By using the KAM(Kolmogorov-Arnold-Moser) theory and time reversal symmetries, we investigate the stability of the equilibrium solutions of the system: x n + 1 = 1 y n , y n + 1 = β x n 1 + y n , n = 0 , 1 , 2 , … , where the parameter β > 0 , and initial conditions x 0 and y 0 are positive numbers. We obtain the Birkhoff normal form for this system and prove the existence of periodic points with arbitrarily large periods in every neighborhood of the unique positive equilibrium. We use invariants to find a Lyapunov function and Morse’s lemma to prove closedness of invariants. We also use the time reversal symmetry method to effectively find some feasible periods and the corresponding periodic orbits.

We present a complete local dynamics and investigate the global dynamics of the following second-order difference equation: , where the parameters , and are nonnegative numbers with condition , , and the initial conditions , are arbitrary nonnegative numbers such that

We present a complete local dynamics and investigate the global dynamics of the following second-order difference equation: , where the parameters , and are nonnegative numbers with condition , , and the initial conditions , are arbitrary nonnegative numbers such that

We investigate the local stability and the global asymptotic stability of the following two dierence equation xn+1 = x nxn 1 +x n 1 Ax 2 +Bxnxn 1 ; x

We find an asymptotic approximations of the stable and unstable manifolds of the saddle equilibrium solution and the periodtwo solutions of the following difference equation xn+1 = p + xn−1/xn, where the parameter p is positive number and the initial conditions x −1 and x0 are positive numbers. These manifolds, which satisfy the standard functional equations of stable and unstable manifolds determine completely global dynamics of this equation.

1Department of Health Informatics, College of Public Health and Health Informatics, King Saud Bin Abdulaziz University for Health Sciences, Riyadh 11481, Saudi Arabia 2Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 17551, Al-Ain, UAE 3Department of Mathematics, National Tsing Hua University, Taiwan 4Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA

We consider the following system of difference equations: where , , , , are positive constants and are initial conditions. This system has interesting dynamics and it can have up to seven equilibrium points as well as a singular point at , which always possesses a basin of attraction. We characterize the basins of attractions of all equilibrium points as well as the singular point at and thus describe the global dynamics of this system. Since the singular point at always possesses a basin of attraction this system exhibits Allee’s effect.

We find the asymptotic approximations of the stable and unstable manifolds of the saddle equilibrium solutions of the following difference equation xn+1 = ax 3 + bx 3 1 + cxn + dxn 1;n = 0; 1;::: where the parameters a;b;c and d are positive numbers and the initial conditions x 1 and x0 are arbitrary numbers. These manifolds determine completely the global dynamics of this equation.

By using the KAM theory, we investigate the stability of the equilibrium solution of a certain difference equation. We also use the symmetries to find effectively the periodic solutions with feasible periods. The specific feature of this difference equation is the fact that we were not able to use the invariant to prove stability or to find feasible periods of the solutions.