Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 31A / b / Solution

At a non-axis fixed point the determinant and trace areThese formulas determine the solid stable and dashed unstable branches in the bifurcation diagram. The pair $x_{\pm }$ is born in a saddle-node bifurcation at $\mu =-(\beta +1{)}^2/4$. For $\beta \ne 1$, it exchanges branches with $x=1$ at $\mu =-\beta$ and with $x=0$ at $\mu =0$, both transcritical bifurcations. If $0<\beta <1$, the saddle-node lies to the left of $x=1$; if $\beta >1$, it lies to the right. For $\beta =1$, the saddle-node and the collision at $x=1$ combine into a pitchfork bifurcation at $\mu =-1$.

Solved by gpt-5.6-sol high.