Electron. J. Diff. Equ., Vol. 2013 (2013), No. 195, pp. 1-8.

Limit cycles for discontinuous generalized Lienard polynomial differential equations

Jaume Llibre, Ana Cristina Mereu

Abstract:
We divide $\mathbb{R}^2$ into sectors $S_1,\dots ,S_l$, with $l>1$ even, and define a discontinuous differential system such that in each sector, we have a smooth generalized Lienard polynomial differential equation $\ddot{x}+f_i(x)\dot{x} +g_i(x)=0$, $i=1, 2$ alternatively, where $f_i$ and $g_i$ are polynomials of degree n-1 and m respectively. Then we apply the averaging theory for first-order discontinuous differential systems to show that for any $n$ and $m$ there are non-smooth Lienard polynomial equations having at least max{n,m} limit cycles. Note that this number is independent of the number of sectors.

Submitted May 7, 2013. Published September 3, 2013.
Math Subject Classifications: 34C29, 34C25, 47H11.
Key Words: Limit cycles; non-smooth Lienard systems; averaging theory.

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Jaume Llibre
Departament de Matematiques
Universitat Autonoma de Barcelona
08193 Bellaterra, Barcelona, Catalonia, Spain
email: jllibre@mat.uab.cat
Ana Cristina Mereu
Department of Physics, Chemistry and Mathematics
UFSCar 18052-780, Sorocaba, SP, Brazil
email: anamereu@ufscar.br

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