Periodic Behavior of Solutions of a Certain Piecewise Linear System of Difference Equations

Wirot Tikjha, Somsri Jintanasonti, Yongwimon Lenbury

Abstract

In this paper, we study the behavior of solutions of piecewise linear system of difference equations x_{n+1}=|x_{n}|-y_{n}-2 and  y_{n+1} = x_{n}+|y_n| with initial condition that (x_{0}, y_{0}) is in R^2 - {(x, y): x < 0  and  y< 0 }. After we observe via a computer program and some direct computations, we found that the system has an equilibrium point and periodic solutions. We also show that the solution of the system is eventually periodic with prime period $3$ by finding the pattern of solutions and formulating the statements that involve the natural numbers and then proving by mathematical induction.