On the hopf bifurcation of the system of dx/dt=y+z, dy/dt=ax+z, dz/dt=bz+x(y-c)

ABSTRACTOne of the useful subjects of bifurcation theory named Hopf bifurcation which had itsorigins in the works of Poincaré around 1892, is expanded to apply a wide spectrum ofdynamical systems and the systems of ordinary differential equations in three dimensionalspace. This thesis consists of three chapters in which two and three dimensional structure ofHopf bifurcation theory was yielded.In the first chapter, basic theorems concerning about Bifurcation analysis theory anddefinitions as bifurcation, stable stationary solution, unstable stationary solution, dissipativesystem, Hopf bifurcation are given in the sense of stability analysis.In the second chapter, the Hopf bifurcation is already known in literature in a twodimensional system of differential equations is investigated. In here the behaviors of twodimensional system starting near the fixed point are pointed out and also the figures are tried toshow related with the subject.In the third chapter, the three dimensional system that is newly described according todynamical systems designing properties, is solved by means of Hopf bifurcation while thesystem parameter or parameters are being changed (i.e., the values of the parameter increaseor decrease in their domains). Here the original system is defined having 3 degree of freedom.Firstly, one of the system parameter in its domain is changed along its axes in order to occurHopf bifurcation. At the and of this, it has shown some figures of the system near the criticalpoint which shows the behavior of the system.It is continued to investigate the system by changing two system parameters and theother remained fixed. At last the necessary transformations are obtained in order to reduce theviithree dimensional system to a two dimensional system. The transforms and their coefficientsare found.At the end of the theoretical work, the operations and their results are calculated withsmall errors by means of the computer program is functional as well as it has a high accuracyand causes to earn much more time. Furthermore giving the original algorithm is clarified to theresearcher who are interested in investigating Hopf bifurcation either in a two dimensional orthree dimensional systems.viii