Algebraic invariant curves for the Liénard equation. (English) Zbl 0895.34026
Summary: K. Odani [J. Differ. Equations 115, No. 1, 146-152 (1995; Zbl 0816.34023)] 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 a simple presentation of A. I. Yablonskij’s [Differ. Equations 2 (1966); translation from Differ. Uravn. 2, 335-344 (1966; Zbl 0173.34603)] example of a quartic limit cycle in a quadratic system.
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
References:
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