Two functions $x(t)$ and $y(t)$ are coupled via: $$\dot x(t) = a y(t)-b,y(t+x(t))=x(t)$$
where $a<0$, $b\neq 0$ is some constant.
I am mostly confused with the second equation. What is the mathematical term for this kind of delay differential equation? What kind of initial condition do I need to specify?
It is obvious that, if $x(0)=y(0)=b/a$, then the system will stay stable for all $t>0$.
Through doing some simulation also back of envelope thinking, it seems that when $x(0)=y(0)=0$, then the system may never converge to an equilibrium/stable solution. However, I have no idea what type of proof technique is required to show that.
Thanks!