The generalized Liénard systems

Mats Gyllenberg; Yan Ping

doi:10.3934/dcds.2002.8.1043

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2002, Volume 8, Issue 4: 1043-1057. Doi: 10.3934/dcds.2002.8.1043

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The generalized Liénard systems

Mats Gyllenberg^1, and
Yan Ping^1,

1.
Department of Mathematics, University of Turku, FIN-20014 Turku

Received: May 31, 2001

Revised: April 30, 2002

Published: September 2002

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Abstract

We consider the generalized Liénard system

$\frac{dx}{dt} = \frac{1}{a(x)}[h(y)-F(x)],$

$\frac{dy}{dt}= -a(x)g(x),\qquad\qquad\qquad\qquad\qquad$ (0.1)

where $a$ is a positive and continuous function on $R=(-\infty, \infty)$, and $F$, $g$ and $h$ are continuous functions on $R$. Under the assumption that the origin is a unique equilibrium, we obtain necessary and sufficient conditions for the origin of system (0.1) to be globally asymptotically stable by using a nonlinear integral inequality. Our results substantially extend and improve several known results in the literature.

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Mathematics Subject Classification: 34D05, 34C05.

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