Assume that we have a dynamical system $\dot x = f(x, u)$ and I want to simulate the system with Euler's Method.
$$x = x + af(x, u)$$
Where $a$ is a small number. This can create unstability. What if I linearize the dynnamical system $\dot x = f(x, u)$ in the point $x$ and then simulate (I think this is the correct way to write the linearized $\dot x = f(x, u)$ )
$$x = x + a\frac{\partial f(x, u)}{\partial x}$$
Will the dynamical simulation with euler's method be secure against unstability?