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?