Consider the differential equation dx=dt D kx x3. (a) If k
Chapter , Problem 21(choose chapter or problem)
Consider the differential equation dx=dt D kx x3. (a) If k 5 0, show that the only critical value c D 0 of x is stable. (b) If k>0, show that the critical point c D 0 is now unstable, but that the critical points c D pk are stable. Thus the qualitative nature of the solutions changes at k D 0 as the parameter k increases, and so k D 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 x0 D kx x3 is the pitchfork diagram shown in Fig. 2.2.13.
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