when you study the local stability of some equilibrium points of a system of differential equations. does computing the jacobian then evaluating it at the equilbrium points and translating the system to the origin then computing the jacobian then evaluating it at the origin yield to the same result ?
I've been trying both processes on numerous examples and the answer seems to be yes. however how could you show that rigorously or at least explain that intuitively ?
Yes, since translation commutes with taking derivatives.
(If $f(x)=g(a+x)$, then $f'(x)=g'(a+x)$ by the chain rule, and in particular $f'(0)=g'(a)$. And it works the same for partial derivatives when you have functions of several variables.)