Discuss the local and global behavior of solutions of r r

Discuss the local and global behavior of solutions of r\' = r - r^3 theta\' = sin^2 (theta) + a at the bifurcation value a = -1.

Solution

The bifurcation values are a = 1 and a = 0. Each of these parameter values yield a flow with two equilibrium points: (0, 0) is a source for both of them while (1, 0) is a sink when a = 1. The flows swirl from the source to the sink.