I've recently been thinking about filters where the feedback loop does not have a delay and have found as long as the filter is linear, deriving a transfer function is fairly trivial. Today I thought I'd try and calculate the output of a sinusoidal oscillator where the output is passed through a gain and bias before being sent to its own frequency control, again with no delay.

While it was fairly easy to write this implicitly

$$y\left(t\right)=\sin\left(2\pi\int_{0}^{t}\left(g\cdot y\left(\tau\right)+b\right)d\tau\right)$$

It's been much harder, at least were my maths is at, to isolate the output $y\left(t\right)$.

After stumbling around for a while I've managed to get to this point

$$y\left(t\right)=2\cdot\frac{\sqrt{1-\left(\frac{g}{b}\right)^{2}}\tan\left(\pi\cdot b\cdot\sqrt{1-\left(\frac{g}{b}\right)^{2}}\cdot t\right)-\frac{g}{b}}{\left(\sqrt{1-\left(\frac{g}{b}\right)^{2}}\tan\left(\pi\cdot b\cdot\sqrt{1-\left(\frac{g}{b}\right)^{2}}\cdot t\right)-\frac{g}{b}\right)^{2}+1}$$

I think this is almost right but I've noticed that firstly, it doesn't start from zero. I think this is because I ended up converting my original equation to a differential equation at some point and lost the integral limits but I'm not sure how to fix this.

The second issue is that it definitely doesn't work when the gain exceeds the bias, $g\gt{b}$, i.e. Through Zero Frequency Modulation.

Is there a more general equation that intersects the origin or starts with a set phase at $t=0$? Is there one that can do TZFM and if so without going via complex numbers? If so how exactly would I go about deriving it?