I have some problems with the following differential equation, it looks a little bit confusing (because of notation), but please take a short look at it, it should be not too difficult.
$y'(t)=f(y(t)), t>0$ for $y\in C^1([0,\infty];\mathbb R^2)$ with $f\in C^1(\mathbb R^2; \mathbb R^2)$. $y_0$ is zero of $f:f(y_0)=0$. The real part of all eigenvalues of the matrix $df(y_0)$ is smaller or equal 0.
Now I have two things:
(i) I introduce $z:C^1([0,\infty]$ which is given by $z(t)=y(t)-y_0$. This should somehow satisfy $z'(t)-Az(t)=\xi (z(t))$, where $lim_{x->0}\frac{\xi (x)}{||x||}=0$ and $A\in GL(2,\mathbb R^2)$
but I do not see why. I tried to rewrite $z'(t)-Az(t)=y'(t)-A(y(t)-y_0)=f(y(t))-A(y(t)-y_0)$ but still do not see the relationship.
(ii) Every solution of $z'(t)-Az(t)=\xi (z(t))$ should satisfy $z(t)=\sum_{i=1}^{2}(z_i(0)z_h^{(i)}(t)+\int_{0}^{t}\xi_i(z(s))z_h^{(i)}(t-s) ds)$
with $z_h^{(i)}$ i=1,2 solutions of the equation $z'_h(t)=Az_h(t)$ with starting values $z_h^{(1)}=(1,0), z_h^{(2)}=(0,1)$
If you could tell how the solution of the differential equation looks like, it should be no problem to see that it satifies the equation above.