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

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An Exploratory Study of the Butterfly Effect Using Agent- Based Modeling An Exploratory Study of the Butterfly Effect Using Agent­ Based Modeling Mahmoud T. Khasawneh, Ph.D. Student, [email protected] Jun Zhang, Ph.D. Student, [email protected] Nevan E. N. Shearer, Ph.D. Student, [email protected] Raed M. Jaradat, Ph.D. Student, [email protected] Elkin Rodriguez-Velasquez, Ph.D. Student, [email protected] Shannon R. Bowling, Assistant Professor, [email protected] 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 phenomenon. Several studies were dedicated to The term the "butterfly effect" is attributed to the work examine the butterfly effect which resulted from the of Edward Lorenz [1]. It is used to describe the format of the ballots in Palm Beach County, Florida sensitive dependence of the behavior of chaotic during the presidential elections in the year 2000 [6, 7, complex systems on the initial conditions of these 8]. The chaos emerging from the confusing systems. The metaphor refers to the notion that a configuration of the dual-column ballot is said to have butterfly flapping its wings somewhere may cause caused 2,000 Democratic voters, a number larger than extreme changes in the ecological system's behavior in then Texas Governor George W. Bush's certified the future, such as a hurricane. 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 2. LITERATURE REVIEW President of the United States. Lorenz is major contributor to the concept of the butterfly effect. He concluded that slight differing initial 3. METHODOLOGY AND MODEL states can evolve into considerably different states. Bewley [2] talked about the high sensitivity observed in DEVELOPMENT nonlinear complex systems, such as fluid convection, A modified version ofthe GasLab© model from the to very small levels of external force. Wang et al. [3] chemistry and physics library in etLogo© was used as a explored an approach for identifying chaotic basis for our analysis ofchaotic complex systems. The phenomena in demands, and studied how a small drift following are the assumptions ofthe modified GasLab© in predicting an initial demand ultimately may cause a model: significant difference to real demand. Palmer [4] • A random seed sets the initial conditions (x-y argued that a hypothetical dynamically-unconstrained coordinates, speed heading). perturbation to a small-scale variable, leaving all other • Two types of agents: particles and diablos (the two large-scale variables unchanged, would take the agents are identical, with the exception of name and system in a completely different direction, off the color). attractor. Yugay and Yashkevich [5] mentioned that the • The two types of agents only interact with their butterfly effect occurs in Long Josephson Junctions own type. They do not interact with each other. (LJJs) as described by a time dependent nonlinear • For the complete duration of the simulation, sine-Gordon equation. This equation states that any particles are in blue, while diablos are in red. alteration within the initial perturbation fundamentally • Agents move in a random heading and certain changes the asymptotic state of the system. Social speed until they collide with another agent of systems can also exhibit the butterfly effect 79 the same kind. Upon collision, a new speed diablo. Our assumption is that this small and heading for the participating change is an equivalent to a butterfly "flapping particles/diablos are set. its wings." • Particles and diablos bounce off a wall and 3. Automation setup for data collection: the code continue moving in the box. was adjusted to avoid the need for doing • A collision occurs if two particles or two diablos manual runs and to enable collection of are on the same patch. sufficient data to test the existence, or lack thereof, of a butterfly effect in the system. All The criterion this paper adopted to test the existence of data points were exported to a text file. a butterfly effect is the average distance between 4. Statistical analysis: a macro was developed to particles and diablos at each tick. The formula for organize the data into an excel spreadsheet in average distance (D) is shown below: order to make the graphs and plot confidence intervals. N 5. Visual demonstration of divergence: a I~(XAI 2 separate model was created to visually -XBJ + (YA1 -YBi / demonstrate the point at which particle i and D = ....:./=..:='------------ N diablo i diverge after starting in the same position. For the purpose of visual (1 ) demonstration, when the distance between particle i and diablo i is equal to half-patch, • N: number of particles/diablos in the system their colors are changed to black to symbolize (for N = 10, 20, 30, ... , 100, 200, 300, ... , the transition from identical systems to 1000) thus obtaining 19 configurations in total random systems. • X : the x-coordinate of particle i Ai In recognition of the importance of systems' complexity • X : the x-coordinate of diablo i Bi in determining the existence of a butterfly effect, we ran • YAi: the y-coordinate of particle i our model(s) for different configurations of • YBi: the y-coordinate of diablo i particles/diablos as mentioned earlier. Moreover, to reduce the effect of randomness and obtain confidence The reason behind using different numbers of intervals for our results, each configuration was run for particles/diablos is to examine the effect of the size of 30 times, each run consisting of 10,000 ticks. the population on the speed at which the butterfly effect emerges in the model. 4. DISCUSSION AND CONCLUSIONS The modeling methodology was divided into five Figures 1, 2, and 3 show the average distance phases: between two random systems for 10, 500, and 1000 1. Creating a model with two random systems: particles. The graphs illustrate that regardless of the The original GasLab© model had only one number of agents we have in the model, the average agent; particles. Another agent, diablos, was distance tends to fluctuate around 38. It is evident that added to the model with identical behavior the variance decreases as the number of agents patterns to those of particles. Because two increases. random systems are created, particles use a 10 Partlel•• and Of.blen (Rando~) 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 Tim" was the basis to test the hypothesis of the butterfly effect. The rationale this paper used Figure 1: Average Distance for 10 ParticieslDiablos to have a slider bar to incorporate extremely small changes to the heading of a single agent, which we randomly chose to be a 80 40 J> 30 lS !"" lin ~o I ~, ~ : 20 - ..... .. E ~~ I -~9S" L- .. s -UPl><'f9S", o ·s 0 4000 ,..,. 6000 8000 10000 Figure 2: Average Distance for 500 ParticieslDiablos Figure 5: Average Distance for 500 Particles/Diablos (Same Settings) 1000 Partiel•• and DI.blos (Random) >0 45 40 !>O55 ~25 .... 20 -MeenCO-'9'~F -UpP'!rgS", 10" • o o 2000 4000 6000 aooo 10000 Figure 3: Average Distance for 1000 ParticieslDiablos Figure 6: Average Distance for 1000 ParticieslDiablos Figures 4, 5, and 6 show the average distance (Same Settings) between two systems with same settings for 10, sao, and 1000 particles/diablos, with the exception of Figures 7, 8, and 9 show the difference in average making a change to the heading of one diablo to distance between the model with random systems and examine the butterfly effect. An observation is that as the model with same settings. In all cases, the model the number of particles/diablos is increased in the with same settings will approach the same conditions model, the system diverges quicker. Similarly to the as the model with random systems. Although the random systems, as the number of particles/diablos is model with same settings quickly approaches the increased, the variance decreases. behavior of the model with random systems, it takes longer to actually reach the same average distance of 10P"rtkl". ,,00 DI"b1o. (S"m" 5e,tl"ll') 38. Moreover, as the number of particles/diablos 70 increases, it takes longer to reach the same average 60 distance for model of random systems. 50 .40 j30 10 Partkl". and OIabios (Diff"r~"nc_,,_)__~ po 10 .0 60 0 0 2000 4000 nnw 6000 10000 j 20 15 r 0 Figure 4: Average Distance for 10 Particles/Diablos (Same Settings) r.... Figure 7: Average Distance for 10 Particles/Diablos (Difference) 81 500 Partideund Diablos (Difference) The most important implication of this study is that 60 chaotic complex systems can actually exhibit the l-l_9~" butterfly effect. Scientists, from all disciplines, should -M(Io,ln acknowledge that when studying complex systems and : -U""",W4 complex phenomena, reaching an understanding of the current state of the systems can be traced back to a ;.
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