Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 68

x

0.7

0.3

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 L

Fig. 2.6 Example 2.3; linear inverse demand and cost functions and identical capacity constrained
firms. Bifurcation diagram of output with respect to capacity L with N D 15 firms. A border

collision occurs as the bifurcation value L D 0:625 is crossed

1

1

m

m

m 1

m

1

x

x

0 1 0 1

(a) (b)

Fig. 2.7 Example 2.3; linear inverse demand and cost functions and identical capacity constrained
firms. The determination of the bounds for the chaotic attractor as capacity L varies. The critical
point m is determined at a point where the map T is not differentiable. ( a ) L D 0:8;( b ) L D 0:9

case of two-dimensional discrete-time dynamical systems represented by piecewise
differentiable maps, like the one obtained in the semi-symmetric case, to which we
turn in the next example.
To conclude our analysis of the mathematical properties of the piecewise linear
symmetric model, we investigate what kind of bifurcation occurs when the inte-
rior equilibrium x loses stability for increasing values of N because the derivative
T 0 .x/ become less than 1. As we demonstrate below, a quite particular kind of