Numerically, you treat it just like a single differential equation except you keep track of two sets of data simultaneously since each x and y depend on each. You could easily adapt Euler to do this, it's just twice as many variables. Could use Runge-Kutta too, again, twice as many variables. I don't see how the distance from the equililbrium points are relevant in terms of numerical methods unless some stability issues are encountered like highly oscillatory or singular. I'd suggest learning how Euler works at least, maybe Runge-Kutta if you like pain, maybe do one or two, then just use Mathematica from then on:

Code:

m = 10;
n = 10;
a = 100;
b = 5;
xstart = 1;
ystart = 1;
sol = NDSolve[{Derivative[1][x][t] ==
(m/n)*(a - b*y[t]),
Derivative[1][y][t] ==
(a/b)*(-m + n*x[t]), x[0] == xstart,
y[0] == ystart}, {x, y}, {t, 0, 10}]
ParametricPlot[Evaluate[{x[t], y[t]} /.
sol], {t, 0, 10}, AspectRatio -> 1]