Geodesic
On the center of the generalized Liénard system
Czechoslovak Mathematical Journal, Tome 52 (2002) no. 4, pp. 817-832 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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].
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].
Classification : 34C05, 34C25
Keywords: generalized Liénard system; local center; global center; the differetial inequality theorem; the first approximation 
@article{CMJ_2002_52_4_a13,
     author = {Zhao, Cheng-Dong and He, Qi-Min},
     title = {On the center of the generalized {Li\'enard} system},
     journal = {Czechoslovak Mathematical Journal},
     pages = {817--832},
     year = {2002},
     volume = {52},
     number = {4},
     mrnumber = {1940062},
     zbl = {1021.34023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2002_52_4_a13/}
}
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
SP  - 817
EP  - 832
VL  - 52
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CMJ_2002_52_4_a13/
LA  - en
ID  - CMJ_2002_52_4_a13
ER  - 
%0 Journal Article
%A Zhao, Cheng-Dong
%A He, Qi-Min
%T On the center of the generalized Liénard system
%J Czechoslovak Mathematical Journal
%D 2002
%P 817-832
%V 52
%N 4
%U http://geodesic.mathdoc.fr/item/CMJ_2002_52_4_a13/
%G en
%F CMJ_2002_52_4_a13
Zhao, Cheng-Dong; He, Qi-Min. On the center of the generalized Liénard system. Czechoslovak Mathematical Journal, Tome 52 (2002) no. 4, pp. 817-832. http://geodesic.mathdoc.fr/item/CMJ_2002_52_4_a13/

[1] Shu-Xiang Yu and Ji-Zhou Zhang: On the center of the Liénard equation. J.  Differential Equations 102 (1993), 53–61. | DOI | MR

[2] Yu-Rong Zhou and Xiang-Rong Wang: On the conditions of a center of the Liénard equation. J.  Math. Anal. Appl. 100 (1993), 43–59. | DOI | MR

[3] P. J.  Ponzo and N.  Wax: On periodic solutions of the system $\dot{x}=y-F(x)$, $\dot{y}=-g(x)$. J.  Differential Equations 10 (1971), 262–269. | DOI | MR

[4] Jitsuro Sugie: The global center for the Liénard system. Nonlinear Anal. 17 (1991), 333–345. | DOI | MR

[5] T.  Hara and T.  Yoneyama: On the global center of generalized Liénard equation and its application to stability problems. Funkc. Ekvacioj 28 (1985), 171–192. | MR

[6] Lawrence Perko: Differential Equations and Dynamical Systems. Springer-Verlag, New York, 1991. | DOI | MR