I was trying to get a quadratic approximation for the stable manifold of a fixed point on an ODE. The original problem and solution are discussed at the question Finding approximation to stable manifold of saddle point.
Here is the original problem, for your convenience:
Consider the system $\dot{x} = x+e^{-y}, \dot{y} = -y.$ This system has a single fixed point, $(−1,0).$ This is a saddle point. The unstable manifold is $y=0$, but the stable manifold is some non-linear curve. Let $(x,y)$ be a point on the stable manifold close to $(−1,>0)$ and define $u=x+1.$ Write the stable manifold as $y=a_1u+a_2u^2+O(u^3)$. To determine the coefficients, derive two expressions for $dy/du$ and equate them.
I found the linear coefficient of the approximation using the Jacobian and the slope of the eigenvector corresponding to the stable manifold. I could do this since the eigenvector is tangent to the manifold.
After that, I wanted to do a similar thing to find the quadratic coefficient, but I'm afraid my kung fu isn't strong enough for that! Do you know of anyways to use the second partials of the DE's to find a matrix that will help me out here?
The linked answer has a solution which uses Taylor series, but I wanted to know if there is a way to find it using something similar to the Jacobian but for quadratic terms.
Any help is much appreciated!
You can explicitly solve the second equation $y(x)=ce^{-t}$. The first equation $$ \frac{d}{dt}(e^{-t}x(t))=e^{-t}e^{-ce^{-t}} $$ can now be integrated as $$ x(t)=de^{t}+\frac{e^{-ce^{-t}}-1}{ce^{-t}} $$ For a finite value for $t\to\infty$ we need $d=0$ so that finally $$x(t)=\frac{e^{-y(t)}-1}{y(t)}\iff e^{-y}=1+xy=1-y+uy.$$
This one can now try to solve, $$ \frac12y-\frac16y^2+\frac1{24}y^3\mp..=u\\ y=2u+\frac13y^2-\frac1{12}y^3\pm... $$ This last equation can now be used as fixed-point formula, increasing the order of the approximation by one in every step, leading to \begin{align} y&=2u+\frac43u^2+O(u^3)\\ y&=2u+\frac43\left(u^2+\frac43u^3+O(u^4)\right)-\frac23u^3+O(u^4)\\ &=2u+\frac43u^2+\frac{10}9u^3+O(u^4) \end{align} etc.