The stable manifold of a fixed point on a non linear map

Question

i have a nonlinear map

$f(x,y) = (x/2, 2y-7x^2)$

the map has a fixed point at (0,0).

$Df(0,0)=\begin{pmatrix}0.5 & 0\\\ 0 & 2\end{pmatrix}$

then, the fixed point is a saddle.

On the book (Alligood K.T., Yorke J.A, T.D.Sauer. Chaos: An Introduction to Dynamical Systems) said that the stable manifold of (0,0) is described by the parabola $y=4x^2$. How do i do to reach that result?

Answer 1

You can do this by verifying three things:

1. $y=4x^2$ is invariant under $f(x,y)$
2. Each point $(x,y)$ that lies on the curve $y=4x^2$ is attracted to $(0,0)$ under iteration of $f(x,y)$.
3. The curve $y=4x^2$ is tangent at $(0,0)$ to the expanding eigenvector of $Df(0,0)$.