# 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)

$$\dot{X} = \dot{M} = \frac{1}{1+(E_0 + Y)^m } -a(M_0 +X) =$$ $$\frac{1}{1+E_0^m + mE_0^{m-1} Y + O(Y^2) } -aM_0 -aX =$$ $$\frac{1}{1+E_0^m} \left( 1-\frac{mE_0^{m-1} }{1+E_0^m } Y + O(Y^2) \right) -aM_0 -aX$$ 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.
• From the second to the third line, I factored out $1/(1+E_0^m )$ and then expanded in a geometric series. Jul 23 '20 at 13:55