Given a dynamical system
$${\dot{x_1}} = -x_1 -x_2^2 \\ {\dot{x_2}} = x_2 -x_1^2 $$
Find the stable and unstable manifold.
So by linearizing the system at the critical point it can be written as
$$\dot x = Ax$$ where A is $$\pmatrix{ -1 & 0 \\ 0 & 1}$$
Denoting the difference between the linearized system and the non linear system as
$$S = F - Ax = \pmatrix{-x_2^2 \\ x_1^2}$$
$A$ can be written as two exponential maps (as the maps are commutative) giving
$$U = \pmatrix{e^{-t} & 0 \\ 0 & 0} \\ V = \pmatrix{0 & 0 \\ 0 & e^t}$$
Now this is where I get stuck. This is an example from my notes but I can't follow it from this point onwards. I was hoping someone could explain what everything I write means after this point.
So continuing with this problem. My notes says
$$G(y) = M^{-1}S(My) = S(y) = \pmatrix{-y^2 \\ y_1^2}$$
but I don't really understand where this comes from. Then it says
$$u(t,a) = u(t)a + \int_0^tu(t-s)G(u(s,a))\,ds - \int_t^\infty v(t-s)G(u(s,a))\,ds \\ = \pmatrix{e^{-t} & 0 \\0 & 0}\pmatrix{a_1 \\ 0} + \int_0^1 \pmatrix{e^{-t+s} & 0 \\0 & 0}\pmatrix{-u_2^2(s) \\ u_1^2(s)} \,ds - \int_t^\infty \pmatrix{0 & 0 \\ 0 & e^{t-s}}\pmatrix{-u_2^2(s) \\ u_1^2(s)}\\ds \\ = \pmatrix{a_1e^{-t} \\ 0} + \int_0^t \pmatrix{-e^{-t+s}u_2^2(s) \\ 0}\, ds - \int_t^\infty \pmatrix{0 \\ e^{t-s}u_1^2}\,ds $$
I really don't understand how you get to this though.
Then by solving using Picard's interation (which I know how to do) you get the answer.
$$x_2 = \frac{-x_1^2}{3} + O(x^5)$$
If anyone has any understanding of what the middle section means with all the integrals that would be extremely useful! Thanks!!
Edit: If anyone knows of any literature that is similar to this. If you point me in the right direction that would also be very helpful!
Edit: I found this link, which has my example in it! It's something called the Perko iteration which has never been mentioned to me before. https://www.cds.caltech.edu/~murray/wiki/images/b/ba/Cds140a-wi11-Week4Notes.pdf
After doing some research/googling into this I found this linnk https://www.cds.caltech.edu/~murray/wiki/images/b/ba/Cds140a-wi11-Week4Notes.pdf which contains the example above.
After going through these notes it explains to solve this problem a perko iteration must be found and the method is given in the notes.