It is well-known that a small delay may or may not cause stable equilibrium to become unstable. Can anyone help that if for $\tau=0$ the equilibrium solution is unstable and if $\tau>0$ is there a chance it will become stable?
For some sites or article I read they always considering stable first for $\tau=0$ but I haven't read unstable first for $\tau=0$.
Not sure if this exactly answers your question, but the idea is generally to find equilibirum points, then to study the intervals around those points (see for instance (http://www.scholarpedia.org/article/Basin_of_attraction)
If your starting point is not stable to begin with, your solution with either diverge, or converge to an other stable point. For example the pendulum in $\theta = \frac{\pi}{2}$ unstable, $-\frac{\pi}{2}$ stable.