Periodic orbits on discrete dynamical systems

In this paper, we discuss the discrete dynamical system xn+1 = βxn − g(xn), N = 0, 1, …, arising as a discrete-time network of single neuron, where β is the internal decay rate, g, is a signal function. First, we consider the case where g is of McCulloch-Pitts nonlinearity. Periodic orbits are discussed according to different range of β. Moreover, we can construct periodic orbits. Then, we consider the case where g is a sigmoid function. Sufficient conditions are obtained for (*) has periodic orbits of arbitrary periods and an example is also given to illustrate the theorem.