Determine values for differential equation system asymptotic stable I have this differential equation system where $k \in \mathbb{R}$ is a number.

How can I determine which values of $\textit{k}$ is the differential system asymptotic stable?
And does anybody have an easy setup for these cases (stability)
Thanks a lot!
 A: Using Mathematica's function RSolve you can solve the system of difference, and not differential, equations:
system = {x[t + 1] == 1/2 x[t] + k y[t] - 1/2 (-1)^t k^t, 
           y[t + 1] == k x[t] + 1/2 y[t] + 1/2 (-1)^t k^t}

RSolve[Join[system,{x[0] == x0, y[0] == y0}], {x[t], y[t]},t][[1]]//FullSimplify

$$x_t=\frac{1}{2} \left(2 (-1)^{t} k^t+\left(\frac{1}{2}-k\right)^t (-2+\text{x0}-\text{y0})+\left(\frac{1}{2}+k\right)^t (\text{x0}+\text{y0})\right),\\
y_t=\frac{1}{2} \left(-2 (-1)^{t} k^t+\left(\frac{1}{2}-k\right)^t (2-\text{x0}+\text{y0})+\left(\frac{1}{2}+k\right)^t (\text{x0}+\text{y0})\right)
$$
to check the stability now you have to check where the terms $k^t, \left(\frac{1}{2}-k\right)^t, \left(\frac{1}{2}+k\right)^t$ converge. All those terms converge for $\lvert k\rvert<\frac{1}{2}$. And they converge to the attractor $y=-x$.
Visually:
data = Table[#, {t, 0, 100}]&/@Flatten[
       Table[{1/2 (2  (-k)^t + (1/2 - k)^t (-2 + x0 - y0) + 
              (1/2 + k)^t (x0 + y0)), 
              1/2 (-2  (-k)^t + (1/2 - k)^t (2 - x0 + y0) + 
              (1/2 + k)^t (x0 + y0))}, 
             {x0, -2, 2, 0.5}, {y0, -2, 2, 0.5}], 1];
(* unstable case*)
ListPlot[data/.k -> 1/8, PlotRange -> All, Joined -> True,
         InterpolationOrder -> 2]




(* unstable case*)
ListPlot[data /. k -> 3/4, Joined -> True, InterpolationOrder -> 2, 
         PlotRange -> {{-10, 10}, {-10, 10}}]




Observe that the orbits go to infinity along the diagonal $y=x$. The bigger the value of $k$ the bigger the "fluctuation":
ListPlot[data /. k -> 2, Joined -> True, InterpolationOrder -> 2, 
         PlotRange -> {{-100, 100}, {-100, 100}}]




Its a mystery to me why they redirected here from the Mathematica.SE :/
