Persistence of Periodic Trajectories of Planar Systems under two Parametric Perturbations
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Abstract
We consider a two parametric family of the planar systems with theformbegin{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 oneperiodic solution or limit cycle. Our aim here is to study thebehavior of the system under two parametric perturbations; infact, using the Poincare - Andronov technique, we imposeconditions on the system which guarantee persistence of theperiodic trajectories. At the end, we apply the result on the Vander Pol equation ; where, we consider the effect of nonlineardamping on the equation. Also the Hopf bifurcation for the Vander Pol equation will be investigated.
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