Assume that at a bifurcation point (xc, Rc) = (0, 0), in addition to the usual criteria
Chapter 14, Problem 14.8.3(choose chapter or problem)
Assume that at a bifurcation point (xc, Rc) = (0, 0), in addition to the usual criteria for a bifurcation point, fR = 0 and fxx = 0. Using a Taylor series analysis, show that as an approximation, dx dt = fxx 2 x2 + fxRRx + fRR 2 R2. (a) If f2 xR > fxxfRR, then the bifurcation is called transcritical. Analyze stability using one-dimensional phase diagrams (assuming fxx > 0). Explain why the transcritical bifurcation is also called exchange of stabilities. (b) If f2 xR < fxxfRR, then show that the equilibrium is isolated to R = Rc only.
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