This paper provides insights about the behavior of chaotic complex systems, and the sensitive dependence of the system on the initial starting conditions. How much does a small change in the initial conditions of a complex system affect it in the long term? Do complex systems exhibit what is called the "Butterfly Effect"? This paper uses an agent-based modeling approach to address these questions. An existing model from NetLogo library was extended in order to compare chaotic complex systems with near-identical initial conditions. Results show that small changes in initial starting conditions can have a huge impact on the behavior of chaotic complex systems. The term the "butterfly effect" is attributed to the work of Edward Lorenz [1]. It is used to describe the sensitive dependence of the behavior of chaotic complex systems on the initial conditions of these systems. The metaphor refers to the notion that a butterfly flapping its wings somewhere may cause extreme changes in the ecological system's behavior in the future, such as a hurricane

Bowling, Shannon R.

Khasawneh, Mahmoud T.

Rodriquez-Velasquez, Elkin

Shearer, Nevan E. N.

Zhang, Jun

NASA Technical Reports Server

79
An Exploratory Study of the Butterfly Effect Using Agent-
Based Modeling
Mahmoud T. Khasawneh, Ph.D. Student, mkhasawn@odu.edu
Jun Zhang, Ph.D. Student, jxzhang@odu.edu
Nevan E. N. Shearer, Ph.D. Student, nesheare@odu.edu
Raed M. Jaradat, Ph.D. Student, rjaradat@odu.edu
Elkin Rodriguez-Velasquez, Ph.D. Student, erodrig@odu.edu
Shannon R. Bowling, Assistant Professor, sbowling@odu.edu
Department of Engineering Management and Systems Engineering
Old Dominion University, Norfolk, Virginia 23529, USA
Abstract. This paper provides insights about the behavior of chaotic complex systems, and the sensitive
dependence of the system on the initial starting conditions. How much does a small change in the initial
conditions of a complex system affect it in the long term? Do complex systems exhibit what is called the
"Butterfly Effect"? This paper uses an agent-based modeling approach to address these questions. An
existing model from NetLogo© library was extended in order to compare chaotic complex systems with
near-identical initial conditions. Results show that small changes in initial starting conditions can have a
huge impact on the behavior of chaotic complex systems.
1. INTRODUCTION
The term the "butterfly effect" is attributed to the work
of Edward Lorenz [1]. It is used to describe the
sensitive dependence of the behavior of chaotic
complex systems on the initial conditions of these
systems. The metaphor refers to the notion that a
butterfly flapping its wings somewhere may cause
extreme changes in the ecological system's behavior in
the future, such as a hurricane.
2. LITERATURE REVIEW
Lorenz is major contributor to the concept of the
butterfly effect. He concluded that slight differing initial
states can evolve into considerably different states.
Bewley [2] talked about the high sensitivity observed in
nonlinear complex systems, such as fluid convection,
to very small levels of external force. Wang et al. [3]
explored an approach for identifying chaotic
phenomena in demands, and studied how a small drift
in predicting an initial demand ultimately may cause a
significant difference to real demand. Palmer [4]
argued that a hypothetical dynamically-unconstrained
perturbation to a small-scale variable, leaving all other
large-scale variables unchanged, would take the
system in a completely different direction, off the
attractor. Yugay and Yashkevich [5] mentioned that the
butterfly effect occurs in Long Josephson Junctions
(LJJs) as described by a time dependent nonlinear
sine-Gordon equation. This equation states that any
alteration within the initial perturbation fundamentally
changes the asymptotic state of the system. Social
systems can also exhibit the butterfly effect
phenomenon. Several studies were dedicated to
examine the butterfly effect which resulted from the
format of the ballots in Palm Beach County, Florida
during the presidential elections in the year 2000 [6, 7,
8]. The chaos emerging from the confusing
configuration of the dual-column ballot is said to have
caused 2,000 Democratic voters, a number larger than
then Texas Governor George W. Bush's certified
margin of victory in Florida, to cast their vote for
another candidate instead of then Vice President AI
Gore, which effectively made George W. Bush the 43rd
President of the United States.
3. METHODOLOGY AND MODEL
DEVELOPMENT
A modified version of the GasLab© model from the
chemistry and physics library in etLogo© was used as a
basis for our analysis of chaotic complex systems. The
following are the assumptions of the modified GasLab©
model:
• A random seed sets the initial conditions (x-y
coordinates, speed heading).
• Two types of agents: particles and diablos (the two
agents are identical, with the exception of name and
color).
• The two types of agents only interact with their
own type. They do not interact with each other.
• For the complete duration of the simulation,
particles are in blue, while diablos are in red.
• Agents move in a random heading and certain
speed until they collide with another agent of
https://ntrs.nasa.gov/search.jsp?R=20100012868 2019-08-29T19:06:30+00:00Z
80
the same kind. Upon collision, a new speed
and heading for the participating
particles/diablos are set.
• Particles and diablos bounce off a wall and
continue moving in the box.
• A collision occurs if two particles or two diablos
are on the same patch.
The criterion this paper adopted to test the existence of
a butterfly effect is the average distance between
particles and diablos at each tick. The formula for
average distance (D) is shown below:
NI~(XAI -XBJ 2 + (YA1 -YBi /
D = ....:./=..:='------------
N
(1 )
• N: number of particles/diablos in the system
(for N = 10, 20, 30, ... , 100, 200, 300, ... ,
1000) thus obtaining 19 configurations in total
• XAi : the x-coordinate of particle i
• XBi : the x-coordinate of diablo i
• YAi: the y-coordinate of particle i
• YBi: the y-coordinate of diablo i
The reason behind using different numbers of
particles/diablos is to examine the effect of the size of
the population on the speed at which the butterfly effect
emerges in the model.
The modeling methodology was divided into five
phases:
1. Creating a model with two random systems:
The original GasLab© model had only one
agent; particles. Another agent, diablos, was
added to the model with identical behavior
patterns to those of particles. Because two
random systems are created, particles use a
different random seed than diablos (for speed,
positioning, and heading).
2. Creating a model with same settings: After the
establishment of a model with two random
systems, we then modified the model again
so that particles and diablos use the same
random seed, thus sharing the same speed,
initial positioning, and heading, creating a
model with same initial settings. This model
was the basis to test the hypothesis of the
butterfly effect. The rationale this paper used
to have a slider bar to incorporate extremely
small changes to the heading of a single
agent, which we randomly chose to be a
diablo. Our assumption is that this small
change is an equivalent to a butterfly "flapping
its wings."
3. Automation setup for data collection: the code
was adjusted to avoid the need for doing
manual runs and to enable collection of
sufficient data to test the existence, or lack
thereof, of a butterfly effect in the system. All
data points were exported to a text file.
4. Statistical analysis: a macro was developed to
organize the data into an excel spreadsheet in
order to make the graphs and plot confidence
intervals.
5. Visual demonstration of divergence: a
separate model was created to visually
demonstrate the point at which particle i and
diablo i diverge after starting in the same
position. For the purpose of visual
demonstration, when the distance between
particle i and diablo i is equal to half-patch,
their colors are changed to black to symbolize
the transition from identical systems to
random systems.
In recognition of the importance of systems' complexity
in determining the existence of a butterfly effect, we ran
our model(s) for different configurations of
particles/diablos as mentioned earlier. Moreover, to
reduce the effect of randomness and obtain confidence
intervals for our results, each configuration was run for
30 times, each run consisting of 10,000 ticks.
4. DISCUSSION AND CONCLUSIONS
Figures 1, 2, and 3 show the average distance
between two random systems for 10, 500, and 1000
particles. The graphs illustrate that regardless of the
number of agents we have in the model, the average
distance tends to fluctuate around 38. It is evident that
the variance decreases as the number of agents
increases.
10 Partlel•• and Of.blen (Rando~)
Tim"
Figure 1: Average Distance for 10 ParticieslDiablos
81
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: 20 I ~, ..~E
- .....
I -~9S" L-
40
J>
30
lS
~o
~~
.. s
o
·s 0 4000 6000
,..,.
-UPl><'f9S",
8000 10000
Figure 2: Average Distance for 500 ParticieslDiablos
1000 Partiel•• and DI.blos (Random)
Figure 5: Average Distance for 500 Particles/Diablos
(Same Settings)
CO-'9'~F-Meen
-UpP'!rgS",
>0
45
40
55!>O
~25
.. 20
..
"10
•
o
o 2000 4000 6000 aooo 10000
Figure 3: Average Distance for 1000 ParticieslDiablos
Figure 4: Average Distance for 10 Particles/Diablos
(Same Settings)
Figures 4, 5, and 6 show the average distance
between two systems with same settings for 10, sao,
and 1000 particles/diablos, with the exception of
making a change to the heading of one diablo to
examine the butterfly effect. An observation is that as
the number of particles/diablos is increased in the
model, the system diverges quicker. Similarly to the
random systems, as the number of particles/diablos is
increased, the variance decreases.
10 Partkl". and OIabios (Diff"r~"nc_,,_) _~
60
Figure 6: Average Distance for 1000 ParticieslDiablos
(Same Settings)
Figures 7, 8, and 9 show the difference in average
distance between the model with random systems and
the model with same settings. In all cases, the model
with same settings will approach the same conditions
as the model with random systems. Although the
model with same settings quickly approaches the
behavior of the model with random systems, it takes
longer to actually reach the same average distance of
38. Moreover, as the number of particles/diablos
increases, it takes longer to reach the same average
distance for model of random systems.
10
j 20
15r 0
100004000 nnw 6000
10P"rtkl". ,,00 DI"b1o. (S"m" 5e,tl"ll')
2000
70
60
50
.40j30
po
.0
0
0
r....
Figure 7: Average Distance for 10 Particles/Diablos
(Difference)
82
500 Partideund Diablos (Difference)
60
l-l_9~"
-M(Io,ln
: -U""",W4
;. ax;; ;; CZ::::: :;;:= ;:;:;zs:::s
The most important implication of this study is that
chaotic complex systems can actually exhibit the
butterfly effect. Scientists, from all disciplines, should
acknowledge that when studying complex systems and
complex phenomena, reaching an understanding of the
current state of the systems can be traced back to a
small perturbation earlier in the system's life cycle.
1000 hrticle. one! ~.bIos (Difference)
Figure 8: Average Distance for 500 Particles/Diablos
(Difference)
Figure 9: Average Distance for 1000 Particles/Diablos
(Difference)
REFERENCES
1. Lorenz, E., 1963, "Deterministic Nonperiodic
Flow", Journal of The Atmospheric Sciences,
20,130-141.
2. Bewley, T. R., 1999, "Linear control and
estimation of nonlinear chaotic convection:
Harnessing the butterfly effect", Physics of
Fluid, 11(5), 1169-1186.
3. Wang, K. J., Wee, H.,M., Gao, S., F., and
Chung, S., L., 2004, "Production and inventory
control with chaotic demands", Omega Journal
of Management Science, 33, 97- 106.
4. Palmer, T. N., 2005, "Quantum Reality,
Complex Numbers, and The Meteorological
Butterfly Effect", Bulletin of the American
Meteorological Society, 86(4), 519-530.
5. Yugay, K. N. and Yashkevich, E. A 2006, "The
Bradbury Butterfly Effect in Long Josephson
Junctions", Journal of Superconductivity and
Novel Magnetism, 19(1/2), 1-9.
6. Sinclair, R. C., Mark, M. M., Moore, S. E.,
Lavis, C. A, and Soldat, AS., 2000,
"Psychology: An electoral butterfly effect",
Nature, 408, 665-666.
7. Wand, J., Shotts, K., Sekhon, J., Mebane Jr.,
W., Herron, M., Brady, H., 2001, "The Butterfly
Did it: Aberrant Vote for Buchanan in Palm
Beach County, Florida", American Political
Science Review. 95 (4),793-810.
8. Wu, D.W., 2002, "Regression Analyses On the
Butterfly Ballot Effect: A Statistical Perspective
of the US 2000 Election", International Journal
of Mathematical Education in Science and
Technology, 33, 309-317.
'0000
100001000
1000
6000
Time
4000lOOO
2000o
10 0
60
Collectively, the results of this paper demonstrate that
there is a butterfly effect in chaotic complex systems. In
fact, as complexity increases, the butterfly effect
emerges quicker but takes a longer time to completely
replicate the model with random systems. Therefore,
an additional experiment was run to determine how
long it takes for the model with same settings to
completely replicate the model with random systems.
As evident in Figure 10, the results of the model
indicate that it actually takes about 2 million ticks to
completely replicate the model of random systems, for
the setting of 1000 particles/diablos.
r-i--"·-lIlalIl5
Figure 10: Simulation Results for 1000
Particles/Diablos After 2 Million Ticks


An Exploratory Study of the Butterfly Effect Using Agent-Based Modeling

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