Preface to the Third Edition vPrefa

Preface to the Third Edition v
Preface to the Second Edition ix
Preface to the First Edition xi
List of Symbols xx
1 Dynamics of First-Order Difference Equations 1
1.1 Introduction 1
1.2 Linear First-Order Difference Equations 2
1.2.1 Important Special Cases 4
1.3 Equilibrium Points 9
1.3.1 The Stair Step (Cobweb) Diagrams 13
1.3.2 The Cobweb Theorem of Economics 17
1.4 Numerical Solutions of Differential Equations 20
1.4.1 Euler’s Method 20
1.4.2 A Nonstandard Scheme 24
1.5 Criterion for the Asymptotic Stability of
Equilibrium Points 27
1.6 Periodic Points and Cycles 35
1.7 The Logistic Equation and Bifurcation 43
1.7.1 Equilibrium Points 43
1.7.2 2-Cycles 45
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Contents
1.7.3 22-Cycles 46
1.7.4 The Bifurcation Diagram 47
1.8 Basin of Attraction and Global Stability (Optional) .... 50
2 Linear Difference Equations of Higher Order 57
2.1 Difference Calculus 57
2.1.1 The Power Shift 59
2.1.2 Factorial Polynomials 60
2.1.3 The Antidifference Operator 61
2.2 General Theory of Linear Difference Equations 64
2.3 Linear Homogeneous Equations with Constant
Coefficients 75
2.4 Nonhomogeneous Equations: Methods of Undetermind
Coefficeints 83
2.4.1 The Method of Variation of Constants
(Parameters) 89
2.5 Limiting Behavior of Solutions 91
2.6 Nonlinear Equations Transformable to Linear Equations . 98
2.7 Applications 104
2.7.1 Propagation of Annual Plants 104
2.7.2 Gambler’s Ruin 107
2.7.3 National Income 108
2.7.4 The Transmission of Information 110
3 Systems of Linear Difference Equations 117
3.1 Autonomous (Time-Invariant) Systems 117
3.1.1 The Discrete Analogue of the Putzer Algorithm . . 118
3.1.2 The Development of the Algorithm for An 119
3.2 The Basic Theory 125
3.3 The Jordan Form: Autonomous (Time-Invariant)
Systems Revisited 135
3.3.1 Diagonalizable Matrices 135
3.3.2 The Jordan Form 142
3.3.3 Block-Diagonal Matrices 148
3.4 Linear Periodic Systems 153
3.5 Applications 159
3.5.1 Markov Chains 159
3.5.2 Regular Markov Chains 160
3.5.3 Absorbing Markov Chains 163
3.5.4 A Trade Model 165
3.5.5 The Heat Equation 167
4 Stability Theory 173
4.1 A Norm of a Matrix 174
4.2 Notions of Stability 176

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Contents
4.3 Stability of Linear Systems
4.3.1 Nonautonomous Linear Systems
4.3.2 Autonomous Linear Systems
4.4 Phase Space Analysis
4.5 Liapunov’s Direct, or Second, Method
4.6 Stability by Linear Approximation
4.7 Applications
4.7.1 One Species with Two Age Classes
4.7.2 Host-Parasitoid Systems
4.7.3 A Business Cycle Model
4.7.4 The Nicholson-Bailey Model
4.7.5 The Flour Beetle Case Study
Higher-Order Scalar Difference Equations
5.1 Linear Scalar Equations
5.2 Sufficient Conditions for Stability
5.3 Stability via Linearization
5.4 Global Stability of Nonlinear Equations
5.5 Applications
5.5.1 Flour Beetles
5.5.2 A Mosquito Model
The Z-Transform Method and Volterra Difference Equations
6.1 Definitions and Examples
6.1.1 Properties of the Z-Transform
6.2 The Inverse Z-Transform and Solutions of Difference
Equations
6.2.1 The Power Series Method
6.2.2 The Partial Fractions Method
6.2.3 The Inversion Integral Method
6.3 Volterra Difference Equations of Convolution Type: The
Scalar Case
6.4 Explicit Criteria for Stability of Volterra Equations . . . .
6.5 Volterra Systems
6.6 A Variation of Constants Formula
6.7 The Z-Transform Versus the Laplace Transform
Oscillation Theory
7.1 Three-Term Difference Equations
7.2 Self-Adjoint Second-Order Equations
7.3 Nonlinear Difference Equations
Asymptotic Behavior of Difference Equations
8.1 Tools of Approximation
8.2 Poincare’s Theorem

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8.2.1 Infinite Products and Perron’s Example 344
8.3 Asymptotically Diagonal Systems 351
8.4 High-Order Difference Equations 360
8.5 Second-Order Difference Equations 369
8.5.1 A Generalization of the Poincaré-Perron Theorem . 372
8.6 Birkhoff’s Theorem 377
8.7 Nonlinear Difference Equations 382
8.8 Extensions of the Poincare and Perron Theorems 387
8.8.1 An Extension of Perron’s Second Theorem 387
8.8.2 Poincare’s Theorem Revisited 389
9 Applications to Continued Fractions and Orthogonal Polynomials 397
9.1 Continued Fractions: Fundamental Recurrence Formula . 397
9.2 Convergence of Continued Fractions 400
9.3 Continued Fractions and Infinite Series 408
9.4 Classical Orthogonal Polynomials 413
9.5 The Fundamental Recurrence Formula for Orthogonal
Polynomials 417
9.6 Minimal Solutions, Continued Fractions, and Orthogonal
Polynomials 421
10 Control Theory 429
10.1 Introduction 429
10.1.1 Discrete Equivalents for Continuous Systems . . . 431
10.2 Controllability 432
10.2.1 Controllability Canonical Forms 439
10.3 Observability 446
10.3.1 Observability Canonical Forms 453
10.4 Stabilization by State Feedback (Design via Pole
Placement) 457
10.4.1 Stabilization of Nonlinear Systems by Feedback . . 463
10.5 Observers 467
10.5.1 Eigenvalue Separation Theorem 468
A Stability of Nonhyperbolic Fixed Points of Maps on the Real
Line 477
A.1 Local Stability of Nonoscillatory Nonhyperbolic Maps . . 477
A.2 Local Stability of Oscillatory Nonhyperbolic Maps .... 479
A.2.1 Results with g(x) 479
B The Vandermonde Matrix 481
C Stability of Nondifferentiable Maps 483

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D Stable Manifold and the Hartman-Grobman-Cushing Theorems 487
D.1 The Stable Manifold Theorem 487
D.2 The Hartman-Grobman-Cushing Theorem 489
E The Levin-May Theorem 491
F Classical Orthogonal Polynomials 499
G Identities and Formulas 501
Answers and Hints to Selected Problems 503
Maple Programs 517
References 523
Index 531