Question: Sketch the bifurcation diagram of the following ODE for x ∈ R
dx/dt = -μ2 - x3 + ∈x
on varying μ, considering the cases ∈ > 0, ∈ = 0 and ∈ < 0. Be careful to indicate the stability of any branches and verify that there are saddle-node bifurcations for ∈ > 0. Verify for μ = ∈ = 0 that there is an equilibrium at x = 0 that is neither linearly stable nor unstable (this is called an Isola bifurcation). Show that the curve of saddle-node bifurcations in the (μ, ∈) plane is 9μ2 = 2√(3∈3/2) and plot this curve.