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Dynamics of Second Order Rational Difference Equations

With Open Problems and Conjectures

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## Book Description

This self-contained monograph provides systematic, instructive analysis of second-order rational difference equations. After classifying the various types of these equations and introducing some preliminary results, the authors systematically investigate each equation for semicycles, invariant intervals, boundedness, periodicity, and global stability. Of paramount importance in their own right, the results presented also offer prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of order greater than one. The techniques and results in this monograph are also extremely useful in analyzing the equations in the mathematical models of various biological systems and other applications.

Each chapter contains a section of open problems and conjectures that will stimulate further research interest in working towards a complete understanding of the dynamics of the equation and its functional generalizations-many of them ideal for research projects or Ph.D. theses. Clear, simple, and direct exposition combined with thoughtful uniformity in the presentation make Dynamics of Second Order Rational Difference Equations valuable as an advanced undergraduate or a graduate-level text, a reference for researchers, and as a supplement to every textbook on difference equations at all levels of instruction.

## Table of Contents

INTRODUCTION AND CLASSIFICATION OF EQUATION TYPES

PRELIMINARY RESULTS

Definitions of Stability and Linearized Stability Analysis

The Stable Manifold Theorem in the Plane

Global Asymptotic Stability of the Zero Equilibrium

Global Attractivity of the Positive Equilibrium

Limiting Solutions

The Riccati Equation

Semicycle Analysis

LOCAL STABILITY, SEMICYCLES, PERIODICITY, AND INVARIANT INTERVALS

Equilibrium Points

Stability of the Zero Equilibrium

Local Stability of the Positive Equilibrium

When is Every Solution Periodic with the same Period?

Existence of Prime Period Two Solutions

Local Asymptotic Stability of a Two Cycle

Convergence to Period Two Solutions when C=0

Invariant Intervals

Open Problems and Conjectures

(1,1)-TYPE EQUATIONS

Introduction

The Case a=g=A=B=0: xn+1= b xn/C xn-1

The Case a=b=A=C=0: xn+1=g xn-1/B xn

Open Problems and Conjectures

(1,2)-TYPE EQUATIONS

Introduction

The Case b=g=C=0: xn+1= a /(A+ B xn)

The Case b=g=A=0: xn+1= a /(B xn+ C xn-1)

The Case a=g=B=0: xn+1= b xn/(A + C xn-1)

The Case a=g=A=0: xn+1= b xn/(B xn+ C xn-1)

The Case a=b=C=0: xn+1= g xn-1/(A+ B xn)

The Case a=b=A=0: xn+1= g xn-1/(B xn+ C xn-1)

Open Problems and Conjectures

(2,1)-TYPE EQUATIONS

Introduction

The Case g=A=B=0: xn+1=(a + b xn)/(C xn-1)

The Case g=A=C=0: xn+1=(a + b xn)/B xn

Open Problems and Conjectures

(2,2)-TYPE EQUATIONS(2,2)- Type Equations

Introduction

The Case g=C=0: xn+1=(a + b xn)/(A+ B xn)

The Case g=B=0: xn+1=(a + b xn)/(A + C xn-1)

The Case g=A=0: xn+1=(a + b xn)/(B xn+ C xn-1)

The Case b=C=0: xn+1=(a + g xn-1)/(A+ B xn)

The Case b=A=0: xn+1=(a + g xn-1)/(B xn+ C xn-1)

The Case a=C=0: xn+1=(b xn+ g xn-1)/(A+ B xn)

The Case a=B=0: xn+1=(b xn+ g xn-1)/(A + C xn-1)

The Case a=A=0: xn+1=(b xn+ g xn-1)/(B xn+ C xn-1)

Open Problems and Conjectures

(2,3)-TYPE EQUATIONS

Introduction

The Case g=0: xn+1=(a + b xn)/(A+ B xn+ C xn-1)

The Case b=0: xn+1=(a + g xn-1)/(A+ B xn+ C xn-1)

The Case a=0: xn+1=(b xn+ g xn-1)/(A+ B xn+ C xn-1)

Open Problems and Conjectures

(3,2)-TYPE EQUATIONS

Introduction

The Case C=0: xn+1=(a + b xn+ g xn-1)/(A+ B xn )

The Case B=0: xn+1=(a + b xn+ g xn-1)/(A+ C xn-1)

The Case A=0: xn+1=(a + b xn+ g xn-1)/(B xn+ C xn-1)

Open Problems and Conjectures

THE (3,3)-TYPE EQUATION The (3,3)- Type Equation: xn+1=(a + b xn+ g xn-1 )/(A+ B xn+ C xn-1)

Linearized Stability Analysis

Invariant Intervals

Convergence Results

Open Problems and Conjectures

APPENDIX: Global Attractivity for Higher Order Equations

BIBLIOGRAPHY

## Author(s)

### Biography

Kulenovic\, Mustafa R.S.; Ladas\, G.