Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 66

1.2

1.2

L= 1 L =1

x

x

0 x

x 1.2 0 1.2

(a) (b)

1

1

x x

0 1 0 1

(c) (d)

Fig. 2.5 Example 2.3; linear inverse demand and cost functions and identical capacity constrained
firms. Illustrating the border collision bifurcation that occurs as the number of firms varies from 8
to 13. ( a ) N D 8;( b ) N D 9;( c ) N D 10;( d ) N D 13. Note how the derivative of the map T at
x crosses a jump discontinuity as N passes through the value 9

1/
2 < 1 on the right branch without
passing through the bifurcation value T 0 .x/ D 1.
In general it is not easy to predict which kind of attractor will emerge from such
a type of bifurcation. In our case, for N>9a stable cycle of period 2 is created
around the unstable equilibrium x (see Fig. 2.5c). The points of the stable 2-cycle
are located on different branches of the piecewise linear map. Hence, the multiplier
of this 2-cycle is given by the product of its derivatives, that is
2 .C 2 / D .1 a/ 1

a.N C

branch and suddenly attains the value 1

:

a.N C 1/
2

For increasing values of N the multiplier .C 2 / crosses the critical value 1 for
N D 11 after which a chaotic attractor suddenly appears (Fig. 2.5d).
Border collision bifurcations may occur in the presence of piecewise smooth
reaction functions for example, due to non-negativity and capacity constraints or
even discontinuous reaction functions. As demonstrated above, they are related