Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 14D / Solution

For parameter-dependent centre manifold theorem, append $\dot{\mu }=0$. At $(x,y,\mu )=(0,0,0)$ the eigenvalues are $0,-1,0$: the $y$ direction is strongly stable and $(x,\mu )$ span the extended centre directions. An extended centre manifold for a parameter is an invariant graph of a function $y=h(x,\mu )$ tangent to $y=0$; nearby trajectories approach its reduced dynamics along stable fibers. The stable eigenvalue remains separated from zero, so local bifurcation and stability are governed by this reduction. This is the parameter-extension content of the requested stable/centre-manifold discussion.

The centre-manifold invariance equation isReflection symmetry permits an invariant graph of a function with $h(-x,\mu )=h(x,\mu )$ and $h(0,\mu )=0$. At quadratic order the left side vanishes and the right side is $-h-x^2$, giving $h=-x^2+O(|\mu |x^2+x^4)$. More explicitly its first mixed coefficient is $2\mu x^2$, found by equating the $\mu x^2$ terms. The reduced equation isThus the origin is locally asymptotically stable for $\mu <0$, becomes a saddle for $\mu >0$, and creates two stable equilibria $x=\pm \sqrt{\mu /2}+O({\mu }^{3/2})$, $y=-\mu /2+O({\mu }^2)$. This is a supercritical pitchfork bifurcation. The exact nearby nonzero branches provide a check: the equilibrium equations givewhich are real and nonzero for small positive $\mu$. The omitted branch $y=1+\sqrt{1+\mu }$ is not near the origin. At $\mu =0$ the leading $-2x^3$ also attracts, although at an algebraic rather than exponential rate.