Chapter 8, Section 8.8, Figure 8.8

Define the functions f and g where f is the left-hand side of Eq.(8.121) and g is the right-hand side.

f[δb_,X_]:= δb+X^2;
g[k_, i_, X_]:=(-1)^(k+i+1)*2*k*X^(k-2);

Plot the f for δb = –10, –6, –3, 0 and 5, and g for k = 1, i = 0, 1.  Note that the δb = –3 curve for f is tangential to the lower curve for g.  This is because δb = –3 is the bifurcation value.  See Fig.8.8a.

Plot[
{f[-10,X],f[-6,X],f[-3,X],f[0,X],f[5,X],
 g[1,0,X],g[1,1,X]},
{X,0,10}, AxesOrigin->{0,-10},Frame->True,
AspectRatio->1,
PlotRange->{{0,4},{-10,10}},PlotPoints->100]

Plot the f for δb = –10, –7, –4, 0, 4 and 7, and g for k = 2, i = 0, –1.  Note that the δb = ±4 curve for f is tangential to the both curves for g.  This is because δb = ±4 are the bifurcation values.  See Fig.8.8b.

Plot[
{f[-10,X],f[-7,X],f[-4,X],f[0,X],f[4,X],f[7,X],
 g[2,0,X], g[2,-1,X]},
{X,0,10}, AxesOrigin->{0,-10},Frame->True,
AspectRatio->1,
PlotRange->{{0,4},{-10,10}},PlotPoints->100]

Plot the f for δb = –10, –5,  0, 5, 9, 15 and 20 and g for k = 3, i = 0, –1.  Note that the δb = 9 curve for f is tangential to the upper curve for g.  This is because δb = 9 is the bifurcation value.  See Fig.8.8c.

Plot[
{f[-10,X],f[-5,X],f[0,X],f[5,X],f[9,X],
f[15,X],f[20,X],
 g[3,0,X], g[3,-1,X]},
{X,0,8}, AxesOrigin->{0,-10},Frame->True,
AspectRatio->1,
PlotRange->{{0,8},{-10,40}},PlotPoints->100]