On the center of the generalized Liénard system

Zhao, Cheng Dong, and He, Qi-Min. "On the center of the generalized Liénard system." Czechoslovak Mathematical Journal 52.4 (2002): 817-832. <http://eudml.org/doc/30747>.

@article{Zhao2002,
abstract = {In this paper, we discuss the conditions for a center for the generalized Liénard system \[ \frac\{\{\rm d\}x\}\{\{\rm d\}t\}=\varphi (y)-F(x), \qquad \frac\{\{\rm d\}y\}\{\{\rm d\}t\}=-g(x), \] or \[ \frac\{\{\rm d\}x\}\{\{\rm d\}t\}=\psi (y), \qquad \frac\{\{\rm dy\}\}\{\{\rm d\}t\}= -f(x)h(y)-g(x), \] with $f(x)$, $g(x)$, $\varphi (y)$, $\psi (y)$, $h(y)\: \mathbb \{R\}\rightarrow \mathbb \{R\}$, $F(x)=\int _0^xf(x)\mathrm \{d\}x$, and $xg(x)>0$ for $x\ne 0$. By using a different technique, that is, by introducing auxiliary systems and using the differential inquality theorem, we are able to generalize and improve some results in [1], [2].},
author = {Zhao, Cheng Dong, He, Qi-Min},
journal = {Czechoslovak Mathematical Journal},
keywords = {generalized Liénard system; local center; global center; the differetial inequality theorem; the first approximation; generalized Liénard system; local center; global center; the differential inequality theorem; the first approximation},
language = {eng},
number = {4},
pages = {817-832},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the center of the generalized Liénard system},
url = {http://eudml.org/doc/30747},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Zhao, Cheng Dong
AU - He, Qi-Min
TI - On the center of the generalized Liénard system
JO - Czechoslovak Mathematical Journal
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 4
SP - 817
EP - 832
AB - In this paper, we discuss the conditions for a center for the generalized Liénard system \[ \frac{{\rm d}x}{{\rm d}t}=\varphi (y)-F(x), \qquad \frac{{\rm d}y}{{\rm d}t}=-g(x), \] or \[ \frac{{\rm d}x}{{\rm d}t}=\psi (y), \qquad \frac{{\rm dy}}{{\rm d}t}= -f(x)h(y)-g(x), \] with $f(x)$, $g(x)$, $\varphi (y)$, $\psi (y)$, $h(y)\: \mathbb {R}\rightarrow \mathbb {R}$, $F(x)=\int _0^xf(x)\mathrm {d}x$, and $xg(x)>0$ for $x\ne 0$. By using a different technique, that is, by introducing auxiliary systems and using the differential inquality theorem, we are able to generalize and improve some results in [1], [2].
LA - eng
KW - generalized Liénard system; local center; global center; the differetial inequality theorem; the first approximation; generalized Liénard system; local center; global center; the differential inequality theorem; the first approximation
UR - http://eudml.org/doc/30747
ER -