Algebraic invariant curves for the Liénard equation

Żoła̧dek, Henryk

doi:10.1090/S0002-9947-98-02002-9

Algebraic invariant curves for the Liénard equation
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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