Elements of Applied Bifurcation TheoryThe years that have passed since the publication of the first edition of this book proved that the basic principles used to select and present the material made sense. The idea was to write a simple text that could serve as a seri ous introduction to the subject. Of course, the meaning of "simplicity" varies from person to person and from country to country. The word "introduction" contains even more ambiguity. To start reading this book, only a moder ate knowledge of linear algebra and calculus is required. Other preliminaries, qualified as "elementary" in modern mathematics, are explicitly formulated in the book. These include the Fredholm Alternative for linear systems and the multidimensional Implicit Function Theorem. Using these very limited tools, a framewo:k of notions, results, and methods is gradually built that allows one to read (and possibly write) scientific papers on bifurcations of nonlinear dynamical systems. Among other things, progress in the sciences means that mathematical results and methods that once were new become standard and routinely used by the research and development community. Hopefully, this edition of the book will contribute to this process. The book's structure has been kept intact. Most of the changes introduced reflect recent theoretical and software developments in which the author was involved. Important changes in the third edition can be summarized as follows. A new section devoted to the fold-flip bifurcation for maps has appeared in Chapter 9. |
Contents
I | 1 |
IV | 5 |
VI | 7 |
VII | 8 |
VIII | 10 |
IX | 11 |
X | 17 |
XI | 18 |
XCVII | 302 |
XCVIII | 305 |
XCIX | 307 |
C | 309 |
CI | 313 |
CII | 315 |
CIII | 316 |
CIV | 323 |
XII | 24 |
XIV | 26 |
XV | 31 |
XVI | 32 |
XVII | 33 |
XVIII | 36 |
XIX | 39 |
XXI | 45 |
XXII | 46 |
XXIII | 49 |
XXIV | 54 |
XXV | 57 |
XXVI | 63 |
XXVII | 67 |
XXVIII | 72 |
XXIX | 75 |
XXX | 77 |
XXXII | 78 |
XXXIII | 81 |
XXXIV | 84 |
XXXV | 89 |
XXXVI | 102 |
XXXVII | 106 |
XXXVIII | 108 |
XXXIX | 114 |
XL | 117 |
XLII | 121 |
XLIII | 122 |
XLIV | 125 |
XLV | 127 |
XLVI | 131 |
XLVII | 136 |
XLVIII | 144 |
XLIX | 145 |
L | 149 |
LI | 154 |
LII | 157 |
LV | 164 |
LVI | 165 |
LVII | 170 |
LVIII | 172 |
LIX | 173 |
LX | 182 |
LXI | 188 |
LXII | 191 |
LXIII | 194 |
LXIV | 195 |
LXVI | 200 |
LXVII | 213 |
LXVIII | 228 |
LXX | 232 |
LXXI | 235 |
LXXII | 237 |
LXXIII | 240 |
LXXIV | 245 |
LXXV | 249 |
LXXVII | 250 |
LXXVIII | 253 |
LXXIX | 262 |
LXXXI | 266 |
LXXXII | 270 |
LXXXIV | 271 |
LXXXV | 273 |
LXXXVI | 275 |
LXXXVII | 278 |
LXXXVIII | 279 |
LXXXIX | 280 |
XC | 282 |
XCI | 285 |
XCII | 291 |
XCIII | 292 |
XCIV | 295 |
XCV | 296 |
XCVI | 299 |
CV | 326 |
CVI | 332 |
CVII | 339 |
CVIII | 345 |
CIX | 351 |
CXI | 358 |
CXII | 368 |
CXIII | 370 |
CXV | 372 |
CXVI | 374 |
CXVII | 376 |
CXVIII | 378 |
CXIX | 382 |
CXX | 384 |
CXXI | 395 |
CXXII | 403 |
CXXIII | 407 |
CXXIV | 412 |
CXXV | 414 |
CXXVI | 418 |
CXXVII | 422 |
CXXVIII | 424 |
CXXIX | 434 |
CXXX | 447 |
CXXXI | 454 |
CXXXII | 466 |
CXXXIII | 479 |
CXXXIV | 480 |
CXXXV | 481 |
CXXXVI | 482 |
CXXXVII | 484 |
CXXXVIII | 485 |
CXXXIX | 486 |
CXL | 487 |
CXLI | 488 |
CXLII | 489 |
CXLIII | 498 |
CXLIV | 502 |
CXLV | 505 |
CXLVI | 506 |
CXLVIII | 508 |
CXLIX | 511 |
CL | 514 |
CLI | 520 |
CLIII | 526 |
CLIV | 529 |
CLV | 537 |
CLVI | 543 |
CLVIII | 549 |
CLIX | 552 |
CLX | 556 |
CLXI | 558 |
CLXII | 559 |
CLXIII | 566 |
CLXIV | 568 |
CLXV | 573 |
CLXVI | 574 |
CLXVII | 576 |
CLXVIII | 579 |
CLXIX | 581 |
CLXX | 587 |
CLXXII | 589 |
CLXXIII | 590 |
CLXXIV | 591 |
CLXXV | 592 |
CLXXVI | 593 |
CLXXIX | 594 |
CLXXX | 595 |
CLXXXI | 596 |
CLXXXII | 597 |
CLXXXIV | 599 |
619 | |
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Common terms and phrases
analysis approximation Bautin bifurcation curve bifurcation diagram Bogdanov-Takens Bogdanov-Takens bifurcation center manifold Chapter closed invariant curve codim 1 bifurcations complex compute Consider continuation continuous-time coordinates corresponding critical cubic terms defined Differential Equations discrete-time dynamical systems eigenspace eigenvalues eigenvectors equilibrium example fixed point flip bifurcation fold bifurcation formula heteroclinic higher-order terms homoclinic bifurcation homoclinic orbit Hopf bifurcation hyperbolic infinite number intersection invariant manifolds invariant set invertible iterations Jacobian matrix Kuznetsov Lemma limit cycle locally topologically equivalent Lyapunov coefficient Math multipliers neighborhood Neimark-Sacker bifurcation nondegeneracy conditions nonlinear nontrivial one-dimensional origin parameter values parameter-dependent phase portraits planar system Poincaré map proof quadratic terms region resonance saddle saddle-focus saddle-node satisfying Shil'nikov smooth function solution stable and unstable sufficiently small tangent Theorem topological normal form torus transformation transversality truncated two-dimensional unique unstable manifolds vector