9
$\begingroup$

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:

$\endgroup$
12
$\begingroup$

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 (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.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.