A family of periodic orbits for the extended Hamiltonian system of the Van der Pol oscillator (Q2107525)

The authors present an analytical study of periodic orbits of the system \[ \dot{x}=p_y, \quad \dot{y}=p_x-\mu (1-x^2)y , \] \[ \dot{p_x}=-y-\mu 2xyp_y, \quad \dot{p_y}=-x+\mu (1-x^2)p_y, \] with the Hamiltonian \(H=p_xp_y+xy-\mu (1-x^2)yp_y.\) The averaging theory is applied to prove the existence of new periodic orbits lying on the energy level \(H=h\) for sufficiently small parameters \(\mu\). As \(\mu \to 0\), the orbits tend to a periodic orbit which is a solution of the extended Van der Pol oscillator with \(\mu =0\).