Introduction to the Modeling and Analysis of Complex Systems

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3.3. WHAT CAN WE LEARN? 33Another thing you can learn from phase space visualizations is the stability of thesystem’s states. If you see that trajectories are converging to a certain point or area inthe phase space, that means the system’s state is stable in that area. But if you seetrajectories are diverging from a certain point or area, that means the system’s state isunstable in that area. Knowing system stability is often extremely important to understand,design, and/or control systems in real-world applications. The following chapters will puta particular emphasis on this stability issue.Exercise 3.4 Where are the attractor(s) in the phase space of the bouncing ballexample created in Exercise 3.3? Assume that every time the ball bounces it losesa bit of its kinetic energy.Exercise 3.5of attraction.For each attractor obtained in Exercise 3.4 above, identify its basinExercise 3.6 For each of the phase spaces shown below, identify the following:• attractor(s)• basin of attraction for each attractor• stability of the system’s state at several locations in the phase spaceExercise 3.7 Consider a market where two equally good products, A and B, arecompeting with each other for market share. There is a customer review website
34 CHAPTER 3. BASICS OF DYNAMICAL SYSTEMSwhere users of each product submit their ratings. Since there is no actual differencein the product quality, the average rating scores are about the same betweenthe two products, but the customers can also see the total number of submittedratings for each product, which shows how popular the product is in the market.Customers tend to adopt a more popular choice. Answer the following questions:• What would the phase space of this system look like?• Are there any attractors? Are there any basins of attraction?• How does the system’s fate depend on its initial state?• If you were in charge of marketing product A, what would you do?
Page 2 and 3: Introduction to the Modeling and An
Page 4 and 5: iiiAbout the TextbookIntroduction t
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Page 10 and 11: PrefaceThis is an introductory text
Page 12 and 13: a comprehensive view of the related
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Page 17 and 18: xviCONTENTS5 Discrete-Time Models I
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Chapter 6Continuous-Time Models I:
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6.2. CLASSIFICATIONS OF MODEL EQUAT
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6.3. CONNECTING CONTINUOUS-TIME MOD
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6.4. SIMULATING CONTINUOUS-TIME MOD
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6.4. SIMULATING CONTINUOUS-TIME MOD
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6.5. BUILDING YOUR OWN MODEL EQUATI
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112 CHAPTER 7. CONTINUOUS-TIME MODE
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Chapter 8Bifurcations8.1 What Are B
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8.2. BIFURCATIONS IN 1-D CONTINUOUS
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8.3. HOPF BIFURCATIONS IN 2-D CONTI
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8.4. BIFURCATIONS IN DISCRETE-TIME
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Chapter 9Chaos9.1 Chaos in Discrete
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9.1. CHAOS IN DISCRETE-TIME MODELS
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Next phase space9.3. LYAPUNOV EXPON
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9.3. LYAPUNOV EXPONENT 159easily co
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9.3. LYAPUNOV EXPONENT 16110Lyapuno
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9.4. CHAOS IN CONTINUOUS-TIME MODEL
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Chapter 10Interactive Simulation of
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10.2. INTERACTIVE SIMULATION WITH P
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10.4. SIMULATION WITHOUT PYCX 181Co
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10.4. SIMULATION WITHOUT PYCX 183re
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Chapter 11Cellular Automata I: Mode
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11.1. DEFINITION OF CELLULAR AUTOMA
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11.1. DEFINITION OF CELLULAR AUTOMA
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11.2. EXAMPLES OF SIMPLE BINARY CEL
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11.3. SIMULATING CELLULAR AUTOMATA
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Chapter 12Cellular Automata II: Ana
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12.2. PHASE SPACE VISUALIZATION 211
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12.2. PHASE SPACE VISUALIZATION 213
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12.3. MEAN-FIELD APPROXIMATION 215E
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12.3. MEAN-FIELD APPROXIMATION 217T
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12.4. RENORMALIZATION GROUP ANALYSI
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228 CHAPTER 13. CONTINUOUS FIELD MO
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OutIn232 CHAPTER 13. CONTINUOUS FIE
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296 CHAPTER 15. BASICS OF NETWORKSg
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326 CHAPTER 16. DYNAMICAL NETWORKS
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Chapter 17Dynamical Networks II: An
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17.1. NETWORK SIZE, DENSITY, AND PE
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17.1. NETWORK SIZE, DENSITY, AND PE
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17.2. SHORTEST PATH LENGTH 37717.2
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17.2. SHORTEST PATH LENGTH 379Eccen
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17.3. CENTRALITIES AND CORENESS 381
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17.4. CLUSTERING 387while the numer
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17.5. DEGREE DISTRIBUTION 3891.00.8
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17.5. DEGREE DISTRIBUTION 391er = n
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17.5. DEGREE DISTRIBUTION 393Pk = [
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17.5. DEGREE DISTRIBUTION 395logkda
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17.6. ASSORTATIVITY 397Assortativit
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17.6. ASSORTATIVITY 399>>> nx.degre
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406CHAPTER 18. DYNAMICAL NETWORKS I
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428 CHAPTER 19. AGENT-BASED MODELSE
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466 BIBLIOGRAPHY[12] “The Human C
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468 BIBLIOGRAPHY[39] H. Sayama, “
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470 BIBLIOGRAPHY[65] P. Holme and J
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Indexaction potential, 202actor, 29
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INDEX 475equilibrium point, 61, 111
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INDEX 477partial differential equat
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Introduction to the Modeling and Analysis of Complex Systems

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