4. Consider the normal form which is a combination of a turning point, or saddle-node

bifurcation, and a pitchfork bifurcation. Let x(t) satisfy the differential equation

dx/dt = c + dx-x³ = v(x,c,d).

When c-0, we have the case of a pitchfork bifurcation, and when d=0, we have the case of a

turning point. We wish to investigate the equilibrium points and their stability as a function of

the parameters c and d. In particular, consider the following:

i)

ii)

Graph the functions f1(x) = -c and f2(x) = dx - x3, and consider their intersections for

the case of d > 0 and d<0. Can you conclude that, (a) for d <0, there is only one zero

of v(x, c, d) for any c, -∞ 0, there is either one zero

or there are three zeros, depending on the value of the parameter c; if c lies in an

interval, -ccr < c < ccr, there are three zeros of v(x, c, d)? Discuss the stability of the

equilibrium points.

Find the relationship between c and d which defines the ccr in i) above, and plot this

function in (d, c) plane (the parameter plane). These curves form the so called

bifurcation set.