Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 331 / 1 / g / Solution

Let $r=Ra-R{a}_c$. The amplitude equation is $\dot{A}=A(r-A^2)$. For $r<0$, $A=0$ is the sole equilibrium and every solution tends monotonically to it. At $r=0$, the origin remains attracting but only algebraically. For $r>0$, the origin is unstable and the two equilibriaare stable: positive initial data tend to $A_{+}$, negative initial data tend to $A_{-}$, and $A(0)=0$ remains zero. A plot of $A(t)$ therefore shows a pitchfork bifurcation normal form at $Ra=R{a}_c$.