On the nonexistence of Liouvillian first integrals for generalized Liénard polynomial differential systems

被引：0

作者：

Guillaume Chèze

Thomas Cluzeau

机构：

[1] Université Paul Sabatier Toulouse 3,Institut de Mathématiques de Toulouse

[2] CNRS UMR 5219,undefined

[3] Université de Limoges,undefined

[4] CNRS,undefined

[5] XLIM UMR 7252,undefined

[6] DMI,undefined

来源：

Journal of Nonlinear Mathematical Physics | 2013年 / 20卷

关键词：

Polynomial vector fields; first integrals; invariant algebraic curves; Liénard polynomial differential systems; 37K10; 34D30; 34C26;

D O I：

暂无

中图分类号：

学科分类号：

摘要：

We consider generalized Liénard polynomial differential systems of the form ẋ = y, ẏ = −g(x) − f(x)y, with f(x) and g(x) two polynomials satisfying deg(g) ≤ deg(f). In their work, Llibre and Valls have shown that, except in some particular cases, such systems have no Liouvillian first integral. In this letter, we give a direct and shorter proof of this result.

引用

页码：475 / 479

页数：4

共 12 条

[1]

Liénard A(1928)Étude des oscillations entretenues Revue générale de l’électricité 23 901-912

[2]

Levinson N(1942)A general equation for relaxation oscillations, Duke Math. J 9 382-403

[3]

Smith O K(1997)Number of limit cycles of the Liénard equation Phys Rev 56 3809-3813

[4]

Giacomini H(1995)The limit cycle of the van der Pol equation is not algebraic J. Differential Equations 115 146-152

[5]

Neukirch S(1998)Algebraic invariant curves for the Liénard equation Trans. Amer. Math. Soc 350 1681-1701

[6]

Odani K(2010)Liouvillian first integrals for Liénard polynomial differential systems Proc. Amer. Math. Soc 138 3229-3239

[7]

Żoładek H(1983)Elementary first integrals of differential equations Trans. Amer. Math. Soc 279 215-229

[8]

Llibre J(1992)Liouvillian first integrals of differential equations Trans. Amer. Math. Soc 333 673-688

[9]

Valls C(undefined)undefined undefined undefined undefined-undefined

[10]

Prelle M J(undefined)undefined undefined undefined undefined-undefined

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