What are the means to consider that a specific function is phase of an oscillator? I hope the experts of the field forgive me for this n00b questions, but I am just trying to understand physics. Assume the following function:
$$\phi(t)=\omega t+\cos(\omega t)$$
The above function has the property that:
$$\phi(t+T)-\phi(t)=\omega T=const.$$
Where $T=\frac{2\pi}{\omega}$.
Now the question is based on these facts, may I consider that $\phi$ actually represents the phase of an oscillator??
NOTE: I know my question is a bit silly because I am not modelling any kind of real physical phenomenon but only introducing a function, but I imagine the function I have written is simple case of an oscillator perturbed by means of a second oscillator.
 A: You can have any function be the phase of an oscillator.  Whatever $f(t)$ is, you can speak of $\cos (f(t))$  Your particular expression would make particular sense for phase modulation of a wave.  The carrier wave is at frequency $\omega$.  Normally the phase modulation would have smaller amplitude than $1$ and probably a lower frequency than $\omega$, but we can plot $\cos(x+\cos(x))$ and get something that is nicely periodic, though distorted from a standard cosine wave.

A: This sort of thing actually shows up both in the engineering aspects of FM radio, as well as in semiclassical representations of intermode coupling in molecular Raman scattering.
Since $\phi$ is periodic, trigonometric functions of $\phi$ such as $\cos(\omega t+\cos(\omega t))$ admit a Fourier series expansion obtainable by the Jacobi-Anger identity. The sidebands are used in FM to transmit information, and in Raman scattering they represent the interaction between two oscillators.
A: For the future readers (if any), I highly suggest the Synchronization book. In page 31 of the book there is a thorough discussion of the phase. I am not quite sure writing the books material would be a copyright infringement but I highly recommend it to the ones who are curious to learn more.
