Regarding the dynamics of a third order nonlinear difference equation

Abstract

In this work, we study qualitative properties of the solutions of the following class of nonlinear third order difference equations xn+1 = pxn-1 + f (xn-1 - xn-2). First we study the relation of attractivity and stability of equilibrium point of this equation and some related equations. Further more we prove the existence of Neimark-Sacker and period doubling (flip) bifurcation for this system by analysing the characteristic equation, and investigate the direction of this bifurcations by using normal form theory. Finally some numerical simulations are carried out to support the analytical results. © 2012 Memarbashi and Ghasemabadi.