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generalized Liénard system; local center; global center; the differetial inequality theorem; the first approximation
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].
[1] Shu-Xiang Yu and Ji-Zhou Zhang: On the center of the Liénard equation. J.  Differential Equations 102 (1993), 53–61. DOI 10.1006/jdeq.1993.1021 | MR 1209976
[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 10.1006/jmaa.1993.1381 | MR 1250276
[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 10.1016/0022-0396(71)90050-7 | MR 0288360
[4] Jitsuro Sugie: The global center for the Liénard system. Nonlinear Anal. 17 (1991), 333–345. DOI 10.1016/0362-546X(91)90075-C | MR 1123207
[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
[6] Lawrence Perko: Differential Equations and Dynamical Systems. Springer-Verlag, New York, 1991. DOI 10.1007/978-1-4684-0392-3 | MR 1083151
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