Theorem 10.3.1 (Hartman–Grobman). If \(x^*\) is a hyperbolic equilibrium, there are neighbourhoods \(V\) of \(x^*\) and \(W\) of \(0\) and a homeomorphism \(h\colon V \to W\) that maps orbits of \(x' = f(x)\) in \(V\) to orbits of the linearisation \(u' = Au\) in \(W\), preserving the direction of time: \(h(\phi _t(x)) = e^{At}h(x)\) while both sides remain in the neighbourhoods.

Theorem 10.3.2 (Stable manifold theorem). Let \(x^*\) be a hyperbolic equilibrium, with \(k\) eigenvalues of negative real part and \(n - k\) of positive real part, and let \(E^s\), \(E^u\) be the corresponding (generalised) eigenspaces. Then near \(x^*\) there are \(C^1\) manifolds \(W^s_{\mathrm {loc}}\) of dimension \(k\) and \(W^u_{\mathrm {loc}}\) of dimension \(n - k\), tangent at \(x^*\) to \(E^s\) and \(E^u\), such that \begin {gather*} W^s_{\mathrm {loc}} = \{x \text { near } x^* : \phi _t(x) \to x^* \text { as } t \to +\infty \},\\ W^u_{\mathrm {loc}} = \{x \text { near } x^* : \phi _t(x) \to x^* \text { as } t \to -\infty \} . \end {gather*}