## Relatório de pesquisa 16/08

Limit Cycles of the Generalized Polynomial Liénard Differential Equations, Jaume Llibre, Ana Cristina Mereu and Marco Antonio Teixeira, submitted Sept. 04.

Abstract
We apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m greater than or equal to 1 there are
differential equations of the form  $\ddot{x}+f(x) \dot{x}+g(x)=0$, with f and g polynomials of degree n and m respectively, having at least [(n+m-1)/2] limit cycles, where [.] denotes the integer part function.

Mathematics Subject Classifications (2000):  34C29, 34C25, 47H11.

Keywords:
limit cycle, periodic orbit, Liénard equation, averaging theory.

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September 04, 2008

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