Slowly oscillating solutions of a Van der Pol oscillator with state-dependent delay

In this paper, we consider a Van der Pol oscillator with state-dependent delay. First, we seek a suitable set E on which we construct a Poincaré operator F . Then, we prove that the operator F is continuous and compact on E  \ {0}. After that, we obtain the ejectivity of the trivial fixed point {0} and the existence of slowly oscillating solution. Finally, we choose an example of state-dependant delay to validate our analytical results through comparison with numerical continuation. These theoretical results generalize and improve some existing results.