Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 56 / 2 / e

Solution

The four nearby portrait types in

u > 0

can be organized by increasing

μ

at fixed

λ > 1/2

close to

1/2

A: $μ < - 1/4$ . There is no interior equilibrium; trajectories approach the stable boundary equilibrium at the origin.
B: $- 1/4 < μ < μ_{H}$ . The lower interior equilibrium is a saddle equilibrium and the upper one a sink equilibrium. The saddle equilibrium's stable manifold separates trajectories approaching the sink equilibrium from those approaching the origin.
C: $μ_{H} < μ < μ_{G}$ . The upper equilibrium is repelling, enclosed by an attracting periodic orbit. The saddle equilibrium's stable manifold separates the periodic orbit's basin from the origin's basin.
D: $μ > μ_{G}$ , still in the local neighborhood with $μ < 0$ . The saddle equilibrium and repelling upper equilibrium remain, but the periodic orbit has disappeared. Generic trajectories ultimately approach the origin.