Invariant manifolds organize nonlinear dynamics near equilibria by separating directions of exponential contraction, expansion, and neutral motion.
Hyperbolic splitting
Let \(f(0)=0\) and \(A=Df(0)\) have no eigenvalues on the imaginary axis. Then \[ \mathbb R^d=E^s\oplus E^u, \] where generalized eigenspaces in \(E^s\) have negative real parts and those in \(E^u\) positive real parts.
Theorem (stable manifold). If \(f\in C^k\), \(k\ge1\), there are local \(C^k\) manifolds \(W_{\mathrm{loc}}^s,W_{\mathrm{loc}}^u\) through zero such that \[ T_0W_{\mathrm{loc}}^s=E^s,\qquad T_0W_{\mathrm{loc}}^u=E^u. \] They are locally invariant; points on \(W^s\) converge exponentially to zero forward in time, and points on \(W^u\) do so backward in time.
Graph-transform mechanism
In coordinates \(x=(\xi,\eta)\in E^s\oplus E^u\), a local stable manifold is a graph \[ \eta=h(\xi),\qquad h(0)=0,\quad Dh(0)=0. \] The time-\(\tau\) map sends Lipschitz graphs to graphs. Forward contraction in \(E^s\) and backward contraction in \(E^u\) make this graph transform a contraction in a suitable function space. Its fixed graph is invariant. This also proves uniqueness in the class of graphs with the prescribed tangent space.
Lyapunov--Perron formulation
Writing \[ \dot \xi=A_s\xi+F_s(\xi,\eta),\qquad \dot\eta=A_u\eta+F_u(\xi,\eta), \] a stable orbit satisfies \[ \xi(t)=e^{tA_s}\xi_0 +\int_0^te^{(t-s)A_s}F_s(x(s))\,\mathrm{d}s, \] \[ \eta(t)=- \int_t^\infty e^{(t-s)A_u}F_u(x(s))\,\mathrm{d}s. \] This integral operator is a contraction on exponentially weighted trajectories for a sufficiently small neighborhood. Evaluating \(\eta(0)\) defines the stable graph.
Center manifolds
Theorem (center manifold, local form). If the spectrum splits into stable, center, and unstable parts, a local center manifold \(W^c\) exists tangent to \(E^c\). Nearby bounded full trajectories lie in it. Stability and bifurcation near the equilibrium reduce to the dynamics on \(W^c\), coupled with exponential behavior off it.
Center manifolds are generally not unique, though their finite Taylor jets are determined to corresponding order. The reduced vector field is the correct place to inspect nonlinear terms when linearization has zero real parts.
Hartman--Grobman theorem
Near a hyperbolic equilibrium, the nonlinear flow is locally topologically conjugate to \(\dot x=Ax\). The conjugacy is generally only a homeomorphism, so it preserves orbit structure but not rates, angles, or differentiable invariants. Stable manifolds refine this topological statement with smooth geometry.