Mathematics of Complexity and Dynamical Systems

Cover......Page 1
Mathematics of Complexity and Dynamical Systems......Page 4
ISBN: 9781461418061......Page 5
Preface......Page 6
Fractals and Multifractals, Section Editor: Daniel ben-Avraham and Shlomo Havlin......Page 8
Perturbation Theory, Section Editor: Giuseppe Gaeta......Page 9
Systems and Control Theory, Section Editor: Matthias Kawski......Page 10
About the Editor-in-Chief......Page 12
Editorial Board Members......Page 14
Section Editors......Page 16
Table of Contents......Page 18
Contributors......Page 24
Introduction......Page 31
Adomian Decomposition Method and Adomian Polynomials......Page 32
Modified Decomposition Method and Noise Terms Phenomenon......Page 33
Solitons, Peakons, Kinks, and Compactons......Page 34
Solitons of the KdV Equation......Page 35
Kinks of the Burgers Equation......Page 36
Peakons of the Camassa–Holm Equation......Page 37
Compactons of the K(n,n) Equation......Page 39
Bibliography......Page 41
Definition of the Subject......Page 43
Random Walks and Normal Diffusion......Page 44
Anomalous Diffusion......Page 46
Anomalous Diffusion on Fractal Structures......Page 48
Percolation Clusters......Page 52
Scaling of PDF and Diffusion Equations on Fractal Lattices......Page 53
Bibliography......Page 54
Definition of the Subject......Page 56
The Mathematics of Swimming......Page 57
The Scallop Theorem Proved......Page 58
Optimal Swimming......Page 59
Future Directions......Page 60
Bibliography......Page 61
Glossary......Page 62
Definition of the Subject......Page 63
Example 1: The Eccentric Cylinder on the Inclined Plane......Page 64
Example 2: The Formation of Traffic Jam......Page 68
Unfoldings......Page 70
The Geometry of the Fold and the Cusp......Page 72
Further Applications......Page 75
Bibliography......Page 77
Definition of the Subject......Page 78
Center Manifold in Ordinary Differential Equations......Page 80
Center Manifold in Discrete Dynamical Systems......Page 85
Normally Hyperbolic Invariant Manifolds......Page 86
Center Manifold in Infinite-Dimensional Space......Page 87
Bibliography......Page 90
Glossary......Page 93
Introduction......Page 94
Picking an Invariant Probability Measure......Page 97
Tractable Chaotic Dynamics......Page 99
Statistical Properties......Page 104
Orbit Complexity......Page 106
Stability......Page 108
Future Directions......Page 110
Bibliography......Page 111
Introduction, History, and Background......Page 118
Fundamental Notions of the Chronological Calculus......Page 120
Systems That Are Affine in the Control......Page 124
Future Directions......Page 128
Bibliography......Page 129
Definition of the Subject......Page 132
Introduction......Page 133
Controllability......Page 134
Stabilization......Page 139
Optimal Control......Page 148
Future Directions......Page 151
Bibliography......Page 152
Glossary......Page 156
Introduction......Page 157
Examples......Page 158
Trees and Graphical Representation......Page 159
Small Divisors......Page 162
Multiscale Analysis......Page 163
Resummation......Page 165
Generalizations......Page 167
Conclusions and Future Directions......Page 170
Bibliography......Page 171
Introduction......Page 173
Discrete Lagrangian and Hamiltonian Mechanics......Page 174
Optimal Control of Discrete Lagrangian and Hamiltonian Systems......Page 182
Controlled Lagrangian Method for Discrete Lagrangian Systems......Page 185
Future Directions......Page 186
Bibliography......Page 187
Definition of the Subject......Page 190
Introduction......Page 191
The Mechanism of Dispersion......Page 192
Strichartz Estimates......Page 196
The Nonlinear Wave Equation......Page 199
The Nonlinear Schrödinger Equation......Page 201
Bibliography......Page 203
Introduction......Page 205
Fractal and Spectral Dimensions......Page 206
Nature of Dynamics on Fractals – Localization......Page 207
Mapping of Physical Systems onto Fractal Structures......Page 208
Relaxation Dynamics on Fractal Structures......Page 209
Transport on Fractal Structures......Page 210
Bibliography......Page 211
Introduction......Page 213
Linear Resonance or Nonlinear Instability?......Page 218
Multibody Systems......Page 221
Continuous Systems......Page 226
Bibliography......Page 232
Glossary......Page 235
Distribution Entropy......Page 236
A Gander at Shannon’s Noisy Channel Theorem......Page 238
The Information Function......Page 239
Entropy of a Process......Page 240
Entropy of a Transformation......Page 241
Determinism and Zero-Entropy......Page 243
Ornstein Theory......Page 244
Topological Entropy......Page 245
Three Recent Results......Page 251
Bibliography......Page 252
Introduction......Page 255
Basics and Examples......Page 256
Ergodicity......Page 258
Ergodic Decomposition......Page 260
Mixing......Page 262
Hyperbolicity and Decay of Correlations......Page 266
Future Directions......Page 267
Bibliography......Page 268
Introduction......Page 271
Ergodic Theorems for Measure-Preserving Maps......Page 272
Generalizations to Continuous Time and Higher-Dimensional Time......Page 276
Pointwise Ergodic Theorems for Operators......Page 278
Subadditive and Multiplicative Ergodic Theorems......Page 279
Entropy and the Shannon–McMillan–Breiman Theorem......Page 281
Amenable Groups......Page 282
Subsequence and Weighted Theorems......Page 283
Ergodic Theorems and Multiple Recurrence......Page 285
Rates of Convergence......Page 287
Future Directions......Page 288
Bibliography......Page 289
Definition of the Subject......Page 294
Introduction......Page 295
Examples......Page 296
Constructions......Page 310
Future Directions......Page 313
Bibliography......Page 314
Introduction......Page 318
Preliminaries......Page 319
Brief Tour Through Some Examples......Page 321
Dimension Theory of Higher-Dimensional Dynamical Systems......Page 324
General Theory......Page 325
Multifractal Analysis......Page 326
Future Directions......Page 327
Bibliography......Page 328
Introduction......Page 332
Basic Facts......Page 333
Connection with Dynamics on the Space of Lattices......Page 334
Diophantine Approximation with Dependent Quantities: The Set-Up......Page 336
Further Results......Page 338
Bibliography......Page 340
Definition of the Subject......Page 343
Ergodic Theory......Page 344
Frequency of Returns......Page 345
Ergodic Ramsey Theory and Recurrence......Page 346
Orbit-Counting as an Analogous Development......Page 348
Diophantine Analysis as a Toolbox......Page 351
Bibliography......Page 353
Ergodic Theory, Introduction to......Page 357
Definition of the Subject......Page 359
Basic Results......Page 360
Panorama of Examples......Page 363
Mixing Notions and multiple recurrence......Page 365
Topological Group Aut(X, )......Page 367
Orbit Theory......Page 368
Spectral Theory for Nonsingular Systems......Page 372
Entropy and Other Invariants......Page 374
Nonsingular Joinings and Factors......Page 376
Applications. Connections with Other Fields......Page 378
Bibliography......Page 382
Glossary......Page 387
Introduction......Page 388
Quantitative Poincaré Recurrence......Page 389
Subsequence Recurrence......Page 390
Multiple Recurrence......Page 392
Connections with Combinatorics and Number Theory......Page 394
Future Directions......Page 395
Bibliography......Page 397
Introduction......Page 399
Basic Definitions and Examples......Page 401
Differentiable Rigidity......Page 403
Local Rigidity......Page 405
Global Rigidity......Page 407
Measure Rigidity......Page 408
Future Directions......Page 409
Bibliography......Page 410
Definition of the Subject......Page 413
Introduction......Page 414
Existence......Page 415
Uniqueness......Page 418
Continuous Dependence on Initial Conditions......Page 420
Extended Concept of Differential Equation......Page 422
Further Directions......Page 423
Bibliography......Page 424
Introduction......Page 425
Control Systems......Page 426
Linear Systems......Page 427
Linearization Principle......Page 429
High Order Tests......Page 431
Controllability and Observability......Page 433
Controllability and Stabilizability......Page 434
Flatness......Page 435
Future Directions......Page 436
Bibliography......Page 437
Definition of the Subject......Page 439
Deterministic Fractals......Page 440
Random Fractal Models......Page 445
Self-Affine Fractals......Page 448
Long-Term Correlated Records......Page 450
Multifractal Records......Page 455
Bibliography......Page 456
Introduction......Page 459
Fractals and Multifractals......Page 460
Aggregation Models......Page 461
Conformal Mapping......Page 467
Harmonic Measure......Page 470
Scaling Theories......Page 471
Bibliography......Page 473
Definition of the Subject......Page 476
Introduction......Page 477
Solving Resistor Networks......Page 478
Conduction Near the Percolation Threshold......Page 481
Voltage Distribution in Random Networks......Page 484
Random Walks and Resistor Networks......Page 488
Bibliography......Page 490
Definition of the Subject......Page 493
Introduction......Page 494
Fractal and Multifractal Time Series......Page 495
Methods for Stationary Fractal Time Series Analysis......Page 498
Methods for Non-stationary Fractal Time Series Analysis......Page 500
Methods for Multifractal Time Series Analysis......Page 506
Statistics of Extreme Events in Fractal Time Series......Page 510
Simple Models for Fractal and Multifractal Time Series......Page 512
Future Directions......Page 514
Bibliography......Page 515
Glossary......Page 518
Introduction......Page 519
Self-similar Branching Structures......Page 520
Fractal Metabolic Rates......Page 523
Physical Models of Biological Fractals......Page 524
Diffusion Limited Aggregation and Bacterial Colonies......Page 525
Measuring Fractal Dimension of Real Biological Fractals......Page 527
Percolation and Forest Fires......Page 528
Critical Point and Long-Range Correlations......Page 530
Lévy Flight Foraging......Page 531
Fractals and Time Series......Page 532
Fractal Features of DNA Sequences......Page 534
Future Directions......Page 537
Bibliography......Page 538
Introduction......Page 542
Examples in Economics......Page 544
Basic Models of Power Laws......Page 551
Market Models......Page 555
Income Distribution Models......Page 558
Bibliography......Page 560
Drainage Networks......Page 562
Earthquakes......Page 563
Floods......Page 564
Earth’s Magnetic Field......Page 565
Bibliography......Page 566
Glossary......Page 567
Definition of the Subject......Page 568
Introduction......Page 569
Dynamical Systems......Page 570
Curves and Dimension......Page 573
Chaos Comes of Age......Page 576
The Advent of Fractals......Page 580
The Merger......Page 582
Bibliography......Page 583
Introduction......Page 589
Percolation......Page 590
Percolation Clusters as Fractals......Page 593
Anomalous Transport on Percolation Clusters: Diffusion and Conductivity......Page 596
Networks......Page 597
Summary and Future Directions......Page 598
Bibliography......Page 599
Definition of the Subject......Page 601
Introduction......Page 602
Scale Laws......Page 603
From Fractal Space to Nonrelativistic Quantum Mechanics......Page 612
From Fractal Space-Time to Relativistic Quantum Mechanics......Page 617
Gauge Fields as Manifestations of Fractal Geometry......Page 618
Bibliography......Page 619
Definition of the Subject......Page 621
Introduction......Page 622
Determining Fractal Dimensions......Page 624
Polymer Chains in Solvents......Page 625
Aggregates and Flocs......Page 626
Aerogels......Page 629
Dynamical Properties of Fractal Structures......Page 630
Spectral Density of States and Spectral Dimensions......Page 631
Future Directions......Page 633
Bibliography......Page 634
Glossary......Page 636
Definition of the Subject......Page 637
Introduction......Page 638
A Wavelet-Based Multifractal Formalism......Page 639
Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription......Page 646
From the Detection of Relication Origins Using the Wavelet Transform Microscope to the Modeling of Replication in Mammalian Geno......Page 651
A Wavelet-Based Methodology to Disentangle Transcriptionand Replication-Associated Strand Asymmetries Reveals a Remarkable Gene......Page 656
Future Directions......Page 660
Bibliography......Page 661
Definition of the Subject......Page 667
Fractality in Real-World Networks......Page 668
Models: Deterministic Fractal and Transfractal Networks......Page 673
Properties of Fractal and Transfractal Networks......Page 676
Future Directions......Page 681
Appendix: The Box Covering Algorithms......Page 682
Bibliography......Page 685
Glossary......Page 687
Introduction......Page 688
One Degree of Freedom......Page 692
Perturbations of Periodic Orbits......Page 694
Invariant Curves of Planar Diffeomorphisms......Page 695
KAM Theory: An Overview......Page 699
Splitting of Separatrices......Page 703
Transition to Chaos and Turbulence......Page 704
Bibliography......Page 708
Glossary......Page 713
Introduction......Page 714
Subsolutions......Page 715
Solutions......Page 717
First Regularity Results for Subsolutions......Page 720
Critical Equation and Aubry Set......Page 722
An Intrinsic Metric......Page 724
Dynamical Properties of the Aubry Set......Page 727
Long-Time Behavior of Solutions to the Time-Dependent Equation......Page 728
Main Regularity Result......Page 731
Future Directions......Page 732
Bibliography......Page 733
Notation......Page 734
Introduction......Page 735
Well-posed Hybrid Dynamical Systems......Page 737
Modeling Hybrid Control Systems......Page 740
Stability Theory......Page 744
Design Tools......Page 747
Applications......Page 750
Discussion and Final Remarks......Page 755
Bibliography......Page 756
Introduction......Page 759
Examples of Conservation Laws......Page 760
Shocks and Weak Solutions......Page 761
Entropy Admissibility Conditions......Page 762
The Riemann Problem......Page 763
Global Solutions......Page 764
Hyperbolic Systems in Several Space Dimensions......Page 765
Future Directions......Page 766
Bibliography......Page 767
Glossary......Page 770
Linear Systems......Page 771
Local Theory......Page 772
Hyperbolic Behavior: Examples......Page 773
Hyperbolic Sets......Page 774
Uniformly Hyperbolic Systems......Page 776
Attractors and Physical Measures......Page 778
Obstructions to Hyperbolicity......Page 779
Partial Hyperbolicity......Page 780
Non-Uniform Hyperbolicity – Linear Theory......Page 781
Non-Uniformly Hyperbolic Systems......Page 783
Bibliography......Page 784
Introduction......Page 785
First Definitions and Examples......Page 786
Linear Systems......Page 788
Nonlinear Systems......Page 794
Bibliography......Page 798
Introduction......Page 801
The Lax Method......Page 803
The AKNS Method......Page 804
Direct Scattering Problem......Page 806
Time Evolution of the Scattering Data......Page 807
Inverse Scattering Problem......Page 808
Solitons......Page 809
Bibliography......Page 811
Definition of the Subject......Page 813
Basic Transformations......Page 814
Basic Tools......Page 815
Isomorphism of Bernoulli Shifts......Page 816
Transformations not Isomorphic to Bernoulli Shifts......Page 818
Classifying the Invariant Measures of Algebraic Actions......Page 819
Flows......Page 820
Non-invertible Transformations......Page 821
Factors of a Transformation......Page 822
Actions of Amenable Groups......Page 823
Bibliography......Page 824
Glossary......Page 826
Introduction......Page 827
Joinings of Two or More Dynamical Systems......Page 828
Self-Joinings......Page 832
Some Applications and Future Directions......Page 834
Bibliography......Page 838
Definition of the Subject......Page 840
Introduction......Page 841
Kolmogorov Theorem......Page 843
Arnold’s Scheme......Page 850
The Differentiable Case: Moser’s Theorem......Page 854
Future Directions......Page 855
B Complementary Notes......Page 859
Bibliography......Page 864
Definition of the Subject......Page 867
Introduction......Page 868
The Generalized Hyperbolic Function–Bäcklund Transformation Method and Its Application in the (2 + 1)-Dimensional KdV Equation......Page 869
The Generalized F-expansion Method and Its Application in Another (2 + 1)-Dimensional KdV Equation......Page 877
The Generalized Algebra Method and Its Application in (1 + 1)-Dimensional Generalized Variable – Coefficient KdV Equation......Page 884
A New Exp-N Solitary-Like Method and Its Application in the (1 + 1)-Dimensional Generalized KdV Equation......Page 897
The Exp-Bäcklund Transformation Method and Its Application in (1 + 1)-Dimensional KdV Equation......Page 903
Bibliography......Page 911
Introduction......Page 914
Inverse Scattering Transform for the KdV Equation......Page 915
Exact N-soliton Solutions of the KdV Equation......Page 916
Further Properties of the KdV Equation......Page 917
Future Directions......Page 918
Bibliography......Page 919
Introduction......Page 920
An Analysis of the Semi-analytical Methods and Their Applications......Page 922
Bibliography......Page 935
Introduction......Page 938
The Adomian Decomposition Method (ADM)......Page 940
The Homotopy Analysis Method (HAM)......Page 941
The Homotopy Perturbation Method (HPM)......Page 942
Numerical Applications and Comparisons......Page 943
Future Directions......Page 947
Bibliography......Page 950
Introduction......Page 954
Least Squares Curve Fitting – A Precursor......Page 955
System Identification......Page 956
Statistical Learning Theory......Page 958
The Complexity of Learning......Page 960
System Identification as a Learning Problem......Page 962
Bibliography......Page 965
Introduction......Page 967
Lyapunov–Schmidt Method for Equilibria......Page 971
Lyapunov–Schmidt Method in Discrete Systems......Page 973
Lyapunov–Schmidt Method for Periodic Solutions......Page 976
Lyapunov–Schmidt Method in Infinite Dimensions......Page 979
Bibliography......Page 980
Definition of the Subject......Page 983
Introduction......Page 984
The Calculus of Variations and the Maximum Principle......Page 985
Variational Problems with Constraints......Page 986
Maximum Principle on Manifolds......Page 988
Abnormal Extrema and Singular Problems......Page 991
Bibliography......Page 992
Introduction: The Dynamical Viewpoint......Page 994
Where do Measure-Preserving Systems Come from?......Page 995
Construction of Measures......Page 997
Finding Finite Invariant Measures Equivalent to a Quasi-Invariant Measure......Page 1000
Finding -finite Invariant Measures Equivalent to a Quasi-Invariant Measure......Page 1001
Some Mathematical Background......Page 1004
Bibliography......Page 1009
Introduction......Page 1011
Symplectic Reduction......Page 1012
Symplectic Reduction – Further Discussion......Page 1016
Reduction Theory: Historical Overview......Page 1020
Cotangent Bundle Reduction......Page 1026
Future Directions......Page 1029
Appendix: Principal Connections......Page 1030
Bibliography......Page 1034
Glossary......Page 1039
Definition of the Subject......Page 1040
Derivation of the Navier–Stokes Equations and Preliminary Considerations......Page 1041
Mathematical Analysis of the Boundary Value Problem......Page 1044
Mathematical Analysis of the Initial-Boundary Value Problem......Page 1056
Future Directions......Page 1069
Bibliography......Page 1070
Introduction......Page 1073
Simple Choreographies and Relative Equilibria......Page 1075
Symmetry Groups and Equivariant Orbits......Page 1076
The 3-Body Problem......Page 1083
Minimizing Properties of Simple Choreographies......Page 1088
Generalized Orbits and Singularities......Page 1090
Asymptotic Estimates at Collisions......Page 1092
Absence of Collision for Locally Minimal Paths......Page 1094
Bibliography......Page 1097
Definition of the Subject......Page 1100
Introduction......Page 1101
Exponential Stability of Constant Frequency Systems......Page 1102
Nekhoroshev Theory (Global Stability)......Page 1104
Applications......Page 1107
Appendix: An Example of Divergence Without Small Denominators......Page 1108
Bibliography......Page 1109
Introduction......Page 1112
Symmetry of Dynamical Systems......Page 1113
Perturbation Theory: Normal Forms......Page 1114
Perturbative Determination of Symmetries......Page 1117
Symmetry Characterization of Normal Forms......Page 1118
Symmetries and Transformation to Normal Form......Page 1119
Generalizations......Page 1120
Symmetry for Systems in Normal Form......Page 1121
Further Normalization and Symmetry......Page 1122
Symmetry Reduction of Symmetric Normal Forms......Page 1124
Conclusions......Page 1125
Additional Notes......Page 1126
Bibliography......Page 1127
Introduction......Page 1132
Problem and Frame of Reference......Page 1133
Equations of Motion......Page 1134
The Shallow Water Theory......Page 1135
Multiple Scale Transformation of Variables......Page 1137
Derivation of the KdV Equation......Page 1138
Bibliography......Page 1139
Non-linear Ordinary Differential Equations and Dynamical Systems, Introduction to......Page 1141
Non-linear Partial Differential Equations, Introduction to......Page 1143
Glossary......Page 1145
Introduction......Page 1146
Examples......Page 1147
Comparison Principle......Page 1148
Existence Results......Page 1149
Boundary Value Problems......Page 1150
Asymptotic Analysis......Page 1151
Other Notions......Page 1152
Future Directions......Page 1153
Bibliography......Page 1154
Definition of the Subject......Page 1156
Introduction......Page 1157
General Problems and Results......Page 1158
Specific Equations......Page 1160
Bibliography......Page 1164
Definition of the Subject......Page 1167
Introduction......Page 1168
Elements of Nonsmooth Analysis......Page 1169
Necessary Conditions in Optimal Control......Page 1171
Dynamic Programming and Viscosity Solutions......Page 1173
Lyapunov Functions......Page 1176
Stabilizing Feedback......Page 1177
Bibliography......Page 1179
Motivation......Page 1182
The Normal Form Procedure......Page 1183
Preservation of Structure......Page 1189
Semi-local Normalization......Page 1192
Non-formal Aspects......Page 1195
Applications......Page 1196
Bibliography......Page 1198
Introduction......Page 1202
Continuation and Discretization of Solutions......Page 1204
Normal Forms and the Center Manifold......Page 1207
Continuation and Detection of Bifurcations......Page 1210
Branch Switching......Page 1215
Connecting Orbits......Page 1219
Future Directions......Page 1221
Bibliography......Page 1222
Introduction......Page 1225
Preliminaries......Page 1226
Linear Systems......Page 1228
Realization Theory......Page 1229
Observers......Page 1231
Bibliography......Page 1233
Introduction......Page 1235
Some Nonlinear Models that Lead to Solitons......Page 1236
Bibliography......Page 1239
Introduction......Page 1242
Periodic Solutions......Page 1244
Poincaré Map and Floquet Operator......Page 1247
Hamiltonian Systems with Symmetries......Page 1251
The Variational Principles and Periodic Orbits......Page 1254
Bibliography......Page 1264
Introduction......Page 1266
Poincaré Operator and Linear Systems......Page 1268
Fixed Point Approach: Perturbation Theory......Page 1269
Fixed Point Approach: Large Nonlinearities......Page 1270
Guiding Functions......Page 1272
Lower and Upper Solutions......Page 1273
Direct Method of the Calculus of Variations......Page 1275
Critical Point Theory......Page 1277
Bibliography......Page 1278
Introduction......Page 1281
Perturbation Techniques......Page 1282
Parametric Excitation of Linear Systems......Page 1284
Nonlinear Parametric Excitation......Page 1288
Applications......Page 1289
Bibliography......Page 1292
Introduction......Page 1295
EvolutiononaFitnessLandscape......Page 1296
Perturbation of Equilibria on a Fitness Landscape......Page 1297
Frequency Dependent Fitness: Game Theory......Page 1298
Equilibria in Evolutionary Game Theory......Page 1299
Perturbations of Equilibria in Evolutionary Game Theory......Page 1300
Bibliography......Page 1304
Introduction......Page 1306
Complex and Real Jordan Canonical Forms......Page 1307
Nilpotent Perturbation and Formal Normal Forms of Vector Fields and Maps Near a Fixed Point......Page 1309
Loss of Gevrey Regularity in Siegel Domains in the Presence of Jordan Blocks......Page 1310
First-Order Singular Partial Differential Equations......Page 1312
Normal Forms for Real Commuting Vector Fields with Linear Parts Admitting Nontrivial Jordan Blocks......Page 1313
Analytic Maps near a Fixed Point in the Presence of Jordan Blocks......Page 1315
Weakly Hyperbolic Systems and Nilpotent Perturbations......Page 1316
Bibliography......Page 1318
Introduction......Page 1320
Mathematics and Physics. Renormalization......Page 1321
Multiscale Analysis......Page 1323
A Paradigmatic Example of PT Problem......Page 1324
Convergence. Scales. Multiscale Analysis......Page 1325
Non Convergent Cases......Page 1327
Conclusion and Outlook......Page 1328
Bibliography......Page 1329
Introduction......Page 1331
Classical Perturbation Theory......Page 1332
Resonant Perturbation Theory......Page 1335
Invariant Tori......Page 1338
Periodic Orbits......Page 1341
Future Directions......Page 1342
Bibliography......Page 1343
Perturbation Theory, Introduction to......Page 1344
Introduction......Page 1347
The Framework......Page 1348
The Leading Order Born–Oppenheimer Approximation......Page 1349
Beyond the Leading Order......Page 1351
Bibliography......Page 1353
Introduction......Page 1355
Preliminaries......Page 1356
Vector Fields near the Boundary......Page 1358
Generic Bifurcation......Page 1361
Singular Perturbation Problem in 2D......Page 1362
Future Directions......Page 1364
Bibliography......Page 1365
Introduction......Page 1367
The Hamiltonian Formalism for PDEs......Page 1368
Normal Form for Finite Dimensional Hamiltonian Systems......Page 1371
Normal Form for Hamiltonian PDEs: General Comments......Page 1372
Normal Form for Resonant Hamiltonian PDEs and its Consequences......Page 1373
Normal Form for Nonresonant Hamiltonian PDEs......Page 1374
Non Hamiltonian PDEs......Page 1377
Extensions and Related Results......Page 1378
Bibliography......Page 1379
Definition of the Subject......Page 1381
Introduction......Page 1382
Presentation of the Problem and an Example......Page 1383
Perturbation of Point Spectra: Nondegenerate Case......Page 1385
Perturbation of Point Spectra: Degenerate Case......Page 1387
The Brillouin–Wigner Method......Page 1392
Symmetry and Degeneracy......Page 1393
Problems with the Perturbation Series......Page 1396
Perturbation of the Continuous Spectrum......Page 1398
Time Dependent Perturbations......Page 1402
Bibliography......Page 1404
Glossary......Page 1406
The WKB Approximation......Page 1407
Semiclassical Approximation in Any Dimension......Page 1413
Propagation of Quantum Observables......Page 1415
Future Directions......Page 1417
Bibliography......Page 1418
Introduction......Page 1419
Poincaré–Dulac Normal Forms......Page 1420
Convergence and Convergence Problems......Page 1421
Lie Algebra Arguments......Page 1424
NFIM and Sets of Analyticity......Page 1426
Hamiltonian Systems......Page 1427
Bibliography......Page 1428
Ising Model......Page 1430
Fractals......Page 1431
Diffusion on Fractals......Page 1432
Ising Model on Fractals......Page 1433
Networks......Page 1434
Bibliography......Page 1435
Definition of the Subject......Page 1437
Introduction......Page 1438
Mathematical Models: What Are They?......Page 1441
Philosophical and Mathematical Structuralism......Page 1444
Three Approaches to Applying Mathematical Models......Page 1447
Validating Mathematical Models......Page 1449
Future Directions......Page 1450
Bibliography......Page 1451
Introduction......Page 1452
Warming Up: Thermodynamic Formalism for Finite Systems......Page 1454
Shift Spaces, Invariant Measures and Entropy......Page 1455
The Variational Principle: A Global Characterization of Equilibrium......Page 1457
The Gibbs Property: A Local Characterization of Equilibrium......Page 1459
Examples on Shift Spaces......Page 1461
Examples from Differentiable Dynamics......Page 1462
Some Ongoing Developments and Future Directions......Page 1465
Bibliography......Page 1466
Glossary......Page 1468
Introduction......Page 1469
Simplest Effective Hamiltonians......Page 1470
Bifurcations and Symmetry......Page 1473
Imperfect Bifurcations......Page 1475
Organization of Bifurcations......Page 1476
Bifurcation Diagrams for Two Degree-of-Freedom Integrable Systems......Page 1477
Bifurcations of fiQuatum Bifurcation Diagramsf......Page 1480
Semi-Quantum Limit and Reorganization of Quantum Bands......Page 1481
Multiple Resonances and Quantum State Density......Page 1483
Bibliography......Page 1484
Definition of the Subject......Page 1487
Fractals and Some of Their Relevant Properties......Page 1488
Random Walks......Page 1490
Diffusion-limited Reactions......Page 1491
Irreversible Phase Transitions in Heterogeneously Catalyzed Reactions......Page 1495
Future Directions......Page 1500
Bibliography......Page 1501
Definition of the Subject......Page 1505
Introduction......Page 1506
Asymptotic Solution of the Van der Pol Oscillator......Page 1510
Canards......Page 1512
Dynamical Systems Approach......Page 1514
Future Directions......Page 1515
Bibliography......Page 1516
Definition of the Subject......Page 1519
Distributed Algorithms on Networks of Processors......Page 1520
Distributed Algorithms for Robotic Networks......Page 1526
Future Directions......Page 1531
Bibliography......Page 1532
Definition of the Subject......Page 1535
Weak-Coupling Limit for Classical Systems......Page 1537
Weak-Coupling Limit for Quantum Systems......Page 1541
Weak-Coupling Limit in the Bose–Einstein and Fermi–Dirac Statistics......Page 1545
Weak-Coupling Limit for a Single Particle: The Linear Theory......Page 1547
Bibliography......Page 1548
Definition of the Subject......Page 1550
Introduction......Page 1551
Completely Integrable Shallow Water Wave Equations......Page 1552
Shallow Water Wave Equations of Geophysical Fluid Dynamics......Page 1556
Computation of Solitary Wave Solutions......Page 1557
Water Wave Experiments and Observations......Page 1558
Bibliography......Page 1560
Introduction......Page 1563
The Fundamental Questions......Page 1564
Lebesgue Measure and Local Properties of Volume......Page 1565
Ergodicity of the Basic Examples......Page 1566
Hyperbolic Systems......Page 1567
Beyond Uniform Hyperbolicity......Page 1571
The Presence of Critical Points and Other Singularities......Page 1574
Bibliography......Page 1576
Introduction......Page 1578
Methods for Soliton Solutions......Page 1579
Bibliography......Page 1581
Solitons......Page 1583
Generalized Solitons and Compacton-like Solutions......Page 1584
Future Directions......Page 1588
Bibliography......Page 1589
Definition of the Subject......Page 1591
Introduction......Page 1592
Historical Discovery of Solitons......Page 1593
Physical Properties of Solitons and Associated Applications......Page 1596
Mathematical Methods Suitable for the Study of Solitons......Page 1602
Bibliography......Page 1603
Definition of the Subject......Page 1606
Introduction: Key Equations, Milestones, and Methods......Page 1607
Extended Definitions......Page 1611
Elastic Interactions of One-Dimensional and Line Solitons......Page 1612
Geometry of Oblique Interactions of KP Line Solitons......Page 1617
Soliton Interactions in Laboratory and Nature......Page 1618
Effects in Higher Dimensions......Page 1621
Applications of Line Soliton Interactions......Page 1624
Bibliography......Page 1626
Solitons, Introduction to......Page 1631
Definition of the Subject......Page 1633
Introduction......Page 1634
Shallow Water Waves and KdV Type Equations......Page 1636
Deep Water Waves and NLS Type Equations......Page 1639
Tsunamis as Solitons......Page 1640
Internal Solitons......Page 1641
Rossby Solitons......Page 1643
Bore Solitons......Page 1645
Bibliography......Page 1646
Glossary......Page 1648
Introduction......Page 1649
Spectral Theory of Weighted Operators......Page 1652
The Multiplicity Function......Page 1655
Rokhlin Cocycles......Page 1656
Rank-1 and Related Systems......Page 1657
Spectral Theory of Dynamical Systems of Probabilistic Origin......Page 1658
Special Flows and Flows on Surfaces, Interval Exchange Transformations......Page 1660
Future Directions......Page 1663
Bibliography......Page 1664
Introduction......Page 1669
Linear Systems......Page 1671
Nonlinear Systems: Continuous Feedback......Page 1673
Discontinuous Feedback......Page 1678
Sensitivity to Small Measurement Errors......Page 1680
Bibliography......Page 1681
Introduction......Page 1683
Mathematical Formulation of the Stability Concept and Basic Results......Page 1684
Stability in Conservative Systems and the KAM Theorem......Page 1693
Structural Stability......Page 1695
Attractors......Page 1697
Generalizations and Future Directions......Page 1698
Bibliography......Page 1699
Glossary......Page 1702
Introduction......Page 1703
Wide-Sense Stochastic Realization......Page 1705
Geometric Stochastic Realization......Page 1709
Dynamical System Identification......Page 1712
Bibliography......Page 1716
Glossary......Page 1719
Origins of Symbolic Dynamics: Modeling of Dynamical Systems......Page 1720
Shift Spaces and Sliding Block Codes......Page 1722
Shifts of Finite Type and Sofic Shifts......Page 1724
Entropy and Periodic Points......Page 1725
The Conjugacy Problem......Page 1727
Other Coding Problems......Page 1729
Coding for Data Recording Channels......Page 1732
Connections with Information Theory and Ergodic Theory......Page 1733
Higher Dimensional Shift Spaces......Page 1735
Bibliography......Page 1737
Introduction......Page 1741
The Generalized Tracking Problem......Page 1742
The Steady-State Behavior of a System......Page 1743
Necessary Conditions for Output Regulation......Page 1746
Sufficient Conditions for Output Regulation......Page 1748
Bibliography......Page 1752
Systems and Control, Introduction to......Page 1754
Acknowledgment......Page 1755
Introduction and History......Page 1756
Dynamic Relations, Invariant Sets and Lyapunov Functions......Page 1759
Attractors and Chain Recurrence......Page 1763
Chaos and Equicontinuity......Page 1767
Minimality and Multiple Recurrence......Page 1773
Bibliography......Page 1776
Introduction......Page 1778
Macroscopic Modeling......Page 1779
Kinetic Modeling......Page 1785
Road Networks......Page 1793
Bibliography......Page 1798
Glossary......Page 1801
Definition of the Subject......Page 1802
Introduction......Page 1804
The Euler Equation of Motion in Rectangular Cartesian and Cylindrical Polar Coordinates......Page 1805
Basic Equations of Water Waves with Effects of Surface Tension......Page 1808
The Stokes Waves and Nonlinear Dispersion Relation......Page 1811
Surface Gravity Waves on a Running Stream in Water......Page 1814
History of Russell’s Solitary Waves and Their Interactions......Page 1816
The Korteweg–de Vries and Boussinesq Equations......Page 1818
Solutions of the KdV Equation: Solitons and Cnoidal Waves......Page 1822
Derivation of the KdV Equation from the Euler Equations......Page 1825
Two-Dimensional and Axisymmetric KdV Equations......Page 1827
The Nonlinear Schrödinger Equation and Solitary Waves......Page 1830
Whitham’s Equations of Nonlinear Dispersive Waves......Page 1832
Whitham’s Instability Analysis of Water Waves......Page 1834
Future Directions......Page 1837
Bibliography......Page 1838
C......Page 1841
E......Page 1842
H......Page 1843
L......Page 1844
P......Page 1845
S......Page 1846
T......Page 1847
Z......Page 1848
A......Page 1849
B......Page 1851
C......Page 1852
D......Page 1856
E......Page 1858
F......Page 1860
G......Page 1863
H......Page 1864
I......Page 1865
L......Page 1867
M......Page 1869
N......Page 1872
O......Page 1874
P......Page 1875
R......Page 1877
S......Page 1879
T......Page 1885
V......Page 1887
Z......Page 1888