Transcendental limit cycles via the structure of arbitrary degree invariant algebraic curves of polynomial planar vector fields. (English) Zbl 1085.34024
This work deals with planar polynomial differential systems and their invariant algebraic curves. The existence of invariant algebraic curves for a differential system allows a better understanding of its dynamical behavior. The author gives a complete, clear and useful explanation of this fact in the introductory section, full of significant citations. The main results and ideas that come from and bring to the study of invariant algebraic curves for planar polynomial differential systems are stated. Besides this explanation, the bibliography contains the most important works related with this topic.
The main result of this paper is a characterization of the structure of invariant algebraic curves in terms of the corresponding planar polynomial differential system. An algorithmic method to determine the existence or nonexistence of invariant algebraic curves for a given polynomial differential system is deduced from this characterization.
Using this method, two examples of planar polynomial differential systems are considered. One of the examples corresponds to the van der Pol oscillator, for which the nonexistence of invariant algebraic curves is proved. In particular, the limit cycle exhibited by this differential system is shown to be nonalgebraic, recovering in this easy way a previous result by Odani. The other example comes from a differential system studied by Dolov also with a limit cycle, which is shown to be nonalgebraic.
The main result of this paper is a characterization of the structure of invariant algebraic curves in terms of the corresponding planar polynomial differential system. An algorithmic method to determine the existence or nonexistence of invariant algebraic curves for a given polynomial differential system is deduced from this characterization.
Using this method, two examples of planar polynomial differential systems are considered. One of the examples corresponds to the van der Pol oscillator, for which the nonexistence of invariant algebraic curves is proved. In particular, the limit cycle exhibited by this differential system is shown to be nonalgebraic, recovering in this easy way a previous result by Odani. The other example comes from a differential system studied by Dolov also with a limit cycle, which is shown to be nonalgebraic.
Reviewer: Maite Grau (Lleida)
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
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