Rapidly and slowly oscillating periodic oscillations of a delayed van der Pol oscillator

In this paper, we introduce a method to prove existence of several rapidly and slowly oscillating periodic oscillations of a delayed van der Pol oscillator. The proof is a combination of careful pen and paper analytic estimates, the contraction mapping theorem and a computer program using interval arithmetic. Using this approach we extend some general existence results obtained by Nussbaum in [Ann. Mat. Pura Appl., 4 (101), 263--306 (1974)].