When something overdamps, it is allegedly an exponential curve. But, the differential equation for harmonic oscillators is
$m y''+k y'+ay = 0$ with the damping coefficient as $ \lambda = a/(2m)$. So, what conditions of these parameters some how cause 0 oscillation at all? Because I thought that in the real world there is always going to be some small oscillation and that it's only an approximation to assume otherwise.
The oscillator is overdamped if the characteristic polynomial has only real roots, i.e. $k^2 - 4 a m \ge 0$. To the extent that the "real world" is modelled by this differential equations, there will then be no oscillation: you can make $y$ overshoot $0$ with an appropriate initial condition, but it won't cross $0$ again.