An Introduction to Difference Equations

Description

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! 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