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An upper bound for the amplitude of limit cycles in Lienard systems with symmetry

Author:
Yang Lijun  Zeng Xianwu  


Journal:
JOURNAL OF DIFFERENTIAL EQUATIONS


Issue Date:
2015


Abstract(summary):

It is well known that the Lienard system (x) over dot =3D y - F (x), (y) over dot =3D -g(x) with symmetry (i.e. F(x) and g(X) are odd functions) has a unique limit cycle under some hypotheses. In this paper we will show that the unique limit cycle locates in the strip region vertical bar x vertical bar < x*, where x* > 0 is uniquely and directly determined by integral(x)(0)* F(x)g(x)dx =3D 0. In other words, an explicit upper bound x* is given for the amplitude (i.e. the maximal value of the x-coordinate) of the unique limit cycle. As a simple application we obtain a uniform estimate A(mu) < root 5=3D2.2361 for each mu > 0, where A(mu) denotes the amplitude of the unique limit cycle in the Lienard system (x) over dot =3D y - mu(x(3)/3 - x), (y) over dot =3D -x for the van der Pol equation (x) over dot + mu(x(2) - 1)(x) over dot + x =3D 0. The upper bound root 5 ATS improves the existing ones. (C) 2014 Elsevier Inc. All rights reserved.


Page:
2701---2710


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