Unstable manifold
= Unstable manifold
{title2=$W^u$}
For a smooth <autonomous differential equation> and a hyperbolic <equilibrium point> $p$, the <unstable manifold> consists of points whose backward trajectories approach $p$:
$$
W^u(p)=\{x:\Phi_t(x)\to p\text{ as }t\to-\infty\},
$$
where the backward solution must exist for all negative time. The local unstable manifold additionally requires that the backward trajectory remain in a chosen small neighborhood. The global unstable manifold is obtained by following this local set forward as far as the <flow map> exists.
Its tangent space at $p$ is the real sum of the <generalized eigenspaces> of the <Jacobian matrix> with <eigenvalues> of positive <real part>. Generalized eigenspaces are needed when the matrix is not diagonalizable. Nonreal <eigenvalues> occur in conjugate pairs, together contributing a real invariant plane. The <stable manifold> uses negative real parts and convergence in forward <time>.
A nonlinear example shows why the tangent space alone is insufficient. For $\dot x=x$, $\dot y=-y+x^2$, the curve $y=x^2/3$ is invariant because its <time derivative> is $2x^2/3=-x^2/3+x^2$. On this curve $x(t)=x_0e^t$ and $y(t)=x_0^2e^{2t}/3$ approach zero backward, so it is the unstable manifold. It is tangent to the $x$-axis at the origin, but it is not that axis. In fact $y-x^2/3$ obeys $\dot w=-w$, so every point off the curve fails to approach zero backward.