Unstable Manifold

The unstable manifold of a saddle fixed point is the set of points that converge to the fixed point under backward iteration. It’s the direction of expansion.

For a linear map with a saddle at the origin, the unstable manifold is the subspace spanned by eigenvectors with $|\lambda |>1$.

---

Linear

Consider the diagonal linear map:

$$A=\left[\begin{array}{cc}2& 0\\ 0& \frac{1}{2}\end{array}\right]$$

Eigenvalues: ${\lambda }_1=2$, ${\lambda }_2=\frac{1}{2}$

Eigenvectors: ${\vec{v}}_1=\left[\begin{array}{c}1\\ 0\end{array}\right]$, ${\vec{v}}_2=\left[\begin{array}{c}0\\ 1\end{array}\right]$

The x-axis is the unstable manifold. Points on the x-axis satisfy:

$${\vec{x}}_n=A^n{\vec{x}}_0=\left[\begin{array}{c}{x}_0\cdot 2^n\\ 0\end{array}\right]\to \infty$$

As $n\to \infty$, points on the x-axis diverge from the origin because $2^n\to \infty$.

But under backward iteration, points on the x-axis converge to the origin:

$${\vec{x}}_{-n}=A^{-n}{\vec{x}}_0=\left[\begin{array}{c}{x}_0\cdot {\left(\frac{1}{2}\right)}^n\\ 0\end{array}\right]\to \left[\begin{array}{c}0\\ 0\end{array}\right]$$

---

Geometric

The unstable manifold represents the set of initial conditions that “came from” the fixed point. If you start on the unstable manifold and iterate backward, you approach the fixed point.

Near a saddle, the unstable manifold looks like the eigenvector direction associated with the expanding eigenvalue.

For saddles in ${\mathbb{R}}^2$:

• The unstable manifold is typically a curve (1-dimensional)
• Points on the manifold diverge from the fixed point under forward iteration
• The unstable manifold acts as a “source direction” from the saddle

---

Nonlinear

For nonlinear maps, the unstable manifold is generally curved (not a straight line). But near the fixed point, it’s tangent to the unstable eigenvector direction.

The Stable Manifold Theorem also guarantees that smooth unstable manifolds exist for hyperbolic fixed points, and they’re tangent to the unstable eigenspace at the fixed point.

---

• Stable Manifold: converges to the fixed point under forward iteration ($n\to \infty$)
• Unstable manifold: converges to the fixed point under backward iteration ($n\to -\infty$)

The unstable manifold expands under forward iteration while the stable manifold contracts.

---

• form the “outflow” from saddle points
• for chaotic systems unstable manifolds of different saddles can intersect stable manifolds (creating homoclinic and heteroclinic tangles)
• intersections are the geometric signature of chaos