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 BIBLIOGRAPHYa=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 BIBLIOGRAPHYp;lt;BR>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 BIBLIOGRAPHYIX: Global Attractivity for Higher Order Equations BIBLIOGRAPHY.