How to prove that a solution for a set of ODE's is symmetric and unique

Question

Consider the following set of ODE's:

$\frac{dx}{dt}=-x+\frac{1}{1+ae^{-by}}$

$\frac{dy}{dt}=-y+\frac{1}{1+ae^{-bx}}$

how can I mathematically prove that there is a symmetric fixed point at $x=y$, and that the solution for this point is unique?

Answer 1

A fixed point means $\frac{dx}{dt}=\frac{dy}{dt}=0$. If you search for a fixed point on $x=y$, you get $$-x+\frac{1}{1+a e^{-bx}}=0$$ You can plug in values at $\pm\infty$ and you see that the function on the left side of the equation changes sign. Since it's continuous, it means it has at least one solution. With suitable $a$ and $b$, I can get more than one, so it's not unique. Try $a=1000$ and $b=20$.