CDS - Examples for Eigenvalues and Eigenvectors in 2D

This is the space generated by eigenvector with both positive eigevalues:
And the flow associated is:

Example of a saddle (linear):

• $v_1$ spans the unstable subspace $(E^U)$
• $v_2$ spans the stable subspace $(E^S)$

Example of a non-linear saddle:

• near to the ss, the linar approximation is tangent to the manyfold
• As you can see the stable manyfold $(W_{\text{LOC}}^S)$ has the flow that points toward the ss, while for the unstable manyfold $(W_{\text{LOC}}^U)$ the flow points to infinty.
• The linar approximation is valid only near the ss.

If we draw the flow of the non-linar system:

• Notice the similiraties with the linear system.
• In this case is like the linear system, but it is distorted.