So suppose I have the system of nonlinear differential equations: $y'(t)=F(y(t))$
It looks like one equates the drive function to a local taylor approximation about $y(t_n)=y_n$ e.g $F(y(t)) \approx F(y_n) + J(y_n)(y(t)-y_n)$, and now the stability analysis would be driven by the properities of $J$.
If I apply the euler method now:
$y_{n+1} = y_n + \Delta t F(y_n)$. The Jacobian term cancels. This is confusing to me because I believe the jacobian should infleunce this calculation for stability somehow, but I can't put the pieces together how.
Can someone tell me what I am missing?
Thank you for the input.