Product of Two systems with the same asymptotically stable fixed points

Question

I am trying to figure out the nature of a new dynamical system that is equal to the product of two dynamical systems with the same asymptotically stable fixed point. For instance, if i have $x' = f(x)$ and $x' = g(x)$ and both have the same fixed point, what could be said about the nature of the fixed point for system $x' = g(x)f(x)$?

Answer 1

The equilibrium of the resulting system is unstable. Let $x^*$ an equilibrium of the two systems. Then, due to the asymptotic stability assumption of $x^*$ there is some $\epsilon>0$ such that

i) $f(x)<0$ and $g(x)<0$ for all $x\in (x^*,x^*+\epsilon)$

ii) $f(x)>0$ and $g(x)>0$ for all $x\in (x^*-\epsilon,x^*)$

This means that $f(x)g(x)>0$ for all $x\in (x^*-\epsilon,x^*+\epsilon)$ except $x^*$ and thus all trajectories starting from $x^*+\epsilon_0$ with $\epsilon_0>0$ will grow away from $x^*$.