Studying the Stability of a Non-linear Autoregressive Model (Polynomial with Hyperbolic Cosine Function)

In this paper we study the statistical properties of one of a non-linear autoregressive model with hyperbolic triangle function(polynomial with hyperbolic cosinefunction)by using the local linearization approximation method to find the stability of the model (singular point and its stability conditions and the stability of limit cycle).Where we started by the model of lower order (first and second and third order) and generalized the idea, and we tried to apply these theory results by using some of examples to explain one of important truth that says (if the model has unstable singular point, then it, maybe, has a stable limit cycle).