Time-reversibility and invariants of some 3-dim systems. (Time-reversibility and ivariants of some 3-dim systems.) (English) Zbl 1527.34059
Summary: We study time-reversibility and invariants of the group of transformations \(x \to x\), \(y \to \alpha y\), \(z \to \alpha^{-1} z\) for three-dimensional polynomial systems with \(0:1:-1\) resonant singular point at the origin. An algorithm to find the Zariski closure of the set of time-reversible systems in the space of parameters is proposed. The interconnection of time-reversibility and invariants of the group mentioned above is discussed.
MSC:
| 34C14 | Symmetries, invariants of ordinary differential equations |
| 15A72 | Vector and tensor algebra, theory of invariants |
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
| 34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |
| 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 |
| 37C05 | Dynamical systems involving smooth mappings and diffeomorphisms |
| 37C10 | Dynamics induced by flows and semiflows |
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| [16] | Tatjana Petek Faculty of Electrical Engineering and Computer Science, University of Maribor, Koroška cesta 46, SI-2000 Maribor, Slovenia Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia E-mail: tatjana.petek@um.si |
| [17] | Valery G. Romanovski Faculty of Electrical Engineering and Computer Science, University of Maribor, Koroška cesta 46, SI-2000 Maribor, Slovenia Center for Applied Mathematics and Theoretical Physics, Mladinska 3, SI-2000 Maribor, Slovenia Faculty of Natural Science and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia E-mail: valerij.romanovskij@um.si |
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