## Algebraic invariant curves for the Liénard equation

HTML articles powered by AMS MathViewer

- by Henryk Żoła̧dek PDF
- Trans. Amer. Math. Soc.
**350**(1998), 1681-1701 Request permission

## Abstract:

Odani has shown that if $\deg g\leq \deg f$ then after deleting some trivial cases the polynomial system $\dot {x}=y, \dot {y}=-f(x)y-g(x)$ does not have any algebraic invariant curve. Here we almost completely solve the problem of algebraic invariant curves and algebraic limit cycles of this system for all values of $\deg f$ and $\deg g$. We give also a simple presentation of Yablonsky’s example of a quartic limit cycle in a quadratic system.## References

- V. I. Arnol′d and Yu. S. Il′yashenko,
*Ordinary differential equations*, Current problems in mathematics. Fundamental directions, Vol. 1, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 7–149, 244 (Russian). MR**823489** - D. Cerveau and R. Moussu,
*Groupes d’automorphismes de $(\textbf {C},0)$ et équations différentielles $ydy+\cdots =0$*, Bull. Soc. Math. France**116**(1988), no. 4, 459–488 (1989) (French, with English summary). MR**1005391**, DOI 10.24033/bsmf.2108 - Kenzi Odani,
*The limit cycle of the van der Pol equation is not algebraic*, J. Differential Equations**115**(1995), no. 1, 146–152. MR**1308609**, DOI 10.1006/jdeq.1995.1008 - Odani K. The integration of polynomial Liénard system in elementary functions (preprint). (1995).
- Stróżyna E. and Żoła̧dek H. The analytic normal form for the nilpotent singularity (preprint). (1996).
- J. C. Wilson,
*Algebraic periodic solutions of $\ddot x+f(x)\dot x+g(x)=0$*, Contributions to Differential Equations**3**(1964), 1–20. MR**159989** - A. I. Jablonskiĭ,
*On limit cycles of a certain differential equation*, Differencial′nye Uravnenija**2**(1966), 335–344 (Russian). MR**0193318** - Henryk Żołądek,
*The classification of reversible cubic systems with center*, Topol. Methods Nonlinear Anal.**4**(1994), no. 1, 79–136. MR**1321810**, DOI 10.12775/TMNA.1994.024 - Henryk Żołądek,
*Quadratic systems with center and their perturbations*, J. Differential Equations**109**(1994), no. 2, 223–273. MR**1273302**, DOI 10.1006/jdeq.1994.1049

## Additional Information

**Henryk Żoła̧dek**- Affiliation: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
- Email: zoladek@mimuw.edu.pl
- Received by editor(s): April 10, 1995
- Received by editor(s) in revised form: August 26, 1996
- Additional Notes: Supported by Polish KBN Grant No 2 P03A 022 08
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**350**(1998), 1681-1701 - MSC (1991): Primary 34C05, 58F21
- DOI: https://doi.org/10.1090/S0002-9947-98-02002-9
- MathSciNet review: 1433130