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.
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