On proving the absence of limit cycles in a dynamical system

Question

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.

Answer 1

$$ \dot{X} = \dot{M} = \frac{1}{1+(E_0 + Y)^m } -a(M_0 +X) = $$ $$ \frac{1}{1+E_0^m + mE_0^{m-1} Y + O(Y^2) } -aM_0 -aX = $$ $$ \frac{1}{1+E_0^m} \left( 1-\frac{mE_0^{m-1} }{1+E_0^m } Y + O(Y^2) \right) -aM_0 -aX $$ 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.