This investigation is concerned with a nonlinear differential equation that governs oscillations having infinitely many limit cycles. Bounds and approximations for the corresponding limit amplitudes are obtained from suitable estimates of first integrals.

This paper deals with the “modified van der Pol differential equations” q¨ − (sgn q˙) (δ/2)q˙ 2 h(q) + κ 2 f(q) = 0, where h(q) and f(q) are suitable functions. It is shown that there exists a unique limit cycle. For the limit amplitude both lower and upper bounds are established in the case of unrestricted values δ. For small values δ the limit amplitude can be calculated immediately.