Stable Manifold

The stable manifold of a saddle fixed point is the set of points that converge to the fixed point under forward iteration. It’s the direction of contraction.

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

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Linear Example

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 y-axis is the stable manifold. Points on the y-axis satisfy:

$${\vec{x}}_n=A^n{\vec{x}}_0=\left[\begin{array}{c}0\\ y_0\cdot {\left(\frac{1}{2}\right)}^n\end{array}\right]\to \left[\begin{array}{c}0\\ 0\end{array}\right]$$

As $n\to \infty$, every point on the y-axis converges to the origin because ${\left(\frac{1}{2}\right)}^n\to 0$.

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Geometric

The stable manifold is the set of initial conditions that “end up” at the fixed point. If you start on the stable manifold and iterate forward, you approach the fixed point.

Near a saddle, the stable manifold looks like the eigenvector direction associated with the contracting eigenvalue.

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

• The stable manifold is typically a curve (1-dimensional)
• Points off the manifold diverge from the fixed point
• The stable manifold acts as a “separatrix” between basins of attraction

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Nonlinear

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

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

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