| Title:
|
On the center of the generalized Liénard system (English) |
| Author:
|
Zhao, Cheng-Dong |
| Author:
|
He, Qi-Min |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
52 |
| Issue:
|
4 |
| Year:
|
2002 |
| Pages:
|
817-832 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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]. (English) |
| Keyword:
|
generalized Liénard system |
| Keyword:
|
local center |
| Keyword:
|
global center |
| Keyword:
|
the differetial inequality theorem |
| Keyword:
|
the first approximation |
| MSC:
|
34C05 |
| MSC:
|
34C25 |
| idZBL:
|
Zbl 1021.34023 |
| idMR:
|
MR1940062 |
| . |
| Date available:
|
2009-09-24T10:57:07Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127767 |
| . |
| Reference:
|
[1] Shu-Xiang Yu and Ji-Zhou Zhang: On the center of the Liénard equation.J. Differential Equations 102 (1993), 53–61. MR 1209976, 10.1006/jdeq.1993.1021 |
| Reference:
|
[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. MR 1250276, 10.1006/jmaa.1993.1381 |
| Reference:
|
[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. MR 0288360, 10.1016/0022-0396(71)90050-7 |
| Reference:
|
[4] Jitsuro Sugie: The global center for the Liénard system.Nonlinear Anal. 17 (1991), 333–345. MR 1123207, 10.1016/0362-546X(91)90075-C |
| Reference:
|
[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 0816825 |
| Reference:
|
[6] Lawrence Perko: Differential Equations and Dynamical Systems.Springer-Verlag, New York, 1991. MR 1083151, 10.1007/978-1-4684-0392-3 |
| . |
