We introduce interdisciplinary research and get students and the audience familiarized with the difference equations; solving them explicitly, determining the long-term behavior of solutions (convergence, boundedness and periodicity). We help to develop intuition in analyzing convergence of solutions in terms of subsequences and analyzing patterns of periodic cycles. Our book helps you learn applications in biology, economics and business, computer science and engineering.
Author(s): Michael A. Radin
Publisher: World Scientific Publishing
Year: 2019
Language: English
Pages: 332
City: Singapore
CONTENTS
Preface
Author Introduction
1. Introduction
1.1. Recursive Sequences
1.2. Order of a Difference Equation and Explicit Solution
1.2.1. Non-autonomous difference equations
1.2.2. Convolution
1.3. Equilibrium Points
1.4. Convergent Sequences (Solutions)
1.5. Periodic Sequences (Solutions)
1.6. Complex Numbers and Periodic Cycles
1.7. Specific Patterns of Periodic Cycles
1.8. Eventually Constant Sequences (Solutions)
1.9. Eventually Periodic Sequences (Solutions)
1.10. Additional Examples of Periodic and Eventually Periodic Solutions
1.11. Divergent (Unbounded) Sequences (Solutions)
1.12. Chapter 1 Exercises
2. First Order Linear Difference Equations
2.1. Homogeneous First Order Linear Difference Equations
2.1.1. Applications of first order linear difference equations in biology
2.1.2. Applications of first order linear difference equations in finance
2.2. Nonhomogeneous First Order Linear Difference Equations
2.3. Non-autonomous First Order Linear Difference Equations
2.3.1. Applications of non-autonomous first order linear difference equations in signal processing
2.4. Periodic Traits of Non-autonomous First Order Linear Difference Equations
2.5. Chapter 2 Exercises
3. First Order Nonlinear Difference Equations
3.1. Local Stability Character of Equilibrium Points
3.1.1. The Beverton–Holt Model
3.1.2. The Logistic Models
3.1.3. The Ricker Model
3.1.4. The Ricker Stock Recruitment Model
3.1.5. The Hassell Model
3.2. The Cobweb Method
3.3. Global Asymptotic Stability (Convergence)
3.4. Periodic Traits of Solutions
3.4.1. Periodic solutions of the Riccati Difference Equation
3.4.2. Periodic solutions of Non-autonomous Riccati Difference Equations
3.4.3. The periodic solutions of the Logistic Difference Equation
3.4.4. Chaos and Chaotic orbits
3.4.5. The Tent-Map
3.4.6. The 3X+1 Conjecture
3.4.7. Autonomous Piecewise difference equation as a Neuron model
3.4.8. Autonomous Piecewise difference equation as a Neuron model when β = 1
3.4.9. The Williamson’s Model
3.4.10. The West Nile Epidemics Model
3.5. Chapter 3 Exercises
4. Second Order Linear Difference Equations
4.1. Homogeneous Second Order Linear Difference Equations
4.1.1. The Fibonacci Sequence
4.1.2. The Riccati Difference Equation as a second order linear difference equation
4.1.3. The Gambler’s Ruin Problem
4.2. Asymptotic Behavior of Second Order Linear Difference Equations
4.3. Nonhomogeneous Second Order Linear Difference Equations with a Constant Coefficient
4.3.1. The National Income
4.4. Nonhomogeneous Second Order Linear Difference Equations with a Variable Geometric Coefficient
4.5. Nonhomogeneous Second Order Linear Difference with a Variable Coefficient nk
4.5.1. Applications of non-autonomous second order linear difference equations in signal processing
4.6. Linear Independence of Solutions
4.7. Periodic Solutions of Second Order Homogeneous Linear Difference Equations
4.8. Periodic Traits of Non-autonomous Second Order Linear Difference Equations
4.9. Third and Higher Order Linear Difference Equations
4.10. Chapter 4 Exercises
5. Second Order Nonlinear Difference Equations
5.1. Local Stability Character of Equilibrium Points
5.1.1. Pielou’s Δ.E. (Model)
5.1.2. Delayed Ricker Model
5.1.3. Harvard School of Public Health population model
5.1.4. Perennial Grass Model
5.2. Global Asymptotic Stability (Convergence)
5.3. Patterns of Periodic Solutions of Second Order Rational Difference Equations
5.4. Periodic Patterns of Second Order Non-autonomous Rational Difference Equations
5.5. Periodic and Eventually Periodic Solutions of Max-Type Difference Equations
5.6. Chapter 5 Exercises
6. Advanced Characteristics and New Research Questions
6.1. Higher Order Linear Difference Equations
6.2. Periodic Traits of Third and Higher Order Linear Difference Equations
6.3. Applications of Higher Order Linear Difference Equations in Signal Processing
6.4. Systems of Linear Difference Equations
6.5. Periodic Traits of Systems of Linear Difference Equations
6.6. Applications of Systems of Linear Difference Equations in Signal Processing
6.7. Systems of Nonlinear Difference Equations
6.7.1. The Host Parasitoid Model
6.7.2. May’s Host Parasitoid Model
6.7.3. System of Beverton−Holt Equations
6.7.4. System of Ricker Equations
6.7.5. Predator-Prey Model
6.7.6. Applications of systems of difference equations in synchronization
6.8. Advanced Periodic Characteristics of Higher Order Nonlinear Difference Equations
6.9. Third and Higher Order Rational Difference Equations
6.10. Third and Higher Order Non-autonomous Rational Difference Equations
6.11. More on Max-Type Difference Equations
6.12. Non-autonomous Piecewise Difference Equations and Systems of Piecewise Difference Equations
6.12.1. Applications of systems of piecewise difference equations in neural networking
6.13. Additional Examples of Periodicity Graphs
6.14. Chapter 6 Exercises
7. Answers to Selected Odd-Numbered Problems
7.1. Answers to Chapter 1 Exercises
7.2. Answers to Chapter 2 Exercises
7.3. Answers to Chapter 3 Exercises
7.4. Answers to Chapter 4 Exercises
7.5. Answers to Chapter 5 Exercises
7.6. Answers to Chapter 6 Exercises
Appendices
A.1. Patterns of Sequences
A.2. Alternating Patterns of Sequences
A.3. Finite Series
A.4. Convergent Infinite Series
A.5. Periodic Sequences and Modulo Arithmetic
A.6. Alternating Periodic Sequences and ModuloArithmetic
Bibliography
Index
