Introduction to the Modeling and Analysis of Complex Systems

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5.7. LINEAR STABILITY ANALYSIS OF DISCRETE-TIME NONLINEAR... 93∂F i∂x 1x eq∂F i∂x 2x eqF i F i x eqDx 1Dx2x 2x 1x eqF i (x eq +Dx) ≈ F i (x eq ) + Dx 1 + Dx 2x 1x 2∂F i∂x 1x eq∂F i∂x 2x eqFigure 5.14: Linearization of a multidimensional nonlinear function around an equilibriumpoint.
94 CHAPTER 5. DISCRETE-TIME MODELS II: ANALYSISaround x = x eq , just like a regular derivative of a scalar function. Note that the orders ofrows and columns of a Jacobian matrix must match. Its i-th row must be a list of spatialderivatives of F i , i.e., a function that determines the behavior of x i , while x i must be usedto differentiate functions for the i-th column.By combining the result above with Eq. (5.57), we obtain∆x t ≈ J∆x t−1 , (5.71)where J is the Jacobian matrix of F at x = x eq . Look at how simple it can get! Thedynamics are approximated in a very simple linear form, which describes the behavior ofthe small perturbations around x eq as the new origin.Now we can calculate the eigenvalues of J to see if this system is stable or not, aroundx eq . If the absolute value of the dominant eigenvalue λ d is less than 1, the equilibrium pointis stable; even if a small perturbation is added to the system’s state, it asymptoticallygoes back to the equilibrium point. If |λ d | > 1, the equilibrium point is unstable; any smallperturbation added to the system’s state grows exponentially and, eventually, the system’sstate moves away from the equilibrium point. Sometimes, an unstable equilibrium pointmay come with other eigenvalues that show stability. Such equilibrium points are calledsaddle points, where nearby trajectories are attracted to the equilibrium point in somedirections but are repelled in other directions. If |λ d | = 1, it indicates that the system maybe neutral (also called Lyapunov stable), which means that the system’s state neitherdiverges away from nor converges to the equilibrium point. But actually, proving thatthe point is truly neutral requires more advanced nonlinear analysis, which is beyond thescope of this textbook. Finally, if the eigenvalues are complex conjugates, oscillatorydynamics are going on around the equilibrium points. Such equilibrium points are calleda stable or unstable spiral focus or a neutral center, depending on their stabilities. Figure5.15 shows a schematic summary of these classifications of equilibrium points for twodimensionalcases.Linear stability analysis of discrete-time nonlinear systems1. Find an equilibrium point of the system you are interested in.2. Calculate the Jacobian matrix of the system at the equilibrium point.3. Calculate the eigenvalues of the Jacobian matrix.4. If the absolute value of the dominant eigenvalue is:• Greater than 1 ⇒ The equilibrium point is unstable.
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Introduction to the Modeling and An
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iiiAbout the TextbookIntroduction t
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New York. He is also a Fellow of th
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PrefaceThis is an introductory text
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a comprehensive view of the related
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Chen, Hal Lewis, Vlad Miskovic, Chu
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xviCONTENTS5 Discrete-Time Models I
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xviiiCONTENTS15.3 Constructing Netw
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Chapter 1Introduction1.1 Complex Sy
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1.1. COMPLEX SYSTEMS IN A NUTSHELL
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1.2. TOPICAL CLUSTERS 7The possibil
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1.2. TOPICAL CLUSTERS 9these topica
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12 CHAPTER 2. FUNDAMENTALS OF MODEL
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Part IISystems with a Small Number
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30 CHAPTER 3. BASICS OF DYNAMICAL S
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36 CHAPTER 4. DISCRETE-TIME MODELS
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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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300 CHAPTER 15. BASICS OF NETWORKS5
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302 CHAPTER 15. BASICS OF NETWORKSE
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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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17.7. COMMUNITY STRUCTURE AND MODUL
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406CHAPTER 18. DYNAMICAL NETWORKS I
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428 CHAPTER 19. AGENT-BASED MODELSE
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430 CHAPTER 19. AGENT-BASED MODELS
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432 CHAPTER 19. AGENT-BASED MODELST
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438 CHAPTER 19. AGENT-BASED MODELSF
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440 CHAPTER 19. AGENT-BASED MODELSY
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452 CHAPTER 19. AGENT-BASED MODELSe
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454 CHAPTER 19. AGENT-BASED MODELSf
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458 CHAPTER 19. AGENT-BASED MODELSi
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460 CHAPTER 19. AGENT-BASED MODELSw
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462 CHAPTER 19. AGENT-BASED MODELSi
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464 CHAPTER 19. AGENT-BASED MODELSB
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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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