This thesis is based on a paper by Hiroki Murakami [2014]. In this thesis, by utilizing the Poincaré–Bendixson theory and the Hopf bifurcation theory, we analyze both rigid-price and flexible-price nonlinear dise- quilibrium Keynesian macroeconomic systems, prove the existence of a persistent business cycle and derive the conditions for global asymptotic stability of the equilibrium. Consequently, we find that a Hopf bifurcation occurs for a lower value of the quantity adjustment parameter in the flexible-price system than in the rigid-price one and that inflation expectation effects may easily destabilize the economic system. Furthermore, we reveal that global asymptotic stability of the flexible-price system is unlikely to be achieved. This thesis is organized as follows. In Chapter 1 Introduction and some economic schools introduced. In Chapter 2 we will introduce some basic basic economic concepts and definitions including ISLM model, philips curve, economic growth, economic equilibrium, some economic functions for our anal- ysis and mathematical economic from keynes point of view. In Chapter 3 we will introduce some basic mathematical definitions and theorems necessary for our analysis in the following chapters including, Hopf bifurcation theorem, Routh-Horwitz criterion, Brock-Scheinkman theorem, Liu’s criterion and Asada-Yoshida’s theorem. In Chapter 4, we will consider a rigid-price disequilibrium system. In this system, aggregate income will be assumed to move in response to the current economic conditions, and capital stock will in- crease or decrease as a result of ex post net investment (capital formation), while the money market will be assumed to be in equilibrium. This system can be viewed as a generalized form of business cycle models of Keynes–Kaldor type. In this chapter, the existence of a stable limit cycle (a persistent business cycle) and the global asymptotic stability of the equilibrium will be rigorously investigated by the Poincaré–Bendixson theorem. In Section 4.2.1, we will explore the monetary side of the economy. In particular, we will derive, from the money market equilibrium condition, the relationship between the rate of interest and real variables such as aggregate income and capital stock. By representing the rate of interest as a function of these real variables, we will see that changes in the rate of interest are implied in those in the real variables. In Chapter 5, we will formalize a flexible-price disequilibrium system by taking account of changes in the price level, and the existence of a periodic orbit will be proved by the Hopf bifurcation theorem. We will verify the existence of a persistent business cycle (a periodic orbit) even under the relatively weak price level effects. By so doing, we will intend to take issue with the orthodox argument that flexibility of prices contributes to the stability of the economic system and verify that the price level effects and the inflation–deflation expectation effects are destabilizing factors. In Section 5.3.1, we will briefly examine the global asymptotic stability of the flexible-price system. By deriving sufficient conditions for global asymptotic stability, we shall see how difficult it is for the equilibrium to attain global asymptotic stability in a flexible-price situation. In Chapter 6, we will conclude our analysis. Moreover, the importance of the Keynesian discretionary policies shall be briefly discussed on the basis of the results in our analysis.