This problem is a subpart of question 8.1.14 in book Nonlinear Dynamics by Strogatz. Here we have 2 differential equations:
$\dot{x_1}=-x_1+F(I-bx_2)$
$\dot{x_2}=-x_2+F(I-bx_1)$
where $F(x)=\frac{1}{1+e^{-x}}$
$I$ and $b$ are positive numbers
We always have a symmetrical fixed point $x_1=x_2=x_\alpha$. This $x_\alpha$ satisfies the following relation: $x=\frac{e^I}{e^I+e^{bx}}$. It is to be established that this symmetric fixed point loses stability at high values of b in a pitchfork bifurcation.
I tried computing the jacobian at the symmetric fixed point, but faced problems in predicting the value of determinant of jacobian at the symmetric fixed point as the parameter b is increased. This is so because:
$\Delta=1-b^2x_\alpha^4e^{-2I}e^{2bx_\alpha}$. where $\Delta$ is determinant of jacobian at symmetric fixed state. While $b$ increases, the problem is that $x_\alpha$ decreases. Hence determination of sign of determinant requires order of decrease of transcendental solution $x_\alpha$ with $b$.
Any help is appreciated.