Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 195, pp. 1-8.
Title: Limit cycles for discontinuous generalized Lienard
polynomial differential equations
Authors: Jaume Llibre (Univ.Autonoma de Barcelona, Catalonia, Spain)
Ana Cristina Mereu (UFSCar, Sorocaba, SP, Brazil)
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 03, 2013.
Math Subject Classifications: 34C29, 34C25, 47H11.
Key Words: Limit cycles; non-smooth Lienard systems; averaging theory.