On proving the absence of limit cycles in a dynamical system

I’m studying this classical paper in nonlinear dynamics in biophysics and I want to understand it properly. I’m stuck at this two-variable problem which I’ve struggled with for too long now.

$$ dot M = frac{1}{1+E^m} – a M $$
$$ dot E = M – b E $$

Where $ dot z = frac{dz}{dt} $. It turns out later that $ m geq 8 $. Note also that, because it’s concentrations of molecules, $ M,E geq 0 $.

This coupled system of equations has a fixed point $ (M_0,E_0) $ where the nullclines $ dot M = 0 $ and $ dot E = 0 $ intersect. This occurs when:

$$ M_0 = b E_0 $$
$$ a b E_0 (1+E_0^m) = 1 $$

So far, so good. Now the article says: "We expand near this point by writing $ M = M_0 + X $,   $ E = E_0 + Y $"

$$ dot X = – m a^2 b^2 E_0^{m+1} Y – a X + O(Y^2) $$
$$ dot Y = X – b Y $$

I get why they expand near the fixed point and I get that $ O(Y^2) $ means neglecting higher-order terms. And I think I can derive the equation for $ dot Y $:

$ dot Y = dot {(E-E_0)} = M_0 + X – b (E_0 + Y) = b E_0 + X – b E_0 – b Y = X – b Y $

But I need your help to understand how to get $ dot X $ just from taylor expansion and the implicit equation for $ E_0 $.

The paper can be found here: http://www.math.us.edu.pl/mtyran/dydaktyka/biomatematyka/griffith_1968_I.pdf
J. S. Griffith "Mathematics of Cellular Control Processes" J. Theoret. Biol. (1968)

Thanks in advance.