### Keywords : limit cycle

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

Abdulghafoor Gasim Salim; Anas Salim Youns Abdullah

AL-Rafidain Journal of Computer Sciences and Mathematics, 2014, Volume 11, Issue 1, Pages 81-91
DOI: 10.33899/csmj.2014.163733

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).

##### Studying the Stability of Some Models Combinatory with Application

AL-Rafidain Journal of Computer Sciences and Mathematics, 2010, Volume 7, Issue 2, Pages 139-157
DOI: 10.33899/csmj.2010.163903

In  this paper we find the statistical properties (Moments) of the harmonic model with additive noise, The stability of the mixed spectra (linear and non-linear models)  for special case (low order) by using the Ozaki linear approximation method is found .The time series of  the mean monthly temperature of Bege City is applied in order to explain the studied method. A mathematical model SARIMA(1,0,0)(2,0,0)s is suggested by the NBIC criterion and other statistical tools (auto-correlation and the residual variance). A one year ahead prediction is made for the studied  time series by using the proposed model .

##### Limit Cycles of Lorenz System with Hopf Bifurcation

Azad I. Amen; Rizgar H. Salih

AL-Rafidain Journal of Computer Sciences and Mathematics, 2008, Volume 5, Issue 1, Pages 81-99
DOI: 10.33899/csmj.2008.163951

We prove that near the bifurcation point unstable limit cycle arises from the Lorenz system. In the analysis, we use the method of local bifurcation theory, especially the center manifold and the normal form theorem. A computer algebra system using Maple to derive all the formulas and verify the results presented in this paper.