Juliet’s love/hate for Romeo at time t {1]
From the cadence of Peterson's discussion however, it is evident that he has trouble relating the model's
structure to its purported behavior. Indeed, at one point he says that
Page 474
System Dynamics '91 Page 475
| telephoned my father, 29 years retired from the math game, to ask if
he had any suggestions for making the Strogatz formulations: even
clearer. Said he:....you’re probably intimidated by the reference to the
Shakespearean tragedy. Substitute the Hatfields and the McCoys, and
see if that doesn't help.’
It is important to note that Peterson's difficulty -- i:e., the inability to intuitively understand mathematical
expressions -- is one that is common to many people.” It is also important to note that Strogatz’s model can
easily be understood via the tools and techniques of system dynamics.
Exercise #1
An interesting exercise in an introductory system dynamics course then, is to hand out Peterson’s
article and assign students the task of helping him to understand the dyadic relationship between Romeo
and Juliet. This exercise forces students to utilize, and perhaps discover for the first time, a surprisingly
large number of system dynamics skills.
The first step students. must take to help Peterson understand Strogatz’s model is to translate [1] into
stocks and flows and a system dynamics language. It is crucial that students master this skill if they are to
begin developing the ability to intuitively understand the traditional mathematics underlying dynamical
systems. Figure 2 shows [1] translated into a STELLA model and Figure 3 lists the corresponding
STELLA equations.
Before the model can be simulated and its behavior compared to the “never-ending cycle of love and
hate” cited by Peterson, students must answer four important questions: 1) What are the model’s
parameters? 2) What are the model's initial conditions? 3) What step size should be used for the
simulation? and 4) What numerical integration technique should be employed? As Peterson provides no
definitive answers to these questions,° students must supply their own. This is desirable, of course,
because it forces them to think critically about the issues involved.
Parameters and Initial Values
In terms of the model's initial values, students can select any real numbers for Romeo and Juliet’s initial
love-hate level except zero, which yields a fixed point. It is up to the instructor to determine whether this
insight should be explicitly told to students or left for self-discovery in the exercise. Either way, students
should come away from the exercise realizing that, if the initial values of both Romeo and Juliet’s love-hate
are zero (i.e., neutral), they will each stay in that position indefinitely. That is, the parameters “a” and “b,”
when multiplied by zero, yield no change in either person's affections.
In terms of the values of the model's parameters, Peterson does provide a hint by quoting Strogatz as
saying that “The parameters a,b are positive, to be consistent with the story.” Thus students are provided
with the signs, but not the magnitudes, of the parameters. Peterson's quote may be a bit misleading,
however, as the parameter “a” in [1] has a negative sign that causes it to influence the system in a direction
opposite to “b,” and hence to cause its oscillations. Moreover, if students select zero -- a nonnegative
rather than a positive value -- for the parameters, the model will again yield a fixed point. No matter what
values are chosen, however, the instructor should make sure that students are able to discuss, either in
written or verbal form, the intuitive meaning of each parameter, and that they can specify the parameters’
dimensions and the dimensions of the model's other variables.
Although students do not need simulation to discover that zero is a poor choice for the model's
parameters and/or initial values, two important properties of [1] are quite difficult to discover without it. The
first involves the relationship between the magnitudes of the model's parameters and the periodicity of its
oscillation. The second involves the relationship between its initial values and the amplitude of its
oscillation. More specifically, through repeated simulation, students will discover that the absolute values
of “a” and “b” are inversely related to the periodicity of the model's oscillation, and that the absolute
magnitude of its initial values are positively related to the amplitude of its oscillation. It is particularly
important that they discover these properties because they are applicable beyond the exercise. That is,
they convey the insight that the character of any linear system's oscillation is determined by its parameters
Page 476 System Dynamics '91
and initial values. To ensure that this revelation occurs, the instructor should debrief students after having
them systematically vary “a,” “b,” r(0), and j(0) over a moderate number of simulation runs.
Step Size, Integration Method, and Integration Error
Strogatz’s harmonic oscillator produces a large amount of integration error when simulated. As a
result, students that select a value for DT that is relatively large (especially if they have also chosen Euler's
method of integration and/or parameter values that are relatively large and hence generate a faster rate of
change) will see a system that apparently generates exploding oscillations. It is all too easy, of course, for
them to believe that this is the model’s actual behavior. The instructor must make sure therefore, that
students cut DT in half after an initial simulation run to see if the system's behavior changes appreciably.
Indeed, in the case of Strogatz’s model, successively reducing DT alters the appearance of each
simulation run dramatically. Moreover, students will find that by reducing the absolute values of “a” and “b”
(for a given DT and integration method), they can decrease the model's integration error. The instructor
may wish, therefore, to tie this portion of the exercise to Forrester’s (1968, p. 6-10) heuristic that DT be
less than one half, but greater than one fifth, of a system's shortest first order delay. -
The existence of, and trade-offs between, various methods of numerical-integration can also be
revealed to students in dramatic fashion in this portion of the exercise. This can be accomplished by
having students simulate the model first with Euler's method and then with a second and/or fourth order
Runge-Kutta method and/or the Adams-Bashforth method, and a constant DT. Students will observe, of
course, that switching from, say, Euler's method to a fourth order Runge-Kutta method significantly
reduces integration error and increases simulation time.
Dynamic Behavior
Figure 4 presents a time series plot of Romeo and Juliet’s emotions. Clearly, with a small enough step
size and/or an accurate enough integration method, Strogatz’s model generates a “never-ending cycle of
love and hate” -- i.e., one with a constant periodicity and amplitude. Given the emphasis on the intuitive
understanding of dynamical behavior in the system dynamics paradigm, it is important that students be
asked to describe, either in written or verbal form, the dynamics underlying this: behavior. Since appealing
to Peterson's article is of no help, students will again be forced to think critically about the dynamical story
being told.
Essentially students must notice, as Peterson notes and Figure 4 reveals, that Romeo is fickle and
that Juliet’s affections mimic and lag behind Romeo's. As Romeo's state of love turns from neutral to hate
(approximately period 2), Juliet’s level of love, although still positive, begins to fall. When Juliet’s love hits
the neutral point (approximately period 5), Romeo apparently feels that he has been a “heel” long enough
and begins to reverse himself. Since it takes some time for Romeo to move back to a state of love,
however, Juliet’s mimicking affections are driven below the neutral point into a state of hate. In fact, the
couple reaches a state of “equal loathing” just after period 6. -
When Romeo's affections finally rise past the neutral point (approximately period 8), Juliet apparently
begins to feel that Romeo is a “reformed man” and begins to reduce her level of hatred towards him.
When she passes the neutral point (approximately period 10.5), however, Romeo's fickleness kicks in and
he reverses field. Apparently he feels that it is now ok to ease up because he has increased his love for
Juliet long enough to repair the damage to their relationship and convince her to love him again. The
cycle, of course, repeats from this point. :
Although getting students to provide a description of the oscillations generated by Strogatz’s model
is an important part of this exercise, the instructor must also make sure that they are able to make the
connection between its mathematical structure and dynamic behavior. Essentially this means getting them
to recognize that the negative value of Romeo's parameter “a” causes the direction of flow in his rate
equation to reverse every time Juliet’s affections cross the neutral line. That is, every time Juliet’s
affections change from positive to negative values, or vice versa. Similarly, students must realize that the
positive value of Juliet’s parameter “b” causes her to move her rate of flow in the same direction as
Romeo's.
System Dynamics '91 Page 477
Causal Loop Diagram
An additional task that the instructor can assign to students in this exercise is to have them draw a
causal loop diagram of Strogatz’s model. Such a diagram can add significantly to a discussion of model
behavior and is particularly useful for helping students see [1] as a system of equations portraying
feedback structure.
Figure 5 is a causal loop diagram of Strogatz’s model. It is recommended that students be asked to
include the flows, as well as the stocks, in the diagram because it enables them to make a smooth
transition to Exercise #2 (below). Moreover, it opens the door for an in-class discussion of “problems with
causal loop diagrams” -- a topic that arises, among other reasons, from the presence of both rates and
levels in the figures (see Richardson 1986).
Inspection of Figure 5 (and, to be safe, Figure 2) reveals that Strogatz’s model is a second order,
major, negative feedback loop. Since a well-known system dynamics heuristic says that oscillation arises in
negative feedback loops with delayed corrective action, the instructor should ask students to find the
delay in the system. This is one of the more conceptually difficult tasks in the exercise as Strogatz’s model
contains no explicit material or informational delays. Nevertheless, a good way to proceed is to note that
Juliet’s flow is an interrupted version of Romeo’s flow. In other words, the integration of Romeo's flow
decouples it from Juliet’s and causes a delay. Of course, the same process occurs when Juliet’s flow is
integrated into her stock of love and hate.
Peterson’s Puzzle
Returning to Figure 1, it is clear that Peterson was struck (and confused) by Strogatz’s. claim that
Romeo and Juliet “manage to achieve simultaneous love [only] one quarter of the time.” Another
interesting task therefore, is to have students determine a “clever” way of showing that Strogatz’s
statement is true. Although there are a number of ways that this can be accomplished, an easy one
involves having students plot the levels of Romeo and Juliet's affections against one another on the
phase plane. This is shown in Figure 6. Inspection of this figure reveals that only one quarter of the
model's orbit passes through the area where both Romeo and Juliet’s stocks have positive values. As one
might imagine, however, students that have never been introduced to the phase plane will (probably)
never think of this solution (despite its ease and clarity). Thus, the instructor must decide how much of a
“push” toward the phase plane students should receive prior to starting the exercise.
Problems with the Simple Model
A final task that the instructor can assign to students in this exercise involves having them point out
problems with Strogatz’s model. Indeed, although it is useful for illustrating many system dynamics
concepts, as a dyadic model of a romantic relationship it leaves much to be desired. Although some
prompting by the instructor may be necessary, generally students will notice things such as Romeo
continuing to increase his level of affection toward love, even after Juliet’s affection passes into a state of
hate (approximately periods 5 to 7 in Figure 4) -- a reaction that would not necessarily be exhibited by real
people. Criticisms such as this can be the source of lively classroom discussion and help students build
model conceptualization and critiquing skills. Moreover, critiquing Strogatz’s simple model is good
preparation for Exercise #2.
Strogatz’s General Model
If one pushes past Peterson's article and examines Strogatz’s original piece, one finds that he offers a
second, more general, model of dyadic relationships [2]. Analogous to Exercise #1 then, students can be
asked to analyze this model with the tools and techniques of system dynamics
dr/dt = ay4*r + aor
djdt = aq+r + ago+j
where: _r(t) = Romeo's love/hate for Juliet at time t
j(t) = Juliet’s love/hate for Romeo at time t [2]
Page 478 System Dynamics '91
Exercise #2
According to Strogatz, much of the fun in analyzing [2] comes from the specification of its parameters.
That is, the parameters ajx (i,k = 1,2) can be either positive or negative, and their signs determine the
“romantic style” of each participant. Thus (a41, a12°> 0) would “characterize an ‘eager-beaver” or someone
stimulated by both his/her partner's love and his/her own affectionate feelings, and (a21 > 0, az2 < 0)
would characterize a “cautious lover” or someone excited by his/her partner's love but frightened by
his/her own feelings. Strogatz recommends that students be asked to name the two other possible
romantic styles (i.e., a11, 412 < 0; and-ag1 < 0, ag9°> 0), and provide “romantic forecasts” for various
pairings of styles. Indeed, in terms of the latter task, he poses the question of whether “a cautious
lover...[can] find true love with an eager-beaver.”
Answering Strogatz’s Question
As in the previous exercise, the first step students must take to answer Strogatz’s question and
analyze his more general model, is to translate [2] into a system dynamics language. Figure 7 shows [2]
translated into a STELLA model and Figure 8 lists the corresponding STELLA equations. Inspection of
Figure 8 shows that the model has been parameterized to represent the relationship between an eager-
beaver (Juliet) and a cautious lover (Romeo).
For the reasons outlined above, it is recommended that the instructor ask students to draw-out the
causal loop diagram that corresponds to Figures 7 and 8. Such a diagram is presented in Figure 9.
Inspection of this figure reveals that pairing an eager-beaver with a cautious lover yields a feedback
structure consisting of a major positive loop, a minor positive loop, and a minor negative loop. In terms of
the model's parameters, aj7 controls the strength of the (Juliet’s) minor negative loop, az2 controls the
strength of the (Romeo's) minor positive loop, and a2 and aaj jointly control the strength of the major
positive loop. Moreover, the sign of each parameter determines, either jointly or individually, the polarities
of the loops. Clearly then, a causal loop diagram of [2] can serve as a vehicle for illustrating the difference
between minor and major feedback loops. and as a backdrop for an analysis of the possible affects of the
former on the latter. Intuitively, students must be counseled to realize that these issues are intertwined
with the issue of parameter selection and hence, with the specification of romantic styles.
Figure 10 presents a time series plot of the interactions between an eager-beaver and a cautious
lover. Inspection of the figure reveals that, given the relative strengths of the loops, the answer to
Strogatz’s question is that it is possible for an eager-beaver and a cautious lover to find true love. Here
again, it is recommended that students be asked to describe, either in written or verbal form, why this is so
-- i.@., why Romeo's love and Juliet’s love both grow exponentially. Essentially, students must recognize
that although Romeo’s minor negative loop (caused by the fear of his own feelings) acts as a drag on the
growth of his love, it is not strong enough to override the effects of the major and minor positive loops.
Students often find this result curious if all of the model's parameters are of equal magnitude (in absolute
value), as in the present case.
Other Romantic Styles
As one might imagine, Strogatz’s general model can be used for numerous tasks beyond the analysis
of the “cautious lover/eager-beaver case.” For example, the romantic styles left undefined by Strogatz can
be defined, incorporated into [2], and simulated.
Figure 11 shows the equations, and Figure 12 the corresponding causal loop diagram, for [2] after it
has been parameterized to represent an eager-beaver (Romeo) paired with a “Cyrano de Bergerac”
(Juliet) -- i.e., a person that is stimulated by his/her own private feelings but repelled by the more public
attention given by his/her object of desire (a21 < 0, age >0). Clearly this combination yields a feedback
structure consisting of a major negative loop and two minor positive loops. The instructor can use this
structure to illustrate the well-known system dynamics heuristic that positive feedback loops tend to
exacerbate the instability generated by negative feedback loops containing delayed corrective action.
The ever-increasing instability generated by pairing an eager-beaver with a Cyrano de Bergerac can
be seen by inspecting the time series plot presented in Figure 13. In this case students should recognize
System Dynamics '91 Page 479
that the positive loops continually give the system's oscillatory tendencies “kicks” or “bursts of energy.”
The instructor can drive home this point by having them increase the strength of the positive loops (i.e.,
the values of ay1 and ago) and re-simulate the model. One caveat, however, is that students will
sometimes attribute the explosive behavior of Figure 13 to integration error, rather than to system
structure. The instructor should make sure, therefore, that the topic of integration error-generated
oscillations versus structurally-generated oscillations gets discussed before or after the completion of the
exercise.
Analogous to Figures 11 and 12, Figure 14 presents the equations and Figure 15 the causal loop
diagram for [2], after it has been parameterized to represent a cautious lover paired with a “cognitive
dissonant” (a21, az2 < 0). Here, a “cognitive dissonant” is person who is basically fickle and moves his/her
emotions in a direction opposite to his/her lover's, but who also has inner feelings that slow down and
work against the fickleness. In this case, as shown in Figure 15, the combination yields a feedback
structure consisting of a major and two minor negative. loops. This structure is useful for illustrating some
ideas from control theory.
The main technical insight students can draw from the “cautious lover-cognitive dissonant structure” is
that the minor negative loops dampen or “control” the oscillations generated by the major negative loop.
This can be seen in time series plot presented in Figure 16. In terms of the romantic relationship, the
important insight is that, after starting in a-state of mutual love and fluctuating between states of love and
hate, Romeo and Juliet end up neutral towards one another -- i.e., in a state of equilibrium or stability. Of
note is that this result is transferable to many dyads, whether they consist of interacting people, firms,
nations, species, etc.
Problems with the General Model
As in Exercise #1, asking students to critique Strogatz’s general model can help them develop
additional system dynamics skills. In the case of the cautious lover-Cyrano de Bergerac pairing (Figures
11-13), for example, students should realize that the explosive behavior generated by the model is not
sustainable in any real system. Moreover, via prompting by the instructor, students should be able to
determine that some limits need to be added to the model to “rein it in” and make it more realistic. Strogatz
in fact suggests that the instructor ask students to add nonlinear terms to [2] “to prevent the possibilities
of unbounded passion or disdain.” Given the emphasis on nonlinearity in the system dynamics paradigm,
and the existence of software that makes it easy to test the dynamic effects of new structures, such an
assignment would certainly be reasonable. It could also serve as a good introduction to the study of limit
cycles.
Additional Twists on the Exercises
Transferability of Structure
In the original newspaper account (Figure 1), Peterson recounts his father’s advice to substitute the
Hatfields and the McCoys for Romeo and Juliet to help [1] make greater sense. The instructor can use this
statement for two purposes in these exercises. The first is as a tool for introducing students to the
concept of generic structures -- i.e., identical feedback structures that arise in different systems. In fact, as
luck would have it, the interactions between the Hatfields and McCoys is one of the examples used by
Richardson (1986, p. 167) in his discussion of problems with causal loop diagrams.4
The second use of the senior Peterson’s advice is to have students analyze whether or not it makes
sense -- i.e., whether substituting the Hatfields and McCoys for Romeo and Juliet in [1] really yields a
“correct” model of a feud. Indeed, as Richardson (1986) points out, a simple, one loop conceptualization
of the interactions between the Hatfields and McCoys is an oversimplification and can lead to problems in
defining loop polarity and predicting dynamical behavior. An interesting classroom discussion therefore
can arise from asking students to define a “correct” feedback structure for a feud and then interpret it in
terms of the Romeo and Juliet story.
As usual, boy + girl = confusion
Harvard University mathematician has devised
a teaching plan that, as he puts it, “relates
mathematics to a topic that’s already on the
minds of many college students: the time-
evolution of a love affair between two people.”
Harvard’s Steven H. Strogatz described the plan in
Mathematics magazine under the sexy title “Love
Affairs and Differential Equations.”
He bases his ill-fated love affair on the story of
Romeo and Juliet, except that it’s not their families
that keep them apart—it’s Romeo’s fickleness.
The more Juliet loves him, the more Romeo
begins to dislike her. But when Juliet loses interest,
his feelings for her warm up. She, on the other hand,
tends to echo him. Her-love grows when he loves
her, and turns to hate when he hates her.
According to Strogatz: “A simple model for their
ill-fated romance is
dr/dt = -aj, dj/dt = br,
where
r(t) = Romeo’s love/hate for Juliet at time t
Figure 1: Clarence Peterson’s Account of Steven Strogatz's Article: Love Affairs and Differential Equations.
J) = Juliet’s love/hate for Romeo at time t.”
It’s important to know that the “positive values r, j
signify love; negative values signify hate.” It also
helps to know that “the parameters a,b are positive,
to be consistent with the story.”
Or so says Strogatz, who goes on to note that “the
sad outcome of their affair, of course, is a never-
ending cycle of love and hate; their governing.
equations are those of a simple harmonic oscillator.”
The news is not all bad. According to the
equation, the harmonically oscillating Romeo and
Juliet “manage to achieve simultaneous love one-
quarter of the time.”
I telephoned my father, 29 years retired from the
math game, to ask if he had any suggestions for
making the Strogatz formulations even clearer.
Said he: “I’m mostly into gardening now, son, but
my guess is that, lacking a Harvard education, you're
probably intimidated by the reference to the
Shakespearean tragedy. Substitute the Hatfields and
the McCoys, and see if that doesn’t help.”
Clarence Petersen
O8p o8ed
16, Sorureukgq wraiskg¢
System Dynamics '91 Page 481
Optional Mathematical Rigor
Strogatz’s models can be used for additional study tasks that involve the traditional tools and
techniques of mathematical dynamics. Due to their relatively advanced nature, they are conceived here as
optional additions to Exercises 1 and 2.
A number of points about dynamical systems can be revealed to students by having them solve [1]
and [2] analytically. These solutions can then be used to: 1) drive home the distinction between simulated
and analytical solutions; 2) show how the behavior of a linear system is merely the sum of the behaviors of
its parts (this can be contrasted with the behavior of a nonlinear system); and 3) show precisely the
parameters that control a linear system's amplitude and periodicity.
Other topics that might be linked to Exercises 1 and 2 include: 1) the calculation of feedback loop
gains in the models; 2) the definition and use of integral and other methods of control in Strogatz’s
general model; 3) the definition and calculation of the dominant loop polarity in each model (see
Richardson 1984); and 4) the calculation of the eigenvalues and eigenvectors for each model and the
subsequent relation of them to system behavior.
Conclusion
The field of system dynamics emphasizes the intuitive understanding of the mathematics underlying
dynamical systems. This paper has offered two exercises that may be useful in helping students develop
their dynamic intuition.
Endnotes
An electronically scanned version of this article is available from the author.
Including many who have had courses in calculus, differential equations, and control theory.
Strogatz (1988) does not either.
Another good example of the transferability of structure involves a comparison of a simple arms race
model (e.g., Forrester 1985) and [2], parameterized to represent two cautious lovers. Both of these
structures consist of a major positive loop and two minor negative loops.
ONS
References
Forrester, J. W. 1968. Principles of Systems. Cambridge, MA: Productivity Press.
Forrester, J. W. 1985. Dynamic Modeling of the Arms Race. System Dynamics Group Memo D-3684-3.
Alfred P. Sloan School of Management. The Massachusetts Institute of Technology.
Peterson, C. 1988. As Usual, Boy + Girl = Confusion. Chicago Tribune. Section 2. Page 1.
Richardson, G. P. 1984. Loop Dominance, Loop Polarity, and the Concept of Dominant Polarity.
Proceedings of the 1984 System Dynamics Conference. Oslo, Norway.
Richardson, G. P. 1986. Problems with Causal Loop Diagrams. System Dynamics Review 2:158-170.
Strogatz, S. H. 1988. Love Affairs and Differential Equations. Mathematics Magazine 61(1): 35.
Page 482
System Dynamics '91
RomeotowHate
‘ChgRemect oveHate
O JulletLoveHate
crgtulerovedtie
ZeroNeual
Figura 2: STELLA Representation of Stogats's Simple Model
1: Romeotovetate 2: Jutetoveriate 3: ZeoNeuiat
1 = : -
2 3.00 ; |
al
0.00 6.00 12.00 18,00
Time
Figure 4: Time Saris Plot of Romeo and Jukets Atectons (Stogat's Simple Model)
1: RomeoLoveHane v, JuletLoveHate
| sumonasons
3 Love One
< Gouna
. 7 Time
q i
3 i
2 :
3
PomeotoveHate
Figure 6; Phase Plot of Romeo and Jules Emotione (Stogat's Simple Mode!)
Baan « etonit- (Chldeona)
INIT Sut ovale 1 {Love-Hate ua i”
Shafemsotovaiate = 1 1 PlomeoLoveHate + #12 JuietLoveHate {Love Hate Units/Timo)
EerNovta 0 Love- Hat Uri}
Fique &: STELLA Equatone fr Stogat's General Model (Cage Beaver Paired wih
‘Cautious Lover) 7
24.00
uliotLovetiate( = JetLoveHaw - 4) + (CholuletLoveHtate)* ot
INIT Rom tao = 1 {LoverHate Unit
CChgFlomeoLoveHate - a'JulelLoveHiate {Love Hate Unite/Time}
an S(wTime)
Bx 5 (tuTime}
ZexoNeual = 0 (Love-Hate Units)
Figure 9: STELLA Equatene for Stogat's Simple Medot
CO,
ovetate Chg Jutet Lovertiate
aes
Figure 5: Cousal Loop Diagram of Stogatz's Simple Model
ZeroNeurrat a2
Figure 7: STELLA Representation of Suogatz's General Model
‘ChoRomee Lovetate
ae (oe)
seathene
Figure 9: Causal Locp Diagram for Strogat't General Mode! (Eager-Beaver
Paved With Cautove Laver)
System Dynamics '91 Page 483
iitLoveHate(y) = JuletLove
(t ) + (ChaduiotLovettat)* dt
INT JtetLoveHato» 1 (Love-H
ChalaletLoveHate ~ a21Romeal veHate + a22"JutotLoveHate {Love-Hale UritwTimej
FiomecLovelatet) = RomectoveHetat =a) + (ChgRomectoveHate) "et
‘RomeoLoveHaie =f (Love-tale
i = att RomeoLoveHat + at JuletLovetate(Love-Hete Units /Tim9)
3] 20.004-
E
‘Cutie
Zorolioutal =O {Love-Hate Unis}
Figure 11: STELLA Equatons for Svogat's General Model (Eage-Boaver Paes With
r i Cyrano de Bergerac)
] oa, H i
3.00 00 12.00 18100 24.00
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1: RemesLoveltte 2: Juletovetiat
sists
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wie oars amid aaa eat 6.00 12.00 18.00 24.00
igure 12: Caveal Loop ts Stops Gene Mod (E59e-Deewr im
aed Wty Cyrerio do Borgere:) Time
Figure 13: Time Serie Pict of Romeo and Julet's Emotons (Eager-Beaver Paired With Cyrano de Bergerac)
Rome Lovetiete:
JteloveHt() = Jamra -49 + (ChpdselLovet)
iNtT hotcovefae «uote nay EST”
Cnpiiotovetine = at bomesese
Ronealoveitay ~ Romeotoetcing a0 s ghee
Inova ewan ChaRomestovetiteg-~ (-)_-pcngdettoverite
stent paren)
E22 Stet
SIL Siac
22.3 patie)
Eecioutal “9 (love Hate Uris Jbl Lovetite
Figute 14: STELLA Equations br Stogatz’s General Model (Cautous Lover Paired With
Cognitive Dissonant)
Figure 15: Causal Loop Diagram for Srogats's General Modal (Cautious Lover
Paked Wit Cognitive Diesenand)
1: RomeoLoveHate 2 JuletLoveHate 3: ZoroNouwat
1 Hy i
2 3.00 }
0.00 12:00 18100 24.00
Tine
Figure 16: Time Series Pot of Romeo and Jukots Emotions (Cautious Lover Paired With Cognitve Dissenant)
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