IJPAM: Volume 44, No. 1 (2008)

PERIOD ANNULI IN THE LIÉNARD TYPE EQUATION

S. Atslega$^1$, F. Sadyrbaev$^2$
$^1$Department of Mathematics
Daugavpils University
1, Parades Str., Daugavpils, LV-5400, LATVIA
e-mail: oglana@tvnet.lv
$^2$Institute of Mathematics and Computer Sciences
University of Latvia
29, Rainis Blvd., Riga, LATVIA
e-mail: felix@latnet.lv


Abstract.We consider the equation

\begin{displaymath}
x''+ \frac{2x}{1+x^2} x'^2 + g(x)=0, \eqno{(i)}
\end{displaymath}

where $g(x) = - x ( x^2 - p^2) (x^2 - q^2).$ Comparison of phase portraits for equations $(i)$ and

\begin{displaymath}x''+ g(x)=0\eqno{(ii)}\end{displaymath}

is made. We describe decomposition of the first quadrant of the $(q,p)$-plane into regions where equations $(i)$ and $(ii)$ have or have not a period annulus, that is, a set of concentric cycles enclosing several critical points of equivalent planar systems.

Received: February 3, 2008

AMS Subject Classification: 34C05, 58F14

Key Words and Phrases: period annulus, phase portrait, Liénard equation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 44
Issue: 1