ABSTRACT
Piece-wise and Max-Type Difference Equations: Periodic and Eventually Periodic Solutions is intended for lower-level undergraduate students studying discrete mathematics.
The book focuses on sequences as recursive relations and then transitions to periodic recursive patterns and eventually periodic recursive patterns. In addition to this, it will also focus on determining the patterns of periodic and eventually periodic solutions inductively. The aim of the author, throughout this book, is to get students to understand the significance of pattern recognition as a mathematical tool.
Key Features
- Can provide possible topics for undergraduate research and for bachelor’s thesis
- Provides supplementary practice problems and some open-ended research problems at the end of each chapter
- Focusses on determining the patterns of periodic and eventually periodic solutions inductively
- Enhances students’ algebra skills before moving forward to upper level courses
- Familiarize students with the topics before they start undergraduate research by providing applications.
Contents
Preface vii
Acknowledgments ix
Author xi
- Introduction 1
- Order and Explicit Solution of a ∆.E. . . . . . . . . . . . . . 5
- Non-Autonomous Difference Equations . . . . . . . . . . . . 6
1.1 Recursive Sequences . . . . . . . . . . . . . . . . . . . . . . . 3
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
- Piece-wise Difference Equations 63
- The Autonomous Neuron Model . . . . . . . . . . . . . . . . 72
- Autonomous Neuron Model when β = 1 . . . . . . . . 79
- Non-Autonomous Neuron Model . . . . . . . . . . . . . . . . 82
- Non-Autonomous Neuron Model when β0β1 = 1 . . . 88
3.1 First-Order Riccati ∆.E. . . . . . . . . . . . . . . . . . . . . 43
3.2 Second-Order Riccati ∆.E. . . . . . . . . . . . . . . . . . . . 50
3.3 Chapter 3 Exercises . . . . . . . . . . . . . . . . . . . . . . . 60
4.1 The Collatz Conjectures . . . . . . . . . . . . . . . . . . . . 64
4.2 The Tent-Map . . . . . . . . . . . . . . . . . . . . . . . . . . 65
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
