Knowledge (XXG)

Normal form (dynamical systems)

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529: 326: 209: 115: 379: 138: 238: 246: 510: 570: 146: 60: 457: 439: 589: 599: 594: 563: 51: 47: 46:
in a system. All systems exhibiting a certain type of bifurcation are locally (around the equilibrium)
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Strogatz, Steven. "Nonlinear Dynamics and Chaos". Westview Press, 2001. p. 52.
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is a simplified form that can be useful in determining the system's behavior.
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Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
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to the normal form of the bifurcation. For example, the normal form of a
204:{\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=r\ln x+x-1} 110:{\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=\mu +x^{2}} 140:
is the bifurcation parameter. The transcritical bifurcation
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Normal Forms and Unfoldings for Local Dynamical Systems
337: 249: 220: 149: 126: 63: 373: 320: 232: 203: 132: 109: 42:Normal forms are often used for determining local 564: 8: 430:Guckenheimer, John; Holmes, Philip (1983), 571: 557: 452:(Second ed.), Springer, Section 2.4, 491: 336: 309: 290: 263: 253: 250: 248: 219: 163: 153: 150: 148: 125: 101: 77: 67: 64: 62: 409: 450:Elements of Applied Bifurcation Theory 240:can be converted to the normal form 7: 525: 523: 543:. You can help Knowledge (XXG) by 264: 254: 164: 154: 78: 68: 25: 527: 18:Normal form (bifurcation theory) 400:more generally in mathematics. 374:{\displaystyle u=x-1,\mu =r+1} 315: 302: 1: 448:Kuznetsov, Yuri A. (1998), 616: 522: 493:10.4249/scholarpedia.1902 434:, Springer, Section 3.3, 331:with the transformation 48:topologically equivalent 501:Murdock, James (2003). 466:Murdock, James (2006). 52:saddle-node bifurcation 539:-related article is a 375: 322: 234: 205: 134: 111: 388:for use of the terms 376: 323: 235: 206: 135: 112: 335: 247: 218: 147: 133:{\displaystyle \mu } 124: 61: 484:2006SchpJ...1.1902M 233:{\displaystyle x=1} 590:Bifurcation theory 371: 318: 230: 201: 130: 107: 600:Mathematics stubs 595:Dynamical systems 552: 551: 512:978-0-387-21785-7 272: 172: 86: 16:(Redirected from 607: 573: 566: 559: 531: 524: 516: 497: 495: 462: 444: 417: 414: 380: 378: 377: 372: 327: 325: 324: 319: 314: 313: 295: 294: 273: 271: 267: 261: 257: 251: 239: 237: 236: 231: 210: 208: 207: 202: 173: 171: 167: 161: 157: 151: 139: 137: 136: 131: 116: 114: 113: 108: 106: 105: 87: 85: 81: 75: 71: 65: 37:dynamical system 21: 615: 614: 610: 609: 608: 606: 605: 604: 580: 579: 578: 577: 520: 513: 500: 465: 460: 447: 442: 429: 426: 424:Further reading 421: 420: 415: 411: 406: 333: 332: 305: 286: 262: 252: 245: 244: 216: 215: 162: 152: 145: 144: 122: 121: 97: 76: 66: 59: 58: 23: 22: 15: 12: 11: 5: 613: 611: 603: 602: 597: 592: 582: 581: 576: 575: 568: 561: 553: 550: 549: 532: 518: 517: 511: 498: 468:"Normal forms" 463: 458: 445: 440: 425: 422: 419: 418: 408: 407: 405: 402: 390:canonical form 386:canonical form 370: 367: 364: 361: 358: 355: 352: 349: 346: 343: 340: 329: 328: 317: 312: 308: 304: 301: 298: 293: 289: 285: 282: 279: 276: 270: 266: 260: 256: 229: 226: 223: 212: 211: 200: 197: 194: 191: 188: 185: 182: 179: 176: 170: 166: 160: 156: 129: 118: 117: 104: 100: 96: 93: 90: 84: 80: 74: 70: 24: 14: 13: 10: 9: 6: 4: 3: 2: 612: 601: 598: 596: 593: 591: 588: 587: 585: 574: 569: 567: 562: 560: 555: 554: 548: 546: 542: 538: 533: 530: 526: 521: 514: 508: 504: 499: 494: 489: 485: 481: 477: 473: 469: 464: 461: 459:0-387-98382-1 455: 451: 446: 443: 441:0-387-90819-6 437: 433: 428: 427: 423: 413: 410: 403: 401: 399: 398:standard form 395: 391: 387: 382: 368: 365: 362: 359: 356: 353: 350: 347: 344: 341: 338: 310: 306: 299: 296: 291: 287: 283: 280: 277: 274: 268: 258: 243: 242: 241: 227: 224: 221: 198: 195: 192: 189: 186: 183: 180: 177: 174: 168: 158: 143: 142: 141: 127: 102: 98: 94: 91: 88: 82: 72: 57: 56: 55: 53: 49: 45: 40: 38: 34: 30: 19: 545:expanding it 534: 519: 505:. Springer. 502: 478:(10): 1902. 475: 472:Scholarpedia 471: 449: 431: 412: 397: 393: 389: 383: 330: 213: 119: 44:bifurcations 41: 32: 26: 537:mathematics 394:normal form 33:normal form 29:mathematics 584:Categories 404:References 384:See also 357:μ 348:− 284:− 278:μ 196:− 184:⁡ 128:μ 92:μ 480:Bibcode 509:  456:  438:  120:where 31:, the 535:This 396:, or 214:near 35:of a 541:stub 507:ISBN 454:ISBN 436:ISBN 54:is 488:doi 27:In 586:: 486:. 474:. 470:. 392:, 381:. 181:ln 572:e 565:t 558:v 547:. 515:. 496:. 490:: 482:: 476:1 369:1 366:+ 363:r 360:= 354:, 351:1 345:x 342:= 339:u 316:) 311:3 307:u 303:( 300:O 297:+ 292:2 288:u 281:u 275:= 269:t 265:d 259:u 255:d 228:1 225:= 222:x 199:1 193:x 190:+ 187:x 178:r 175:= 169:t 165:d 159:x 155:d 103:2 99:x 95:+ 89:= 83:t 79:d 73:x 69:d 20:)

Index

Normal form (bifurcation theory)
mathematics
dynamical system
bifurcations
topologically equivalent
saddle-node bifurcation
canonical form
ISBN
0-387-90819-6
ISBN
0-387-98382-1
"Normal forms"
Bibcode
2006SchpJ...1.1902M
doi
10.4249/scholarpedia.1902
ISBN
978-0-387-21785-7
Stub icon
mathematics
stub
expanding it
v
t
e
Categories
Bifurcation theory
Dynamical systems
Mathematics stubs

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