Piece-wise and Max-type difference equations: periodic and eventually periodic solutions

Cover......Page 1
Half Title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Table of Contents......Page 6
Preface......Page 8
Acknowledgments......Page 10
Author......Page 12
1 Introduction......Page 14
 1.1 Recursive Sequences......Page 16
 1.2 Order and Explicit Solution of a Δ.E......Page 18
 1.3 Non-Autonomous Difference Equations......Page 19
 1.4 Periodic Sequences......Page 20
 1.5 Alternating Periodic Cycles......Page 25
 1.6 Specific Patterns of Periodic Cycles......Page 26
 1.7 Eventually Periodic Sequences......Page 27
 1.8 Piece-wise Sequences......Page 31
 1.9 Chapter 1 Exercises......Page 33
2 Linear Difference Equations......Page 38
 2.1 Autonomous Linear Difference Equations......Page 39
  2.2.1 Multiplicative Form of Eq.(2.5)......Page 40
  2.2.2 Additive Form of Eq.(2.5)......Page 44
 2.3 Chapter 2 Exercises......Page 53
 3.1 First-Order Riccati Δ.E......Page 56
 3.2 Second-Order Riccati Δ.E......Page 63
 3.3 Chapter 3 Exercises......Page 73
4 Piece-wise Difference Equations......Page 76
 4.1 The Collatz Conjectures......Page 77
 4.2 The Tent-Map......Page 78
 4.3 The Autonomous Neuron Model......Page 85
  4.3.1 Autonomous Neuron Model when β=1......Page 92
 4.4 Non-Autonomous Neuron Model......Page 95
  4.4.1 Non-Autonomous Neuron Model when β[sub(0)]β[sub(1)] =1......Page 101
 4.5 The Williamson Model......Page 105
 4.7 Chapter 4 Exercises......Page 106
 5.1 The Autonomous Case(Eq.[5.1])......Page 110
  5.1.1 Eventually Periodic with Period-2......Page 113
  5.1.2 Eventually Periodic with Period-4......Page 122
  5.1.3 Eventually Periodic with Period-3......Page 130
  5.1.4 Eventually Constant with K=1......Page 138
 5.2 The Non-Autonomous Case(Eq.[5.2])......Page 143
  5.2.1 Eventually Periodic with Period-2......Page 145
  5.2.2 Eventually Periodic with Period-4......Page 153
  5.2.3 Eventually Periodic with Period-6......Page 157
 5.3 Chapter 5 Exercises......Page 160
 6.2 Alternating Patterns of Sequences......Page 162
 6.4 Convergent Infinite Series......Page 163
  6.5.1 Alternating Periodicity......Page 164
 6.6 Patterns as an Initial Value Problem......Page 165
 6.7 Specific Periodic Patterns......Page 166
Bibliography......Page 168
Index......Page 170