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Quadratic systems with a symmetrical solution.
- Source :
-
Electronic Journal of Qualitative Theory of Differential Equations . 2018, p1-18. 18p. - Publication Year :
- 2018
-
Abstract
- In this paper we study the existence and uniqueness of limit cycles for socalled quadratic systems with a symmetrical solution: ... where (x, y) ∊ R², t ∊ R, aij, bij ∊ R, i.e. a real planar system of autonomous ordinary differential equations with linear and quadratic terms in the two independent variables. We prove that a quadratic system with a solution symmetrical with respect to a line can be of two types only. Either the solution is an algebraic curve of degree at most 3 or all solutions of the quadratic system are symmetrical with respect to this line. For completeness we give a new proof of the uniqueness of limit cycles for quadratic systems with a cubic algebraic invariant, a result previously only available in Chinese literature. Together with known results about quadratic systems with algebraic invariants of degree 2 and lower, this implies the main result of this paper, i.e. that quadratic systems with a symmetrical solution have at most one limit cycle which if it exists is hyperbolic. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 14173875
- Database :
- Academic Search Index
- Journal :
- Electronic Journal of Qualitative Theory of Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 133435182
- Full Text :
- https://doi.org/10.14232/ejqtde.2018.1.32