An introduction to difference equations

✍ Scribed by Elaydi S.

• Publisher: Springer
• Year: 2005
• Tongue: English
• Leaves: 563
• Series: Undergraduate Texts in Mathematics
• Edition: 3ed
• Category: Library

✦ Synopsis

This book integrates both classical and modern treatments of difference equations. It contains the most updated and comprehensive material, yet the presentation is simple enough for the book to be used by advanced undergraduate and beginning graduate students. This third edition includes more proofs, more graphs, and more applications. The author has also updated the contents by adding a new chapter on Higher Order Scalar Difference Equations, and also recent results on local and global stability of one-dimensional maps, a new section on the various notions of asymptoticity of solutions, a detailed proof of Levin-May Theorem, and the latest results on the LPA flour-beetle model.

✦ Table of Contents

Preface to the Third Edition......Page 6
Preface to the Second Edition......Page 10
Preface to the First Edition......Page 12
Contents......Page 16
List of Symbols......Page 22
1.1 Introduction......Page 24
1.2 Linear First-Order Di.erence Equations......Page 25
1.3 Equilibrium Points......Page 32
1.4 Numerical Solutions of Di.erential Equations......Page 43
1.5 Criterion for the Asymptotic Stability of Equilibrium Points......Page 50
1.6 Periodic Points and Cycles......Page 58
1.7 The Logistic Equation and Bifurcation......Page 66
1.8 Basin of Attraction and Global Stability (Optional)......Page 73
2.1 Di.erence Calculus......Page 80
2.2 General Theory of Linear Di.erence Equations......Page 87
2.3 Linear Homogeneous Equations with Constant Coe.cients......Page 98
2.4 Linear Nonhomogeneous Equations: Method of Undetermined Coe.cients......Page 106
2.5 Limiting Behavior of Solutions......Page 114
2.6 Nonlinear Equations Transformable to Linear Equations......Page 121
2.7 Applications......Page 127
3.1 Autonomous (Time-Invariant) Systems......Page 140
3.2 The Basic Theory......Page 148
3.3 The Jordan Form: Autonomous (Time-Invariant) Systems Revisited......Page 158
3.4 Linear Periodic Systems......Page 176
3.5 Applications......Page 182
4 Stability Theory......Page 196
4.1 A Norm of a Matrix......Page 197
4.2 Notions of Stability......Page 199
4.3 Stability of Linear Systems......Page 207
4.4 Phase Space Analysis......Page 217
4.5 Liapunov’s Direct, or Second, Method......Page 227
4.6 Stability by Linear Approximation......Page 242
4.7 Applications......Page 252
5 Higher-Order Scalar Di.erence Equations......Page 268
5.1 Linear Scalar Equations......Page 269
5.2 Su.cient Conditions for Stability......Page 274
5.3 Stability via Linearization......Page 279
5.4 Global Stability of Nonlinear Equations......Page 284
5.5 Applications......Page 291
6 The Z-Transform Method and Volterra Di.erence Equations......Page 296
6.1 De.nitions and Examples......Page 297
6.2 The Inverse Z-Transform and Solutions of Difference Equations......Page 305
6.3 Volterra Di.erence Equations of Convolution Type: The Scalar Case......Page 314
6.4 Explicit Criteria for Stability of Volterra Equations......Page 318
6.5 Volterra Systems......Page 322
6.6 A Variation of Constants Formula......Page 328
6.7 The Z-Transform Versus the Laplace Transform ^5......Page 331
7.1 Three-Term Di.erence Equations......Page 336
7.2 Self-Adjoint Second-Order Equations......Page 343
7.3 Nonlinear Di.erence Equations......Page 350
8.1 Tools of Approximation......Page 358
8.2 Poincar´e’s Theorem......Page 363
8.3 Asymptotically Diagonal Systems......Page 374
8.4 High-Order Di.erence Equations......Page 383
8.5 Second-Order Di.erence Equations......Page 392
8.6 Birkho.’s Theorem......Page 400
8.7 Nonlinear Di.erence Equations......Page 405
8.8 Extensions of the Poincar´e and Perron Theorems......Page 410
9.1 Continued Fractions: Fundamental Recurrence Formula......Page 420
9.2 Convergence of Continued Fractions......Page 423
9.3 Continued Fractions and In.nite Series......Page 431
9.4 Classical Orthogonal Polynomials......Page 436
9.5 The Fundamental Recurrence Formula for Orthogonal Polynomials......Page 440
9.6 Minimal Solutions, Continued Fractions, and Orthogonal Polynomials......Page 444
10.1 Introduction......Page 452
10.2 Controllability......Page 455
10.3 Observability......Page 469
10.4 Stabilization by State Feedback (Design via Pole Placement)......Page 480
10.5 Observers......Page 490
A.1 Local Stability of Nonoscillatory Nonhyperbolic Maps......Page 500
A.2 Local Stability of Oscillatory Nonhyperbolic Maps......Page 502
Appendix B: The Vandermonde Matrix......Page 504
Appendix C: Stability of Nondi.erentiable Maps......Page 506
D.1 The Stable Manifold Theorem......Page 510
D.2 The Hartman–Grobman–Cushing Theorem......Page 512
Appendix E: The Levin–May Theorem......Page 514
Appendix F: Classical Orthogonal Polynomials......Page 522
Appendix G: Identities and Formulas......Page 524
Answers and Hints to Selected Problems......Page 526
Maple Programs......Page 540
Cobweb Program......Page 541
Bifurcation Diagram Program......Page 542
Phase Space Diagram with Four Initial Points......Page 544
References......Page 546
Index......Page 554