Dynamics of Second Order Rational Difference Equations

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