.. [

For the second derivative, the term in square brackets can be factorised to give

eiP i e;Pi e;Pi 2a

(5.57) a -SC· .e.P.W.e 'J

We can now examine the behaviour of these derivatives for a range of values of Wj ' To do this, it is helpful to keep the basic model equation for Sij in mind and the most convenient form for it involves combining (5.1) and (5.2) by substituting for Ai:

We can then note the following features of the first derivative, using Equations (5.54) and (5.57). (1) As Wj + 00, (5.57) shows that the term wje-SCij in the sum

in the denominator dominates and is equal to the same term in the numerator. So, Sij + eiPi and the term in square brackets in (5.54) then gives us our main result:

aW j (2) As Wj + 0, Sij + 0 and hence so also does Si/eiPi' Sij in

Equation (5.54) has a factor w; and so aSij has a factor W~-l. Thus: aw:- J

I

For finite values of Wj (greater than zero of course), we note from Equation (5.56) that the first factor is positive, the first factor in brackets is negati ve and therefore the overall sign depends on the second factor in brackets. This involves us in a study of the term a2a 1 which appears in that factor. This is plotted in Figure 5.5. This shows clearly that for a < 1, the second factor is always positive and hence a

aw: zero if a > 1. J

For a > 1, the term a2a 1 is positive and so the sign of the factor depends on Si/e;Pi relative to it. If we put

(5.61)

e;Pi

This information now allows us to plot the shape of the Si· - W

-~ ) (5.62) e;Pi

(5.63)

(5.64)

(5.65)

which, writing Sij in full from (5.57) can be given as

(5.68)

After some manipulation, this gives

--,-X",--,1:,,--w_~e_-e_C_ik_ ] 1/0.J<.#j J (1 - x)e-eCij

(5.69)

The Sjj - Wj curves, with the points of inflexion in different places, are shown in Figure 5.7. This suggests that, when they a re added together, there may be more than one poi nt of i nflexion in the D. - W. curve. For 0. < 1, a2 Dj < 0 since it is

J J 2 made up of all such negative terms. aW j Thus, we can now collect these results together and present the forms of the Dj - Wj curves for different 0. values in Figure 5.8. Note that the upper bound is now, of course Ee.P. . Sij ill

by adding the lines OJ = kW j to the curves of Figure 5.8, we can examine the equilibrium points which are the intersections of the O( 1) curve and the 0(2) line for each case and investi-

The next point to note is that the line does not always intersect the curve: two cases are shown on each of the plots

in Figure 5.9 - the dashed line obviously being the nonintersecting case. An additional example has been added to Figure 5.9 - case (e) - which shows an intermediate case of the line intersecting part of the curve, but not all of its as in case (d). This will be the basis of our initial discussion of bifurcation properties. First, however, we concentrate on the intersecting case and examine stability.