An excellent introduction to the world of Difference Equations. Professor Elaydi presents the area with care, in a pedagogical way, with rigor, plenty of examples and applications. This version of the ebook is very nice, with ocr layer and a super detailed bookmarks. Enjoy it!
Author(s): Saber Elaydi
Edition: 3rd
Publisher: Springer
Year: 2005
Language: English
Pages: 547
Tags: Difference Equations, Mathematics
Cover......Page 1
Title......Page 3
Preface to the Third Edition......Page 5
Contents......Page 13
List of Symbols......Page 18
1.1 Introduction......Page 20
1.2 Linear First-Order Difference Equations......Page 21
Exercises 1.1 and 1.2......Page 26
1.3 Equilibrium Points......Page 28
Exercises 1.3......Page 36
1.4 Numerical Solutions of Differential Equations......Page 39
Exercises 1.4......Page 45
1.5 Criterion for the Asymptotic Stability of Equilibrium Points......Page 46
Exercises 1.5......Page 53
1.6 Periodic Points and Cycles......Page 54
Exercises 1.6......Page 59
1.7 The Logistic Equation and Bifurcation......Page 62
Exercises 1.7......Page 68
1.8 Basin of Attraction and Global Stability (Optional)......Page 69
Exercises 1.8......Page 73
II. Linear Difference Equations of Higher Order......Page 74
2.1 Difference Calculus......Page 75
Exercises 2.1......Page 81
2.2 General Theory of Linear Difference Equations......Page 82
Exercises 2.2......Page 91
2.3 Linear Homogeneous Equations with Constant Coefficients......Page 93
Exercises 2.3......Page 98
2.4 Linear Nonhomogeneous Equations: Method of Undetermined Coefficients......Page 101
Exercises 2.4......Page 106
2.5 Limiting Behavior of Solutions......Page 109
Exercises 2.5......Page 115
2.6 Nonlinear Equations Transformable to Linear Equations......Page 116
Exercises 2.6......Page 121
2.7 Applications......Page 122
Exercises 2.7......Page 130
3.1 Autonomous (Time-Invariant) Systems......Page 135
Exercises 3.1......Page 141
3.2 The Basic Theory......Page 143
Exercises 3.2......Page 151
3.3 The Jordan Form: Autonomous (Time-Invariant) Systems Revisited......Page 153
Exercises 3.3......Page 168
3.4 Linear Periodic Systems......Page 171
Exercises 3.4......Page 176
3.5 Applications......Page 177
Exercises 3.5......Page 188
IV. Stability Theory......Page 191
4.1 A Norm of a Matrix......Page 192
Exercises 4.1......Page 193
4.2 Notions of Stability......Page 194
Exercises 4.2......Page 201
4.3 Stability of Linear Systems......Page 202
Exercises 4.3......Page 209
4.4 Phase Space Analysis......Page 212
Exercises 4.4......Page 221
4.5 Liapunov’s Direct, or Second, Method......Page 222
Exercises 4.5......Page 235
4.6 Stability by Linear Approximation......Page 237
Exercises 4.6......Page 245
4.7 Applications......Page 247
Open problems......Page 261
V. Higher-Order Scalar Difference Equations......Page 262
5.1 Linear Scalar Equations......Page 263
5.2 Sufficient Conditions for Stability......Page 268
Exercises 5.1 and 5.2......Page 272
5.3 Stability via Linearization......Page 273
Exercises 5.3......Page 276
5.4 Global Stability of Nonlinear Equations......Page 278
Exercise 5.4......Page 283
5.5 Applications......Page 285
VI. The Z-Transform Method and Volterra Difference Equations......Page 290
6.1 Definitions and Examples......Page 291
Exercises 6.1......Page 297
6.2 The Inverse Z-Transform and Solutions of Difference Equations......Page 299
Exercises 6.2......Page 307
6.3 Volterra Difference Equations of Convolution Type: The Scalar Case......Page 308
Exercises 6.3......Page 311
6.4 Explicit Criteria for Stability of Volterra Equations......Page 312
Exercises 6.4......Page 315
6.5 Volterra Systems......Page 316
Exercises 6.5......Page 321
6.6 A Variation of Constants Formula......Page 322
Exercises 6.6......Page 324
6.7 The Z-Transform Versus the Laplace Transform......Page 325
7.1 Three-Term Difference Equations......Page 329
Exercises 7.1......Page 334
7.2 Self-Adjoint Second-Order Equations......Page 336
Exercises 7.2......Page 341
7.3 Nonlinear Difference Equations......Page 343
Exercises 7.3......Page 348
8.1 Tools of Approximation......Page 350
Exercises 8.1......Page 353
8.2 Poincaré’s Theorem......Page 355
Exercises 8.2......Page 363
8.3 Asymptotically Diagonal Systems......Page 366
Exercises 8.3......Page 374
8.4 High-Order Difference Equations......Page 375
Exercises 8.4......Page 382
8.5 Second-Order Difference Equations......Page 384
Exercises 8.5......Page 388
8.6 Birkhoff’s Theorem......Page 392
Exercises 8.6......Page 396
8.7 Nonlinear Difference Equations......Page 397
Exercises 8.7......Page 401
8.8 Extensions of the Poincar´e and Perron Theorems......Page 402
Term Projects 8.8......Page 410
9.1 Continued Fractions: Fundamental Recurrence Formula......Page 411
9.2 Convergence of Continued Fractions......Page 414
Exercises 9.1 and 9.2......Page 419
9.3 Continued Fractions and Infinite Series......Page 422
Exercises 9.3......Page 425
9.4 Classical Orthogonal Polynomials......Page 427
9.5 The Fundamental Recurrence Formula for Orthogonal Polynomials......Page 431
Exercises 9.4 and 9.5......Page 434
9.6 Minimal Solutions, Continued Fractions, and Orthogonal Polynomials......Page 435
Exercises 9.6......Page 440
10.1 Introduction......Page 443
10.2 Controllability......Page 446
Exercises 10.1 and 10.2......Page 457
10.3 Observability......Page 460
Exercises 10.3......Page 468
10.4 Stabilization by State Feedback (Design via Pole Placement)......Page 471
Exercises 10.4......Page 479
10.5 Observers......Page 481
Exercises 10.5......Page 487
Appendix A. Stability of Nonhyperbolic Fixed Points of Maps on the Real Line......Page 489
A.1 Local Stability of Nonoscillatory Nonhyperbolic Maps......Page 490
A.2 Local Stability of Oscillatory Nonhyperbolic Maps......Page 492
Appendix B. The Vandermonde Matrix......Page 494
Appendix C Stability of Nondifferentiable Maps......Page 496
D.1 The Stable Manifold Theorem......Page 499
D.2 The Hartman–Grobman–Cushing Theorem......Page 501
Appendix E. The Levin–May Theorem......Page 503
Appendix F. Classical Orthogonal Polynomials......Page 510
Appendix G. Identities and Formulas......Page 511
Exercises 1.3......Page 512
Exercises 1.7......Page 513
Exercises 2.2......Page 514
Exercises 2.4......Page 515
Exercises 3.1......Page 516
Exercises 3.3......Page 517
Exercises 3.4......Page 518
Exercises 4.4......Page 519
Exercises 5.3......Page 520
Exercises 6.3......Page 521
Exercises 7.2......Page 522
Exercises 9.3......Page 523
Exercises 10.3......Page 524
Exercises 10.5......Page 525
Maple Programs......Page 526
References......Page 532
Index......Page 539
