Stability properties of discretization of ODE I am trying to find some conditions which guarantee that a continuous time dynamical system and it's discretization have the same behavior with regard to equillibrium points. Specifically that if the original system has a unique equillibrium point to which it converges exponentially for all initial settings, then that point is also globally stable in the discrete system. 
For instance, we may write (wlog) $$\dot{x} = -x + f(x)$$
and the corresponding discrete system would be
$$x_{n+1} = f(x_{n})$$
One condition which will ensure this is that $f(x)$ has small Jacobian in an appropriate norm, i.e. $$\|\frac{d f}{d x}\| < 1$$ since then we can deduce that $f$ is Lipschitz and apply the contraction mapping theorem. But this seems like a very weak condition and I was wondering if there is anything stronger, or perhaps references to some literature which addresses such things.

Might as well answer my own question here. This example shows a situation where they might not have the same behavior.
As an example, consider the system
$$ \dot{x} = -8x$$
along with discretizations
$$T_{\epsilon}(x) = x - \epsilon 8x$$
for $\epsilon = \frac{1}{4}$ we get
$x_{n+1} = x_{n} - 2x_{n} = -x_{n}$, and therefore the system will be oscillating. For any larger $\epsilon$, the system will be exhibit growing oscillations. For example $\epsilon = \frac{1}{2}$ leads to $x_{n+1} = x_{n} - 4x_{n} = -3x_{n}$. In terms of starting point $x_{0}$ we have $x_{n} = (-1)^{n}3^{n}x_{0}$.
