About a System of Piecewise Linear Difference Equations with Many Periodic Solutions

We want to draw attention to the system of first order piecewise linear difference equations of two equations \(x_{n+1} = |x_n| - y_n - b\) and \(y_{n+1} = x_n - |y_n| - d\) , \(n=0,1,2,..., (x_0, y_0)\in \textbf{R}^2\) , where the parameters b and d are any positive real numbers. We will show that this system has interesting behavior compared to other similar systems. We show that there exists an unstable equilibrium \((d,-b)\) . It has been shown that there are no solutions with period 2 and 3, but depending on the values of parameters b and d there are solutions with periods 5, 6, 7, 11, 12, 13, 16, 17, 18, 19, 20, 24, 25, 27, 30, 36. We have a hypothesis that all solutions are eventually periodic solutions.