PERSISTENCE OF PERIODIC TRAJECTORIES OF PLANAR SYSTEMS UNDER TWO PARAMETRIC PERTURBATIONS

Title & Authors
PERSISTENCE OF PERIODIC TRAJECTORIES OF PLANAR SYSTEMS UNDER TWO PARAMETRIC PERTURBATIONS

Abstract
We consider a two parametric family of the planar systems with the form $\small{\dot{x}=P(x,\;y)+{\in}_1p_1(x,\;y)+{\in}_2p_2(x,\;y)}$, $\small{\dot{y}=Q(x,\;y)+{\in}_1p_1(x,\;y)+{\in}_2p_2(x,\;y)}$, where the unperturbed equation($\small{{\in}_1={\in}_2=0}$) 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;
Language
English
Cited by
1.
EXTENSION OF CHICONE'S METHOD FOR PERTURBATION SYSTEMS OF THREE PARAMETERS WITH APPLICATION TO THE LIENARD SYSTEM, International Journal of Bifurcation and Chaos, 2012, 22, 03, 1250065
2.
Geometric Method for Free Oscillator under Two Parametric Perturbation, Journal of Dynamical Systems and Geometric Theories, 2011, 9, 1, 49
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