Unstable manifold ($W^u$)

For a smooth autonomous differential equation and a hyperbolic equilibrium point $p$, the unstable manifold consists of points whose backward trajectories approach $p$: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 $x(t)=x_0{e}^t$ and $y(t)=x_0^2{e}^{2t}/3$ 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.