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Vietnam Journal of Mathematics 38:2 (2010) 203-210  

Continuation of Periodic Solutions for a Class of Lienard Equations under Two Parametric Nonautonomous Perturbations

Z. Afsharnezhad and M. Karimi Amaleh

Abstract.  We consider the two parametric family of perturbed Lienard equations $$\ddot{x}+f(x)\dot{x}+g(x,\dot{x},t,\varepsilon)=0.\qquad\qquad\qquad\qquad (*)$$ Here $\varepsilon$ is a parameter and $f(x),\ g(x,\dot{x},t,\varepsilon)$ are polynomials with respect to x, y and Cr, r > 1 with respect to t. Equation (*)is an effect of nonlinear forcing on the Lienard equation. Our aim is to show the persistence of periodic solutions of Lienard equation (if there is any) under perturbations. Therefore first we find some condition under which the Lienard equation has at least one periodic orbit, and then we investigate the persistence of the periodic orbit under perturbation (equation (*)). The techniques that we use are techniques of Chicone and Melnikov

2000 Mathematics Subject Classification: 34A36, 34C27, 37G15.

Keywords: Perturbation, periodic orbit, persistency, bifurcation map.

 

 

 

 

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