Piece-wise and Max-Type Difference Equations

Contents

Preface vii

Acknowledgments ix

Author xi






Introduction 1
1.1 Recursive Sequences . . . . . . . . . . . . . . . . . . . . . . . 3






Order and Explicit Solution of a ?.E. . . . . . . . . . . . . . 5



Non-Autonomous Difference Equations . . . . . . . . . . . . 6



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








Alternating Periodic Cycles . . . . . . . . . . . . . . . . . . . 12



Specific Patterns of Periodic Cycles . . . . . . . . . . . . . . 13



Eventually Periodic Sequences . . . . . . . . . . . . . . . . . 14



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

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






Linear Difference Equations 25





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



2.2 Non-Autonomous Linear ?.E.'s . . . . . . . . . . . . . . . . 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






Riccati Difference Equations 43
3.1 First-Order Riccati ?.E. . . . . . . . . . . . . . . . . . . . . 43

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

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




Piece-wise Difference Equations 63
4.1 The Collatz Conjectures . . . . . . . . . . . . . . . . . . . . 64

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






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





Autonomous Neuron Model when ? = 1 . . . . . . . . 79




Non-Autonomous Neuron Model . . . . . . . . . . . . . . . . 82





Non-Autonomous Neuron Model when ?0?1 = 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










Eventually Periodic with Period-2 . . . . . . . . . . . 100



Eventually Periodic with Period-4 . . . . . . . . . . . 109



Eventually Periodic with Period-3 . . . . . . . . . . . 117



Eventually Constant with K = 1 . . . . . . . . . . . . 125





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










Eventually Periodic with Period-2 . . . . . . . . . . . 132



Eventually Periodic with Period-4 . . . . . . . . . . . 140



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








Convergent Infinite Series . . . . . . . . . . . . . . . . . . . . 150



Periodicity and Modulo Arithmetic . . . . . . . . . . . . . . 151





Alternating Periodicity . . . . . . . . . . . . . . . . . . 151




Patterns as an Initial Value Problem . . . . . . . . . . . . . 152



Specific Periodic Patterns . . . . . . . . . . . . . . . . . . . . 153



Bibliography 155

Index 157