Center manifold theorem

Remarks

The usual proof first cuts $f$ off outside a small ball, so that the nonlinear part is globally small in $C^1$. For the modified equation, a global center manifold is the graph of a map $E^c \to E^s \oplus E^u$. This map is found as a fixed point of a contraction on a space of functions with controlled growth; this is the Lyapunov–Perron method. The center manifold need not be unique. For $x' = x^2$, $y' = -y$, every curve made of $y = c e^{1/x}$ for $x < 0$ and $y = 0$ for $x \ge 0$ is a center manifold of the origin. The finiteness of $k$ matters. For a $C^\infty$ system there is a $C^k$ center manifold for each finite $k$, but in general no $C^\infty$ one, and the neighbourhood may shrink as $k$ grows. If no eigenvalue of $Df(x_0)$ has positive real part, the stability of $x_0$ is the same as for the restriction of the equation to $W^c$.

Going deeper

For a student

The theorem is the main tool for studying non-hyperbolic equilibria, where the Hartman–Grobman theorem gives no information. One computes a Taylor expansion of the center manifold as a graph over $E^c$ by matching coefficients in the invariance equation. Then one studies the reduced equation on $E^c$, which in a bifurcation is often one- or two-dimensional. This reduction underlies the normal forms of the saddle-node and Hopf bifurcations. It complements the stable manifold theorem, which treats the contracting and expanding directions.