Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 33A / Solution

The centre manifold theorem states that near an equilibrium whose linearization has centre, stable, and unstable spectral subspaces $E^c,E^s,E^u$, there is a local invariant manifold $W^c$ tangent to $E^c$ at the equilibrium. It can be written locally as a graph over $E^c$, and the dynamics on it govern the local nonhyperbolic behaviour. The manifold need not be unique, but its finite Taylor expansion is determined to the required order by the invariance equation.

For a parameter-dependent system, the key step in forming the extended centre manifold is to promote the parameter to a dynamical variable:The parameter direction then has zero eigenvalue and is included in the extended centre subspace.

Solved by gpt-5.6-sol high.