Journal of Kirkuk University – Scientific Studies, vol.6, No.2, 2011 184 The Stability Analysis of the Shimizu–Morioka System with Hopf Bifurcation Rizgar H. Salih Department of Mathematics, College of Science - University of Koya. Received: 11/12/2007, Accepted: 2/4/2009 Abstract In this work, we study new system with a rich structure (the Shimizu- Morioka system), which is exhibiting the Lorenz-like dynamics. ) 1 ( ... 0 , x z z y ) z 1 ( x y y x 2                Where the dot denotes . dt d the system obtained a Hopf bifurcation (Supercritical and subcritical) for some values of  . For the analysis we use the center manifold and normal form theorem. A computer algebra system using Maple (version 9) was used to derive all the formulas and verifying the results presented in this work [Char, David]. Introduction The Shimizu-Morioka mode was considered in which complex behavior of trajectories has been discovered [Shimizu] by means of computer simulation. This equations were put forward in [Shimizu] as a model for studying the dynamics of the Lorenz system for large Rayleigh number. A detailed exposition of the plethora of bifurcational phenomena in that system can be found in (Shilinikov 1989, 1991).It was shown in (Sil’nikov 1993, 1991) that there are two types of Lorenz-like attractors in this model. The first is an orientable Lorenz-like attractor and the second is non orientable containing a countable set of saddle periodic orbits with negative multipliers. Basically there are two ways of investigating periodic solutions of more than two coupled ODE. One is to use the fixed-point theorem to establish the existence, but not the stability of periodic solutions in the large. The other method is to investigate the bifurcation of an isolated equilibrium point, as