Hello Stack Exchange community. I am currently having an issue finding the local stable/unstable manifolds of this system. After going at it for a few hours I believe the person who wrote this problem made a mistake as I am unable to find a stable/unstable manifold at $(0,0)$ of the system..
System:
- $X' = -X + Y + 3Y^2 $
- $Y' = Y$
- Eigenvalues: $-1, 1$
- Stable Eigenvalues: $-1$
- Unstable Eigenvalues: $1$
- Nonlinear Part: $3Y^2$
After using the Lyapunov-Peron method (https://www.cds.caltech.edu/~murray/wiki/images/b/ba/Cds140a-wi11-Week4Notes.pdf) and iterating 3 times, I am unable to find an actual stable manifold of degree 2 or 3.
- $X_0 = (0,0)$
- $X_1 = (e^{-t},0)$ (first iteration)
- $X_2 = (e^{-t},0)$ (second iteration)
- $X_3 = (e^{-t},0)$ (third iteration)
The stable and unstable aspect of the iterative method keep going to $0$ as the unstable part is $0$ as the nonlinear part of the $Y'$ equation is $0$.
That being said the nonlinear part of the $X'$ equation ($X' = -X + Y + 3Y^2$) becomes $0$ as it relies on the previous iteration. The previous iteration $(e^{-t},0)$ makes it $0$ as the $y$ vector is $0$ and the $x$ is $e^{-t}$.
For the $y$ vector of the previous iteration do I just directly plug in the solution of the $Y'$ ODE, i.e. $e^t$?
P.S. Sorry if it is a bit messy. I have never used Latex before.