On the motion of a generalized van der Pol oscillator

A generalized van der Pol oscillator is considered, with positive real power nonlinearities in the restoring and damping force, including fractional powers. An analytical approach based on the Krylov–Bogoliubov method is adjusted to derive analytical expressions for the amplitude of a limit cycle for small values of the damping coefficient. These expressions are also derived for some integer power nonlinearities in the equation of motion and the results obtained compared with the existing results from the literature. Relaxation oscillations are studied for larger values of the damping coefficient. Matched asymptotic expansions are used and the influence of the powers of the restoring and damping force on the period of these oscillations is investigated. It is shown that not only can the period increase with the damping power, but it can also have a decreasing trend for some cases and the condition for this to hold is obtained.