MOUELLAH, Boumediene2025-10-282025-10-282025-06-02http//localhost:8080/jspui/handle/123456789/13413In this work, we show a new way of using the averaging theory of first order for studying families of periodic orbits in the extension of the Van der Pol oscillator to a Hamiltonian system of the form (1 ) , 2 , (1 ) , , 2 2 y y x y x y p x x p p y xyp y p x y x p                   where px and py are the conjugate momenta of the variables x and y , respectively and  is a small parameter. In the second part of this work, we provide the maximum number of limit cycles for continuous and discontinuous planar piecewise differential systems formed by linear Hamiltonian saddles and separated either by one or two parallel straight lines.enHamiltonian system, Van der Pol oscillator, Periodic orbit, Differential system, Averaging theory.Thesis