Similar to many ordinary differential equation systems, we can see that there exists a bifurcation point
[beta] = [[beta].sup.*] for the system (2).
A1 and A2 points are Hopf bifurcation points
separately characterized by the 1st and 2nd modes, respectively, whereas A3 point is a double-Hopf bifurcation point
characterized by the combination of both modes.
And, if [f.sub.z](*) has complex values on one side of the OEB bifurcation point
, the above condition will be modified as follows:
Bansal, "New technique for computation of closest Hopf bifurcation point
using real-coded genetic algorithm," IET Generation, Transmission & Distribution, vol.
According to , we know that a Hopf bifurcation point
u must satisfy the following conditions.
Figure 5 shows the price bifurcation diagrams of the two manufacturers and the retailer with change of [k.sub.1] when [k.sub.2] = 0.001, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the first bifurcation point
is [k.sub.1] = 0.00129, and the equilibrium value is (46.73, 26.4, 48.89, and 40.6).
When [[alpha].sub.2] = 1.46 and v = 0.157, the stable region of a expands from the original 2-cycle bifurcation point
[[alpha].sub.1] = 1.16546 in Figure 2 to [[alpha].sub.1] = 1.6091 in Figure 24, which implies that once system is under control, chaos is delayed or eliminated completely.