Consider the differential equation dx/dt = x + kx3 containing the parameter k. Analyze (as...

Consider the differential equation dx/dt = x + kx3 containing the parameter k. Analyze (as in Problem) the dependence of the number and nature of the critical points on the value of k, and construct the corresponding bifurcation diagram.

Problem

Consider the differential equation dx/dt = kx − A3 (a) If k > 0, show that the only critical value c − 0 of x is stable. (b) If k > 0. show that the critical point c = 0 is now unstable, but that the critical points stable. Thus the qualitative nature of the solutions changes at k − 0 as the parameter k increases, and so k = 0 is a bifurcation point for the differential equation with parameter k. The plot of all points of the form (k, c) where c is a critical point of the equation x′ = kx- A"3 is the “pitchfork diagram” shown in Fig.

FIGURE. Bifurcation diagram for dx/dt = kx − x3.