Edit:" please see the attached figure. The blue part is all I have. Also, you can neglect the noise term. Assume the signal is deterministic. Typical values of the unknown parameters $\alpha$ and $\omega_0$ are in the the range [0.25,5]. Note that all the parameters are unknown $A \omega_0, \alpha, \phi_0$. The setup hints at curve fitting which I did but don't prefer."
I have a small part from a chirp signal
$$s(t)= A \cos\left(\omega_0 t+ \frac\alpha2 t^2 + \phi_0\right)+n(t)$$
with very low start frequency $\omega_0$ and chirp rate $\alpha$.
The signal is sampled at high sampling frequency $f_s$. But the available time domain signal is short, i.e. barely a complete cycle. The amplitude of the signal and the phase shift are also unknown. Are there ways to figure out the chirp parameters other than curve fitting or spectrogram?