The Cournot duopoly Kopel Model Pomeau-Manneville type-I intermittency Given a critical transition value less than , attractor is a periodic orbit. (Laminar state) 2. slightly larger than , laminar state is intermittently interrupted by a finite duration burst. (Burst state) 3. , the mean time between burst state is 4. The mean time is: (Power law scaling) Laminar states are interrupted by burst states randomly The duration of laminar states between neighboring burst is uncertain.

The Cournot duopoly Kopel Model Pomeau-Manneville type-I intermittency: Bifurcation diagram Periodic Chaotic Fig. 8 bifurcation diagram of x

The Cournot duopoly Kopel Model Pomeau-Manneville type-I intermittency: Time series We will observe the dynamics on the edge of the transition from chaos to period: (a) (b) Fig. 9 Time series of type-I intermittency in duopoly Kopel model (1). (a) pre-bifurcation time series of with , and after the transient chaotic behavior the system finally evolves into periodic oscillations; (b) post-bifurcation intermittent time series of with , the laminar state and the burst state appear in turn.

The Cournot duopoly Kopel Model Pomeau-Manneville type-I intermittency The duration time of laminar state between two bursts is at totally random and unpredictable. The statistical method: obtain the relationship between the average duration lifetime of laminar state and the correlative parameter over a long time series The dependence of the average time lag between two burst on the parameter is that obeys the power-law scaling. For PM type-I intermittency, .

The Cournot duopoly Kopel Model Pomeau-Manneville type-I intermittency: Power law scaling Very close to 0.5 Fig. 10 Average duration time of lamina states as a function of the distance from the critical Point in log-log plot, .

The Cournot duopoly Kopel Model Crisis-induced intermittency Caused by interior crisis. The crisis-induced intermittency is characterized by time series containing one weak chaotic laminar state that is randomly interrupted by another strong chaotic burst state. In Kopel model, we observe the attractor-widening crisis. Given a critical transition value less than , attractor is chaotic. slightly larger than , the orbit bursts out of the old region and bounced around in the new region of another chaotic attractor. 3. The mean time between burst state is (Power law scaling)

The Cournot duopoly Kopel Model Crisis-induced intermittency: Bifurcation diagram Chaotic attractor suddenly widens Fig. 11 bifurcation diagram of x

The Cournot duopoly Kopel Model Crisis-induced intermittency: Time series and attractors (a) (b) (c) (d) Fig. 12 Time series and phase portraits for Kopel model near the crisis point. (a) and (c) The time series and chaotic attractor when before crisis; (b) and (d) The time series and chaotic attractor when after crisis.

The Cournot duopoly Kopel Model Crisis-induced intermittency: Power law scaling Fig .13 Average duration time between bursts as a function of the distance from the crisis point in log-log plot, .

The Cournot duopoly Kopel Model Controlling Chaos Over the past decade, there has been a great deal of interest in controlling chaos in economic system because of the complexity of economics. Unstable fluctuations have always been regarded as unfavorable phenomena in traditional economics Many methods have been applied to control some economic models: The OGY chaos control method The delayed feedback control method The adaptive feed back control method …… In this work, a very simple state feedback controller is designed to control chaos in Kopel model.

The Cournot duopoly Kopel Model Controlling Chaos By adding a state feedback controller, the dynamic process (1) turns to be (4) where is the control law and is the feedback gain. From the bifurcation diagram, we can see that the Kopel model evolves to chaos when the parameter is bigger than 3. In the following, we just consider the case for , where the Nash-equilibrium is unstable. Our objective: Control the chaotic dynamics to the equilibrium , where the two duopolists share the market equally.

The Cournot duopoly Kopel Model Controlling Chaos The Jacobian matrix of the controlled Kopel system (4) at is (5) and its characteristic equation is (6) The equilibrium is locally asymptotically stable if and only if all solutions of (6) fall inside the unit circle.

The Cournot duopoly Kopel Model Controlling Chaos Theorem 2. Consider the controlled economic system described by (4), the equilibrium is locally asymptotically stable if and only if (7)

The Cournot duopoly Kopel Model Simulation Parameters Initial conditions Control law

The Cournot duopoly Kopel Model Fig. 15 Controlled trajectory of duopolists’ productions and the feedback controller , with parameters .

The Cournot duopoly Kopel Model Conclusions: This simple Cournot duopoly Kopel model has very rich nonlinear dynamics: economic cycle, period-doubling to chaos, PM type-I intermittency, crisis-induced intermittency Given a rigorous computer-assisted proof for the existence of chaos

The Analysis of two Economic Systems The Cournot duopoly Kopel economic Model A Business cycle model