In a physical problem I need to investigate the following nonlinear differential equation


$$\ddot x+\omega^2\left (1+\frac{m^2\dot x^2}{p^2}\right)x=0,$$


where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found the following:


* Ronald E. Mickens, [Generalized harmonic oscillators](https://doi.org/10.1006/jsvi.2000.2989), Journal of Sound and Vibration, Volume 236, Issue 4, Pages 730-732, 28 September 2000.



Nonlinear oscillator with velocity-dependent frequency
This type of ODE,
$$\ddot{x}+f(x)\dot{x}^2+g(x)=0$$
is known as a Liénard equation of the second kind. It has been studied for example in [Monotonicity of the period function of the Liénard equation of second kind](https://arxiv.org/abs/1608.02910) (2016).


A particular case with nice properties is
$$\ddot{x}-\frac{f'(x)}{f(x)}\dot{x}^2+f(x)\int\_0^x\frac{1}{f(u)}du=0,$$
see [Design of nonlinear isochronous oscillators.](https://link.springer.com/article/10.1007/s11071-015-1972-x)


