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Guckenheimer J., Holmes Ph. Ч Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Vol. 42
Guckenheimer J., Holmes Ph. Ч Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Vol. 42

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Ќазвание: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Vol. 42

јвторы: Guckenheimer J., Holmes Ph.


From the reviews: "This book is concerned with the application of methods from dynamical systems and bifurcation theories to the study of nonlinear oscillations. Chapter 1 provides a review of basic results in the theory of dynamical systems, covering both ordinary differential equations and discrete mappings. Chapter 2 presents 4 examples from nonlinear oscillations. Chapter 3 contains a discussion of the methods of local bifurcation theory for flows and maps, including center manifolds and normal forms. Chapter 4 develops analytical methods of averaging and perturbation theory. Close analysis of geometrically defined two-dimensional maps with complicated invariant sets is discussed in chapter 5. Chapter 6 covers global homoclinic and heteroclinic bifurcations. The final chapter shows how the global bifurcations reappear in degenerate local bifurcations and ends with several more models of physical problems which display these behaviors." Book Review - Engineering Societies Library, New York "An attempt to make research tools concerning 'strange attractors' developed in the last 20 years available to applied scientists and to make clear to research mathematicians the needs in applied works. Emphasis on geometric and topological solutions of differential equations. Applications mainly drawn from nonlinear oscillations." American Mathematical Monthly

язык: en

–убрика: ћатематика/

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√од издани€: 2002

 оличество страниц: 484

ƒобавлена в каталог: 25.04.2008

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ѕредметный указатель
$C^{k}$ eigenvalence      38 (Definitions 1.7.2Ч1.7.3)
$C^{k}$ perturbation of size $\varepsilon$      38 (Definition 1.7.1)
$C^{k}$-conjugacy      38 (Definitions 1.7.2Ч1.7.3)
$\alpha$-limit point, $\alpha$-limit set      34 235 236
Address      307
Anosov diffeomorphisms      20Ч21 261Ч262
Area preserving maps      216Ч226
Arnold diffusion      220
Asymptotic measure      283
Asymptotically stable (fixed point)      3Ч4
Attracting set      34 75 91 92 256Ч259
Attractor      36 256Ч257 Definition 259Ч261
Autonomous (differential equation, dynamical system)      2
Autonomous averaged system      (see Averaged system)
Averaged system      161
Averaging (method of)      68 153 166Ч184 206
Averaging theorem      168 (Theorem 4.1.1)
Badly approximated (irrational numbers)      303
Band mergings      348
Basin of attraction      (see Domain of attraction)
BendixsonТs criterion      44 (Theorem
Bi-infinite (symbol) sequence      233
Bifurcation diagram      105Ч106 118Ч120
Bifurcation set      71 119
Bifurcation value (bifurcation point)      71 105 117 119
Blowing up (a degenerate singularity)      362Ч364 368
Bouncing ball (model for the dynamics of)      102Ч116 (Section 2.4)
Bouncing ball (model for the dynamics of), hyperbolic invariant set      242Ч245
Boundary layer      69
Branch of equilibria      118Ч119
Branched manifold      262
Bridge (of a Cantor set)      333
Cantor book      219
Cantor set      88 110 229 232 258 266 285Ч286 332Ч334
Capacity      285 (Definition 5.8.3)
Catastrophe theory      356Ч357
Center      4
Center manifold      123Ч138 (Section 3.2)
Center manifold theorem      121 (Theorem 3.2.1)
Center manifold, approximation of      130Ч138 (Theorems 3.2.2Ч3.2.3)
Center manifold, definition of      124
Center manifold, loss of smoothness of      124Ч127 380Ч381
Center manifold, nonuniqueness of      124Ч125
Center subspace (eigenspace) for flows      10Ч12
Center subspace (eigenspace) for maps      11
Central point, restrictive central point      272
Chain recurrent point, set      236 (Definition 5.2.3)
Characteristic (Floquet) multipliers, exponents      25
Closed orbit      15Ч16
Codimension (of a bifurcation point)      123
Codimension (of an embedded submanifold)      120
coin tossing      227Ч228
Continuous dependence on initial conditions      7
Critical point (one-dimensional maps)      306
Cross section (Poincare section)      23 26
Defining sequence (of a Cantor set)      333
DenjoyТs theorem      301Ч302 (Theorem 6.2.5)
Differential equation      1Ч2
Diophantine conditions      140 220 302Ч303
Discrete dynamical system (map, mapping)      16Ч22
Dissipative saddle point      326
Distortion (of a one-dimensional map)      341 (Definition
Domain of attraction      34 111Ч114
Drifting oscillations      71
DuffingТs equation      22 82Ч91 (Section 173Ч177 191Ч193 198Ч201 267
Eigenspace      9Ч12
Eigenvector      9Ч12
Entrainment      71 305
Entrainment domain      74
Entropy (topological entropy)      281Ч288 (Definitions 5.8)
Equilibrium point, equilibrium solution      3 12
Equivalence relations      38Ч42
Equivariant vector fields      144 150
Ergodic theory      281Ч283
Exchange of stability      (see Transcritical bifurcation)
Existence and uniqueness theorem (differential equations)      3 7
Exponential smallness (of Melnikov functions)      211Ч212
First return (Poincare) map      23Ч32
Fixed point      3 12
Flip (period doubling) bifurcation      86 105 111 157Ч160 (Theorem 311 346Ч349
Floquet theory and Poincare maps      24Ч25
Flow      2
fold      (see Saddle-node bifurcation)
Frequency response function (forced oscillations)      174Ч175
Full family      272 307
Fundamental solution matrix      9
Galerkin method      84 417
Gap (of a feantor set)      333
General position (transversal intersection)      120
Generalized Hopf bifurcation      359
Generic properties (reference to)      41
Global bifurcation      88 117 149 289Ч352 (Chapter 353Ч420 (Chapter
Globally asymptotically stable (fixed point)      5
Gradient system, gradient vector field      49Ч50
GronwallТs inequality      169 (Lemma
Hamiltonian system      46 182Ч185 193Ч226
Hartman-Grobman (linearization) theorem for flows      13
Hartman-Grobman (linearization) theorem for maps      18
Hausdorff dimension      285Ч288 (Definitions 5.8.4) 333
Henon map      245 (Exercise 267 268Ч269 273
Heteroclinic orbit, point      22 45 46 290Ч291
Homoclinic bifurcation      190Ч191 292Ч293 (Theorem 325Ч331
Homoclinic cycle      45
Homoclinic orbit (point)      22 45 47 86 88 101Ч102 114 182Ч193 222 252 291Ч295 313 318Ч342
Homoclinic tangency      183Ч184 190Ч193 325Ч342 (Sections
Homoclinic tangle      222 247Ч248
Hopf bifurcation and averaging      178Ч180 (Theorem 4.3.1)
Hopf bifurcation for flows      73 93 150Ч156
Hopf bifurcation for maps      160Ч165 (Theorem 3.5.2) 305Ч306
Hopf bifurcation, generalized Hopf bifurcation      358Ч360
Horizontal curve, strip      239Ч240
Horseshoe (map)      106Ч111 188 230Ч235 318Ч325
Hyperbolic fixed point      17
Hyperbolic set structure      228 238Ч248 (Definition
Indecomposable (invariant set)      237 (Definition
Independence      281
Index (Poincare index)      50Ч53
Induced map      341
information      281
Initial condition      2
Initial condition, continuous dependence on      7 (Theorem 1.0.4)
Integrable (Hamiltonian) system      185 216
Intermittency      343Ч346
Invariant coordinate      307
Invariant manifold      (see Invariant set Stable Unstable
Invariant measure, probability measure      271 280
Invariant set      33 108Ч109 235
Invariant subspace (eigenspace)      10Ч12
Invariant torus      58Ч60 183 182 218Ч222 298Ч299
Inverse limit construction      264
Isocline      53Ч56
Isoenergetic nondegeneracy      219 (Exercise
Itinerary      307
Jordan form      9
k-jets (truncated Taylor series)      360Ч364
Kneading sequence      307
Kneading theory (kneading calculus      306Ч311 (Section 6.3)
Kolmogorov-Arnold-Moser (KAM) theory      138 182 218Ч226 (Theorem 303 350
Lambda lemma      247 (Theorem
Liapunov exponent (Liapunov number)      283Ч288 (Definition
Liapunov function      4Ч7 49
Limit set      15 34
Linearization (Hartman-Grobman theorem)      12Ч13 (Theorem
Lipschitz, Lipschitz constant      3
Local bifurcation      117 117Ч165 (Chapter
Lorenz attractor      (see Lorenz equations)
Lorenz equations      92Ч102 (Section 128Ч130 273Ч279 (Section 312Ч318 (Section
Lozi map      267
Markov partition      248Ч255 (Definition 314
Mathieu equation      29Ч32
Measure      (see Asymptotic measure Invariant Probability
Melnikov function      187
Melnikov function for subharmonics      195
Melnikov function, exponential smallness of      211Ч212
Melnikov theory (MelnikovТs method      184Ч226 (Sections 369Ч370
Morse sequence      346Ч347
Morse Ч Smale system      64
Multiplicative ergodic theorem      284
Neutral stability      4
Newhouse sinks      88 91 111 331Ч342
Nondegenerate (hyperbolic) fixed point      13
Nonintegrable Hamiltonian system      224
Nonwandering point, set      33 236 (Definition
Normal forms (of degenerate k-jets) for flows      138Ч145 (Section
Normal forms (of degenerate k-jets) for maps (Hopf bifurcation)      161Ч165
Normal forms (of degenerate k-jets) for multiple bifurcations      358 365 377 397 (Chapter
Normal modes (modes of vibration)      58 83 215
One-dimensional maps      268Ч273 (Section 306Ч311 (Section
Orbit      2
Orientation preserving/reversing (map)      20
Panel flutter      414Ч419
PeixotoТs theorem      60Ч62 (Theorem
Period doubling bifurcation      86 105 111 157Ч160 (Theorem 311
Period doubling sequences (cascades)      346Ч349
Period three implies chaos      309 (Exercise 311
Periodic orbit      15Ч16
Phase locking      71 305
Pitchfork bifurcation      85 93 149Ч150 157
Plykin attractor      264Ч266
Poincare index      50Ч53
Poincare map      22Ч32 (Section
Poincare map, averaging and      161Ч171
Poincare Ч Bendixson theorem      44 (Theorem 69
Potential function      49
Power spectrum      87 88 349
Preturbulence      315
Probability measure      280
Proposition      5 3 3
Pseudo-orbits      250
Quadratic tangencies      (see Homoclinic tangencies)
Quasiperiodic orbits      160 349Ч352
Rectangle (Markov partition)      249 (Definition
Reduced Hamiltonian system      214
Reduced system (restricted to the center manifold)      130Ч138
Reduction of Hamiltonian systems      213Ч215
Reflection hyperbolic (fixed point)      105
Relaxation oscillations      69
Renormalization      342Ч352
Repelling set      34
Repellor      36
Resonance of order k (forced oscillations)      173 195 199
Resonant terms (in normal forms)      140 162
Resonant wedge      303
Restrictive central point      273
Rotation number      163 295Ч306 (Section
Rotation number, definition of      296 349Ч352
Saddle (points) of the first (second) kind for maps      105
Saddle connection, saddle-loop (heteroclinic and homoclinic orbits)      46Ч48 73Ч74 86 101Ч102 185 290Ч295 313
Saddle point      4
Saddle-node bifurcation      73 105 111 145 146Ч149 (Theorem 157
Saddle-node bifurcation and averaging      178Ч180 (Theorem
Saddle-node bifurcation and averaging in MelnikovТs theory      197 (Theorem 4.6.3) 343Ч346
SarkovskiiТs theorem      311 (Theorem
Schwarzian derivative      270 306
Semiflow      94
Sensitive dependence on initial conditions      80 98 188
Shadowing      250Ч251
Shadowing lemma      251
Shift map (symbolic dynamics)      78 234 251
SilnikovТs (homoclinic) theorem      318Ч325 (Theorem
SingerТs theorem (one-dimensional maps)      300 (Theorem
Singularity theory      354Ч355
Sink      4
Smale horseshoe      (see Horseshoe)
Smale Ч Birkhoff homoclinic theorem      188 252 253 (Theorem
Small divisors      163 220 302Ч303
Solenoid      264
Solution curve      2
Source      4
Stability dogma      259
Stable fixed point      3Ч4
Stable foliation      96Ч97 264Ч267 274 312
Stable manifold theorem for flows      13Ч14 (Theorem 1.3.2)
Stable manifold theorem for hyperbolic sets      246Ч247 (Theorem 5.2.8)
Stable manifold theorem for maps      18 (Theorem
Stable manifold, global      14 18
Stable manifold, local      13 18
Stable subspace (eigenspace) for flows      10Ч12
Stable subspace (eigenspace) for maps      17
Stationary solution      12
Stochastic layer      222 226
Strange Attractor      86Ч91 255Ч259 (Definition
Structural stability      38Ч42 (Section (Definition 88 258Ч259
Subcritical and supercritical bifurcations      93 150 158Ч160
Subharmonic      27 74 193Ч212 223
Subharmonic bifurcation      197 (see also Flip bifurcation)
Subharmonic Melnikov function      195
Subshift of finite type      251 264
Symbol sequences      78 233
Symbolic dynamics      77Ч79 228Ч230 232Ч234 264 317Ч318
Symmetry groups (and bifurcations)      (see Equivariant vector fields)
Thermohaline convection      412Ч414 417Ч419
Thickness (of Cantor sets)      333
Time average      282 (Definition
Time-dependent perturbation      260
Time-dependent stable      260
Topological conjugacy      (see Topological equivalence)
Topological dimension      261
Topological equivalence      38 (Definition 1.7.2)
Topologically transitive (invariant set)      237 (Definition
Trajectory      2
Transcritical bifurcation (exchange of stability)      145 149Ч150 197
Transition matrix      78 251 253 264Ч266 314Ч315
Transversal homoclinic orbits      88Ч89 113Ч114 183Ч184 188 222 247Ч248
Transversal intersection (of invariant manifolds)      50
Transversality (theorem)      120Ч123
Trapping region      34Ч35 114
Twist mapping      218 219Ч223
Twist theorem (for area preserving maps)      219Ч220 (Theorem
Unfolding (of a bifurcation point)      123 354Ч358 365
Universal structures      342Ч352
Universal unfolding      365 371
Unstable manifold global      14 18
Unstable manifold, local      13 18
Unstable subspace (eigenspace) for flows      10Ч12
Unstable subspace (eigenspace) for maps      17
Van der PolТs equation      67Ч82 (Section 253Ч255
Vector field      2
Vertical curve, strip      239Ч240
Wild hyperbolic sets      191 331Ч342 (Section
Zero      3 12
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