Journal of the
Korean Mathematical Society JKMS

We consider a two parametric family of the planar systems with the form \begin{eqnarray*} \dot{x}&=&P(x,y)+\epsilon_1 p_1(x,y)+\epsilon_2 p_2(x,y),\\ \dot{y}&=&Q(x,y)+\epsilon_1 q_1(x,y)+\epsilon_2 q_2(x,y), \end{eqnarray*} where the unperturbed equation $(\epsilon_1=\epsilon_2=0)$ is assumed to have at least one periodic solution or limit cycle. Our aim here is to study the behavior of the system under two parametric perturbations; in fact, using the Poincare - Andronov technique, we impose conditions on the system which guarantee persistence of the periodic trajectories. At the end, we apply the result on the Van der Pol equation; where, we consider the effect of nonlinear damping on the equation. Also the Hopf bifurcation for the Van der Pol equation will be investigated.

Keywords: periodic trajectory, Poincare map, perturbation, Van der Pol

MSC numbers: Primary 34C25; Secondary 34E