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We present a complete local dynamics and investigate the global dynamics of the following second-order difference equation: xn+1=Axn2+Exn-1+F/axn2+exn-1+f,  n=0,1,2,…, where the parameters A,E,F,a,e, and f are nonnegative numbers with condition A+E+F>0, a+e+f>0, and the initial conditions x-1, x0 are arbitrary nonnegative numbers such that axn2+exn-1+f>0,  n=0,1,2,…. [ABSTRACT FROM AUTHOR]

This paper is concerned with local stability, oscillatory character of positive solutions to the system of the two nonlinear difference equations xn+1=A+xn-1p/ynp and yn+1=A+yn-1p/xnp, n=0,1,…, where A∈(0,∞), p∈[1,∞), xi∈(0,∞), and yi∈(0,∞), i=-1,0. [ABSTRACT FROM AUTHOR]

We focus on a system of a rational m-order difference equation xn+1 = xn-m+1/(A + yn yn-1 ... yn-m+1, yn+1 = (yn-m+1)/(B + xn xn-1... xn-m+1), n = 0,1, ..., where A, B, x0, x-1, ..., x-m+1, y0, y-1, ..., y-m+1 ∈ (0, ∞). We investigate the dynamical behavior of positive solution for the system. [ABSTRACT FROM AUTHOR]