Consider an autonomous system ($y$ is a vector)
$\dfrac{dy}{dt}=f(y)$ , $y(0)=x$
having the solution
$y=y(t,x)$
How am I suppose to compute the derivative
$\dfrac{\partial y}{\partial x}$
at ($x_i,T_i)$ (the guessed initial conditions and guessed period) when I DON'T know the explicit function $y=y(t,x)$. Because the solution of the original ODE is purely numerical. That ODE simply cannot be solved analytically, I only have the numbers. I'm trying to use that to correct the set of guessed initial conditions I found. That derivative I'm trying to get is part of the 1st-order Taylor expansion.
Thanks in advance!