Piece-wise and Max-Type Difference Equations: Periodic and Eventually Periodic Solutions

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.

Table of Contents:
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 β = 1 . . . . . . . . 79 Non-Autonomous Neuron Model . . . . . . . . . . . . . . . . 82 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 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