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.



On proving the absence of limit cycles in a dynamical system
$$
\dot{X} = \dot{M} = \frac{1}{1+(E\_0 + Y)^m } -a(M\_0 +X) =
$$
$$
\frac{1}{1+E\_0^m + mE\_0^{m-1} Y + O(Y^2) } -aM\_0 -aX = 
$$
$$
\frac{1}{1+E\_0^m} \left( 1-\frac{mE\_0^{m-1} }{1+E\_0^m } Y + O(Y^2) \right) -aM\_0 -aX
$$
Now use $1/(1+E\_0^m) =abE\_0 $ as well as $aM\_0 = abE\_0 $, yielding
$$
\dot{X} =-a^2 b^2 m E\_0^{m+1} Y +O(Y^2 ) -aX
$$
as desired.


