Strogatz Exercise 6.1.14: how to approximate stable manifold of a saddle point with a series I'm working through Strogatz's Nonlinear Dynamics and Chaos and am stuck on assignment 6.1.14.
We have the following system:
$$ \dot{x} = x + e^{-y}$$
$$ \dot{y} = -y $$
which has one fixed point, a saddle at (-1, 0). Its unstable manifold is the $x$-axis, the goal of the assignment is to approximate the stable manifold.
Here is what the assignment says:

a) Let $(x, y)$ be a point on the stable manifold, and assume that $(x,y)$ is close to $(-1, 0)$. Introduce a new variable $u = x + 1$, and write the stable manifold as $y = a_1 u + a_2 u^2 + O(u^3)$. To determine the two coefficients, derive two expressions for $dy/du$ and equate them.

I was thinking I could find one expression for $dy/du$ by dividing $\dot{y}$ and $\dot{u}$, where
$$ \dot{u} = (u - 1) + e^{-y} $$
so that
$$ \frac{dy}{du} = \frac{\dot{y}}{\dot{u}} = \frac{-y}{(u - 1) + e^{-y}} $$
and I thought maybe I can assume, because $(x,y)$ is very close to the fixed point and is located on the stable manifold, that
$$ \frac{dy}{du} = \frac{y}{u} $$
i.e. the slope of the manifold is in the direction of the fixed point. However, then I don't see how to express $y$ as a polynomial of $u$.
Any ideas?
 A: I think the idea is to linearise the system.
Indeed, $\dfrac{dy}{du} = \dfrac{\dot{y}}{\dot{u}}$, but $\dot{u} \approx (u-1) + (1 - y + \mathcal{O}(y^2))$ from Taylor expansion.
The system can therefore by simplified as $$\frac{d y}{d u} = \frac{-y}{u-y}$$
If we substitue $y \approx a_1u + a_2u^2$ back, we would get $$\frac{d y}{d u} \approx \frac{-a_1u - a_2u^2}{u -  a_1u - a_2u^2} = \frac{a_1}{a_1-1} - \frac{a_2}{(1-a_1)^2}\cdot\frac{u^2}{u-\frac{a_2}{1-a_1}u^2} \approx \frac{a_1}{a_1-1}$$
which is just a constant polynomial in $u$. So if we equate this with $\dfrac{dy}{du} \approx a_1 + 2a_2u$, we would get $$a_1 = \frac{a_1}{a_1-1}$$ and $$a_2=0$$
So the stable manifold is approximately $y = a_1u = 2(x-1)$, at the region very close to the fixed point $(-1,0)$.
I did find this to be quite difficult, as it involves a lot of approximations, but then the question also gives a hint by telling us to 

assume that $(x,y)$ is close to $(−1,0)$

And while I was reading onto further chapters, I guessed this might be a introduction to linearisation in 2D systems, so that would make perfect sense.
