Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 31E / c

The off-axis equilibria exist for $|a|\le 1$ and coalesce with an axial equilibrium at each endpoint. Thus the stationary bifurcations occur atFor the positive bifurcation, $a_0=y_0=1$. Putso thatWrite the extended center manifold as $Y=h(x,\mu )$. Its invariance equation isUnder the scaling $\mu =O(x^2)$, comparison of the terms of order $x^2$ givesSubstitution into the $x$ equation givesWith the reversed parameter $\nu =-2\mu =2(1-a)$, this is the pitchfork bifurcation normal form $\dot{x}=\nu x-x^3$. Therefore the positive stationary bifurcation is a supercritical pitchfork in the parameter $\nu =2(1-a)$, or equivalently the stable daughter equilibria exist on the $a<1$ side.