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In this paper, we explore local stability, attractor, periodicity character, and boundedness solutions of the second-order nonlinear difference equation. Finally, obtained results are verified numerically.

We study some qualitative properties of the solutions of a system of difference equations, which describes an economic model. The study of the local stability of the equilibrium points is carried out. We give some important results of the invariant and the boundedness of the solutions to the considered system. The global convergence of the solutions is presented and investigated.

We obtain in this paper the solutions of the following recursive sequences 𝑥 𝑛 + 1 = 𝑥 𝑛 𝑥 𝑛 − 3 / 𝑥 𝑛 − 2 ( ± 1 ± 𝑥 𝑛 𝑥 𝑛 − 3 ) , 𝑛 = 0 , 1 , … , where the initial conditions are arbitrary real numbers and we study the behaviors of the solutions and we obtained the equilibrium points of the considered equations. Some qualitative behavior of the solutions such as the boundedness, the global stability, and the periodicity character of the solutions in each case have been studied. We presented some numerical examples by giving some numerical values for the initial values and the coefficients of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program Mathematica to confirm the obtained results.

We study some qualitative behavior of solutions of some max-type difference equations with periodic coefficients. Some new results of the periodicity character of solutions of that type of difference equations will be established.

This paper is devoted to investigate the global behavior of the following rational difference equation:yn+1=αyn-t/(β+γ∑i=0kyn-(2i+1)p∏i=0kyn-(2i+1)q), n=0,1,2,…, whereα,β,γ,p,q∈(0,∞)andk,t∈{0,1,2,…}with the initial conditionsx0, x-1,…, x-2k, x-2max {k,t}-1∈ (0,∞). We will find and classify the equilibrium points of the equations under studying and then investigate their local and global stability. Also, we will study the oscillation and the permanence of the considered equations.