I'm trying to analyze the a differential equation of the form:
$$ \dot{x} = -K_1\sin(\omega t + x) - K_2\sin(x) $$
where $K_1 > 0$, $K_2 > 0$, and $\omega >0$ are positive real constants and $x=x(t) \in \mathbb{R}$. If this equation did not depend on $t$, I can see immediately how I would identify the fixed points of this system and proceed with the usual stability analysis techniques. However, the dependence of $\dot{x}$ on $t$ throws me off. Is there a general method of dealing with time-varying systems like this? I would appreciate any references.
In my search, one method I found involved treating $t$ as an independent state variable, generating another equation. However, I don't see how this helps analysis.
Ultimately what I'm trying to accomplish is to determine how varying the constants can result in bifurcations (specifically, I can observe a stable equilibrium point for certain $K_1$ and $K_2$ and a limit cycle for other choices). I can confirm their presence through simulations, but I would like to obtain something more analytical.
