filters - Separate Complicated Signal into Exponential Components

This morning, I asked a similar question about separating simple signals - with constant frequency - into exponential components. User @AndreasH suggested that a hilbert transform can do this as follows:

$$\cos(\omega_1 t) + j ~\textrm{Hilbert}[\cos(\omega_1 t)] = e^{j\omega_1 t}$$

This works great when $\omega_1$ is a real constant, and when time is linear. Is there a way to do it for a more complicated signal? For example,

$$x(t)=\cos(\omega_o t^2 )$$

When I perform a hilbert transform on this one, I get something with a lot of error functions.

How can I get $y(t)=e^{j\omega_o t^2}$ from $x(t)$?