Piece-wise and Max-Type Difference Equations

Contents

Preface vii

Acknowledgments ix

Author xi

1. Introduction 1

1.1 Recursive Sequences . . . . . . . . . . . . . . . . . . . . . . . 3

1. Order and Explicit Solution of a â.E. . . . . . . . . . . . . . 5
2. Non-Autonomous Difference Equations . . . . . . . . . . . . 6

1.4 Periodic Sequences . . . . . . . . . . . . . . . . . . . . . . . . 7

1. Alternating Periodic Cycles . . . . . . . . . . . . . . . . . . . 12
2. Specific Patterns of Periodic Cycles . . . . . . . . . . . . . . 13
3. Eventually Periodic Sequences . . . . . . . . . . . . . . . . . 14

1.8 Piece-wise Sequences . . . . . . . . . . . . . . . . . . . . . . 18

1.9 Chapter 1 Exercises . . . . . . . . . . . . . . . . . . . . . . . 20

1. Linear Difference Equations 25

1. Autonomous Linear Difference Equations . . . . . . . . . . . 26

2.2 Non-Autonomous Linear â.E.âs . . . . . . . . . . . . . . . . 27

2.2.1 Multiplicative Form of Eq. (2.5) . . . . . . . . . . . . . 27

2.2.2 Additive Form of Eq. (2.5) . . . . . . . . . . . . . . . . 31

2.3 Chapter 2 Exercises . . . . . . . . . . . . . . . . . . . . . . . 40

1. Riccati Difference Equations 43

3.1 First-Order Riccati â.E. . . . . . . . . . . . . . . . . . . . . 43

3.2 Second-Order Riccati â.E. . . . . . . . . . . . . . . . . . . . 50

3.3 Chapter 3 Exercises . . . . . . . . . . . . . . . . . . . . . . . 60

2. Piece-wise Difference Equations 63

4.1 The Collatz Conjectures . . . . . . . . . . . . . . . . . . . . 64

4.2 The Tent-Map . . . . . . . . . . . . . . . . . . . . . . . . . . 65

1. The Autonomous Neuron Model . . . . . . . . . . . . . . . . 72

1. Autonomous Neuron Model when β = 1 . . . . . . . . 79
2. Non-Autonomous Neuron Model . . . . . . . . . . . . . . . . 82

1. Non-Autonomous Neuron Model when β0β1 = 1 . . . 88

4.5 The Williamson Model . . . . . . . . . . . . . . . . . . . . . 92

4.6 The West Nile Epidemics Model . . . . . . . . . . . . . . . . 93

4.7 Chapter 4 Exercises . . . . . . . . . . . . . . . . . . . . . . . 93

5 Max-Type Difference Equations 97

5.1 The Autonomous Case (Eq. [5.1]) . . . . . . . . . . . . . . . 97

1. Eventually Periodic with Period-2 . . . . . . . . . . . 100
2. Eventually Periodic with Period-4 . . . . . . . . . . . 109
3. Eventually Periodic with Period-3 . . . . . . . . . . . 117
4. Eventually Constant with K = 1 . . . . . . . . . . . . 125

5.2 The Non-Autonomous Case (Eq. [5.2]) . . . . . . . . . . . . . 130

1. Eventually Periodic with Period-2 . . . . . . . . . . . 132
2. Eventually Periodic with Period-4 . . . . . . . . . . . 140
3. Eventually Periodic with Period-6 . . . . . . . . . . . 144

5.3 Chapter 5 Exercises . . . . . . . . . . . . . . . . . . . . . . . 147

6 Appendices 149

6.1 Patterns of Sequences . . . . . . . . . . . . . . . . . . . . . . 149

6.2 Alternating Patterns of Sequences . . . . . . . . . . . . . . . 149

6.3 Finite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

1. Convergent Infinite Series . . . . . . . . . . . . . . . . . . . . 150
2. Periodicity and Modulo Arithmetic . . . . . . . . . . . . . . 151

1. Alternating Periodicity . . . . . . . . . . . . . . . . . . 151
3. Patterns as an Initial Value Problem . . . . . . . . . . . . . 152
4. Specific Periodic Patterns . . . . . . . . . . . . . . . . . . . . 153

Bibliography 155

Index 157