This work deals with the existence of stable periodic solutions for the following discrete dynamical systems: \(x(n+1)= \beta x(n)-g(x(n))\), where is \(\beta\) a positive constant and \(g\) is a nonlinear function and \(g\) may be not continuous. The author gives several results about the existence of periodic orbits of arbitrary periods according to different range of \(\beta\). He discussed two cases, the first one when \(g\) is of McCulloch-Pitts and the second case when \(g\) is a sigmoid function. For illustration an example is given an the end of the work.