1. The Cournot duopoly Kopel Model
Pomeau-Manneville type-I intermittency
Given a critical transition value
less than , attractor is a periodic orbit. (Laminar state)
slightly larger than , laminar state is intermittently interrupted by a finite duration burst. (Burst state)
, 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.

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

3. 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.

4. 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,

5. 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,

6. 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)

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

8. 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.

9. 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,

10. 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.

11. 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.

12. 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.

13. 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)

14. The Cournot duopoly Kopel Model
Simulation
Parameters
Initial conditions
Control law

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

16. 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

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