摘要：In this paper, we study qualitative properties of solutions of the following nonlinear third order difference equation: x n + 1 = a x n + b x n − 1 + f ( x n − x n − 1 ) + g ( x n − 1 − x n − 2 ) . $$x_{n+1}= ax_{n} + bx_{n-1}+f(x_{n}-x_{n-1})+g(x_{n-1}-x_{n-2}). $$ In economics, this equation was known as Metzler equation. We study the stability of the solutions and existence of bifurcations. We prove that there exist two bifurcations for this system by analyzing the characteristic equation: (1) Neimark-Sacker bifurcation, (2) period doubling (flip) bifurcation. Next we investigate the direction of these bifurcations by using normal form theory.
