initially

Hence the Equation (7.10) can be wri tten for each mode in turn as

o J/[ 1 dX2]Xl [~ ~+C+X2 - Xl (7.20) [ OdX2 J/[ 1 dX 2 ] (7.21)X2

c + X2 ~ + C + X2 - X2 The equilibrium conditions are

xk = 0, k = 1.2 (7.22)

Setting the right hand side of (7.20) to zero gives, after some mani pula ti on

(Xl - O)(dxt + (ad + 1 )Xl - C - 0) 0 (7.23) This involves subs ti tuti ng for x2 using (7.9) as

Equation (7.23) shows that there are three equi libri urn va lues for Xl' and Equation (7.24) can be used to give the corresponding values for X2' Ore is clearly

xi D, x~ = 0 (7.25) Then, the other two solutions for Xl are obtained by solving the quadratic equation part of (7.23):

< 0 (7.27) -Because Xl < 0, this is not relevant and so is not considered

xi < D (7.28) which, of course, is equivalent to requiring

The next step in the argUirent is to examine the stability of the solutions. It can easily be seen from a consideration of the signs of xk in (7.10) for the different solutions that xi is stable when D < Dc and not otherwise; and vice versa for xi. Thus, if we plot Xl equilibrium values against D (and x, using (7.24)), we obtain the plots of Figure 7.12. This is a familiar kind of bifurcation plot. 214

7 .4~3 Model 2 : addition of psychological factors The attractiveness factors now take the form

Ak = vkkFk where the factors Fk are

Fk = 8k + akx k

(7.32)

The 8k 'S are taken as measuring the effects of publicity and the ak's, of imitation: the more people who use a mode, the more popular it becomes. Much simpler forms are taken for the velocity forms:

p q (7.35) Thus

8j Aj - + aJl (7.36 )

A2 82x2 + a2x~ (7.37) and Equation (7.10) can be written for this case as

8j 8j D( - + aj)/( - + aj + 82X2 + a2x~) - Xj