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Say you have a linear chirp, which is a bit like a sinusoid with a gradually increasing period, but instead of the linearly increasing period, could you pick an arbitrary value, like the red line in the image below?
How would you write the definition of this? Say that the period variable is called AP (arbitrary period).
For the linear chirp I refer to wikipedia.
If I understand correctly, you are trying to create a chirp an arbitrary instantaneous frequency, $f(t)$.
In that case, the phase of the chirp is given by $\varphi(t) = 2 \pi \int f(t)dt$ and your signal in time domain is given by $s(t) = \sin(\varphi(t))$.
For example, in case of a linear chirp you mentioned, $f(t) = f_0 + \beta t$ (where $\beta$ is the sweep rate) and therefore $s(t) = \sin \left(2 \pi (f_0 t+\frac{1}{2}\beta t^2) \right)$.
