Introduction to the Modeling and Analysis of Complex Systems

Recommendations

Info

13.6. REACTION-DIFFUSION SYSTEMS 263Exercise 13.17 Conduct simulations of the Turing pattern formation with severaldifferent parameter settings, and discuss how the parameter variations (especiallyfor the diffusion constants) affect the resulting dynamics.Exercise 13.18 Discretize the Keller-Segel slime mold aggregation model(Eqs. (13.27) and (13.28)) (although this model is not a reaction-diffusion system,this is the perfect time for you to work on this exercise because you can utilizeCode 13.8). Implement its simulation code in Python, and conduct simulationswith µ = 10 −4 , D = 10 −4 , f = 1, and k = 1, while varying χ as a control parameterranging from 0 to 10 −3 . Use a = 1 and c = 0 as initial conditions everywhere, withsmall random perturbations added to them.Belousov-Zhabotinsky reaction The Belousov-Zhabotinsky reaction, or BZ reactionfor short, is a family of oscillatory chemical reactions first discovered by Russian chemistBoris Belousov in the 1950s and then later analyzed by Russian-American chemist AnatolZhabotinsky in the 1960s. One of the common variations of this reaction is essentiallyan oxidation of malonic acid (CH 2 (COOH) 2 ) by an acidified bromate solution, yet this processshows nonlinear oscillatory behavior for a substantial length of time before eventuallyreaching chemical equilibrium. The actual chemical mechanism is quite complex, involvingabout 30 different chemicals. Moreover, if this chemical solution is put into a shallowpetri dish, the chemical oscillation starts in different phases at different locations. Interplaybetween the reaction and the diffusion of the chemicals over the space will result inthe self-organization of dynamic traveling waves (Fig. 13.18), just like those seen in theexcitable media CA model in Section 11.5.A simplified mathematical model called the “Oregonator” was among the first to describethe dynamics of the BZ reaction in a simple form [50]. It was originally proposedas a non-spatial model with three state variables, but the model was later simplified tohave just two variables and then extended to spatial domains [51]. Here are the simplified“Oregonator” equations:ɛ ∂uu − q= u(1 − u) −∂t u + q fv + D u∇ 2 u (13.58)∂v∂t = u − v + D v∇ 2 v (13.59)
264 CHAPTER 13. CONTINUOUS FIELD MODELS I: MODELINGFigure 13.18: Belousov-Zhabotinsky reaction taking place in a petri dish. Imagefrom Wikimedia Commons (“Reakcja Biełousowa-Żabotyńskiego zachodzaca wszalce Petriego” by Ms 1001 – Own work. Trimmed to fit. Licensed under Publicdomain via Wikimedia Commons – http://commons.wikimedia.org/wiki/File:Zdj%C4%99cie012.jpg)
Page 2 and 3:
Introduction to the Modeling and An
Page 4 and 5:
iiiAbout the TextbookIntroduction t
Page 6:
New York. He is also a Fellow of th
Page 10 and 11:
PrefaceThis is an introductory text
Page 12 and 13:
a comprehensive view of the related
Page 14:
Chen, Hal Lewis, Vlad Miskovic, Chu
Page 17 and 18:
xviCONTENTS5 Discrete-Time Models I
Page 19 and 20:
xviiiCONTENTS15.3 Constructing Netw
Page 22 and 23:
Chapter 1Introduction1.1 Complex Sy
Page 24 and 25:
1.1. COMPLEX SYSTEMS IN A NUTSHELL
Page 26 and 27:
1.2. TOPICAL CLUSTERS 7The possibil
Page 28:
1.2. TOPICAL CLUSTERS 9these topica
Page 31 and 32:
12 CHAPTER 2. FUNDAMENTALS OF MODEL
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 46:
Part IISystems with a Small Number
Page 49 and 50:
30 CHAPTER 3. BASICS OF DYNAMICAL S
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
36 CHAPTER 4. DISCRETE-TIME MODELS
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 118 and 119:
Chapter 6Continuous-Time Models I:
Page 120 and 121:
6.2. CLASSIFICATIONS OF MODEL EQUAT
Page 122 and 123:
6.3. CONNECTING CONTINUOUS-TIME MOD
Page 124 and 125:
6.4. SIMULATING CONTINUOUS-TIME MOD
Page 126 and 127:
6.4. SIMULATING CONTINUOUS-TIME MOD
Page 128:
6.5. BUILDING YOUR OWN MODEL EQUATI
Page 131 and 132:
112 CHAPTER 7. CONTINUOUS-TIME MODE
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 150 and 151:
Chapter 8Bifurcations8.1 What Are B
Page 152 and 153:
8.2. BIFURCATIONS IN 1-D CONTINUOUS
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
8.3. HOPF BIFURCATIONS IN 2-D CONTI
Page 162 and 163:
8.3. HOPF BIFURCATIONS IN 2-D CONTI
Page 164 and 165:
8.4. BIFURCATIONS IN DISCRETE-TIME
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Chapter 9Chaos9.1 Chaos in Discrete
Page 174 and 175:
9.1. CHAOS IN DISCRETE-TIME MODELS
Page 176 and 177:
Next phase space9.3. LYAPUNOV EXPON
Page 178 and 179:
9.3. LYAPUNOV EXPONENT 159easily co
Page 180 and 181:
9.3. LYAPUNOV EXPONENT 16110Lyapuno
Page 182 and 183:
9.4. CHAOS IN CONTINUOUS-TIME MODEL
Page 184 and 185:
Page 186 and 187:
Page 188:
Page 192 and 193:
Chapter 10Interactive Simulation of
Page 194 and 195:
10.2. INTERACTIVE SIMULATION WITH P
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
10.4. SIMULATION WITHOUT PYCX 181Co
Page 202 and 203:
10.4. SIMULATION WITHOUT PYCX 183re
Page 204 and 205:
Chapter 11Cellular Automata I: Mode
Page 206 and 207:
11.1. DEFINITION OF CELLULAR AUTOMA
Page 208 and 209:
11.1. DEFINITION OF CELLULAR AUTOMA
Page 210 and 211:
11.2. EXAMPLES OF SIMPLE BINARY CEL
Page 212 and 213:
11.3. SIMULATING CELLULAR AUTOMATA
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
11.5. EXAMPLES OF BIOLOGICAL CELLUL
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Chapter 12Cellular Automata II: Ana
Page 230 and 231:
12.2. PHASE SPACE VISUALIZATION 211
Page 232 and 233: 12.2. PHASE SPACE VISUALIZATION 213
Page 234 and 235: 12.3. MEAN-FIELD APPROXIMATION 215E
Page 236 and 237: 12.3. MEAN-FIELD APPROXIMATION 217T
Page 238 and 239: 12.4. RENORMALIZATION GROUP ANALYSI
Page 244: 12.4. RENORMALIZATION GROUP ANALYSI
Page 247 and 248: 228 CHAPTER 13. CONTINUOUS FIELD MO
Page 251 and 252: OutIn232 CHAPTER 13. CONTINUOUS FIE
Page 281: 262 CHAPTER 13. CONTINUOUS FIELD MO
Page 315 and 316: 296 CHAPTER 15. BASICS OF NETWORKSg
Page 317 and 318: 298 CHAPTER 15. BASICS OF NETWORKSi
Page 319 and 320: 300 CHAPTER 15. BASICS OF NETWORKS5
Page 321 and 322: 302 CHAPTER 15. BASICS OF NETWORKSE
Page 323 and 324: 304 CHAPTER 15. BASICS OF NETWORKSC
Page 325 and 326: 306 CHAPTER 15. BASICS OF NETWORKSC
Page 327 and 328: 308 CHAPTER 15. BASICS OF NETWORKSt
Page 329 and 330: 310 CHAPTER 15. BASICS OF NETWORKSE
Page 331 and 332: 312 CHAPTER 15. BASICS OF NETWORKSs
Page 333 and 334:
314 CHAPTER 15. BASICS OF NETWORKSn
Page 335 and 336:
316 CHAPTER 15. BASICS OF NETWORKSL
Page 337 and 338:
318 CHAPTER 15. BASICS OF NETWORKSJ
Page 339 and 340:
320 CHAPTER 15. BASICS OF NETWORKSF
Page 341 and 342:
322 CHAPTER 15. BASICS OF NETWORKSr
Page 343 and 344:
324 CHAPTER 15. BASICS OF NETWORKSE
Page 345 and 346:
326 CHAPTER 16. DYNAMICAL NETWORKS
Page 347 and 348:
Page 349 and 350:
Page 351 and 352:
Page 353 and 354:
Page 355 and 356:
Page 357 and 358:
Page 359 and 360:
Page 361 and 362:
Page 363 and 364:
Page 365 and 366:
Page 367 and 368:
Page 369 and 370:
Page 371 and 372:
Page 373 and 374:
Page 375 and 376:
Page 377 and 378:
Page 379 and 380:
Page 381 and 382:
Page 383 and 384:
Page 385 and 386:
Page 387 and 388:
Page 390 and 391:
Chapter 17Dynamical Networks II: An
Page 392 and 393:
17.1. NETWORK SIZE, DENSITY, AND PE
Page 394 and 395:
17.1. NETWORK SIZE, DENSITY, AND PE
Page 396 and 397:
17.2. SHORTEST PATH LENGTH 37717.2
Page 398 and 399:
17.2. SHORTEST PATH LENGTH 379Eccen
Page 400 and 401:
17.3. CENTRALITIES AND CORENESS 381
Page 402 and 403:
Page 404 and 405:
Page 406 and 407:
17.4. CLUSTERING 387while the numer
Page 408 and 409:
17.5. DEGREE DISTRIBUTION 3891.00.8
Page 410 and 411:
17.5. DEGREE DISTRIBUTION 391er = n
Page 412 and 413:
17.5. DEGREE DISTRIBUTION 393Pk = [
Page 414 and 415:
17.5. DEGREE DISTRIBUTION 395logkda
Page 416 and 417:
17.6. ASSORTATIVITY 397Assortativit
Page 418 and 419:
17.6. ASSORTATIVITY 399>>> nx.degre
Page 420 and 421:
17.7. COMMUNITY STRUCTURE AND MODUL
Page 422:
17.7. COMMUNITY STRUCTURE AND MODUL
Page 425 and 426:
406CHAPTER 18. DYNAMICAL NETWORKS I
Page 427 and 428:
Page 429 and 430:
Page 431 and 432:
Page 433 and 434:
Page 435 and 436:
Page 437 and 438:
Page 439 and 440:
Page 441 and 442:
Page 443 and 444:
Page 445 and 446:
Page 447 and 448:
428 CHAPTER 19. AGENT-BASED MODELSE
Page 449 and 450:
430 CHAPTER 19. AGENT-BASED MODELS
Page 451 and 452:
432 CHAPTER 19. AGENT-BASED MODELST
Page 453 and 454:
434 CHAPTER 19. AGENT-BASED MODELSI
Page 455 and 456:
436 CHAPTER 19. AGENT-BASED MODELSt
Page 457 and 458:
438 CHAPTER 19. AGENT-BASED MODELSF
Page 459 and 460:
440 CHAPTER 19. AGENT-BASED MODELSY
Page 461 and 462:
442 CHAPTER 19. AGENT-BASED MODELS3
Page 463 and 464:
444 CHAPTER 19. AGENT-BASED MODELSn
Page 465 and 466:
Page 467 and 468:
Page 469 and 470:
450 CHAPTER 19. AGENT-BASED MODELSF
Page 471 and 472:
452 CHAPTER 19. AGENT-BASED MODELSe
Page 473 and 474:
454 CHAPTER 19. AGENT-BASED MODELSf
Page 475 and 476:
Page 477 and 478:
458 CHAPTER 19. AGENT-BASED MODELSi
Page 479 and 480:
460 CHAPTER 19. AGENT-BASED MODELSw
Page 481 and 482:
462 CHAPTER 19. AGENT-BASED MODELSi
Page 483 and 484:
464 CHAPTER 19. AGENT-BASED MODELSB
Page 485 and 486:
466 BIBLIOGRAPHY[12] “The Human C
Page 487 and 488:
468 BIBLIOGRAPHY[39] H. Sayama, “
Page 489 and 490:
470 BIBLIOGRAPHY[65] P. Holme and J
Page 492 and 493:
Indexaction potential, 202actor, 29
Page 494 and 495:
INDEX 475equilibrium point, 61, 111
Page 496 and 497:
INDEX 477partial differential equat
show all

Introduction to the Modeling and Analysis of Complex Systems

Create successful ePaper yourself

Delete template?

Save as template?