Family of quadratic differential systems with invariant hyperbolas:a complete classification in the space R^12
In this article we consider the class QS of all non-degenerate quadratic systems. A quadratic polynomial differential system can be identified with a single point of R^{12} through its coefficients. In this paper using the algebraic invariant theory we provided necessary and sufficient conditio...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Texas State University
2016-06-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2016/162/abstr.html |
| _version_ | 1819231365814550528 |
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| author | Regilene D. S. Oliveira Alex C. Rezende Nicolae Vulpe |
| author_facet | Regilene D. S. Oliveira Alex C. Rezende Nicolae Vulpe |
| author_sort | Regilene D. S. Oliveira |
| collection | DOAJ |
| description | In this article we consider the class QS of all
non-degenerate quadratic systems. A quadratic polynomial
differential system can be identified with a single point of
R^{12} through its coefficients. In this paper using
the algebraic invariant theory we provided necessary and
sufficient conditions for a system in QS to have at least
one invariant hyperbola in terms of its coefficients. We also
considered the number and multiplicity of such hyperbolas. We
give here the global bifurcation diagram of the class QS
of systems with invariant hyperbolas. The bifurcation diagram is
done in the 12-dimensional space of parameters and it is
expressed in terms of polynomial invariants. The results can
therefore be applied for any family of quadratic systems in this
class, given in any normal form. |
| first_indexed | 2024-12-23T11:43:49Z |
| format | Article |
| id | doaj.art-2fdd1d42398141a684eedf95d9d277af |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-23T11:43:49Z |
| publishDate | 2016-06-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-2fdd1d42398141a684eedf95d9d277af2022-12-21T17:48:24ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912016-06-012016162,150Family of quadratic differential systems with invariant hyperbolas:a complete classification in the space R^12Regilene D. S. Oliveira0Alex C. Rezende1Nicolae Vulpe2 Univ. de Sao Paulo, Brazil Univ. de Sao Paulo, Brazil Academy of Sciences of Moldova, Moldova In this article we consider the class QS of all non-degenerate quadratic systems. A quadratic polynomial differential system can be identified with a single point of R^{12} through its coefficients. In this paper using the algebraic invariant theory we provided necessary and sufficient conditions for a system in QS to have at least one invariant hyperbola in terms of its coefficients. We also considered the number and multiplicity of such hyperbolas. We give here the global bifurcation diagram of the class QS of systems with invariant hyperbolas. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of polynomial invariants. The results can therefore be applied for any family of quadratic systems in this class, given in any normal form.http://ejde.math.txstate.edu/Volumes/2016/162/abstr.htmlQuadratic differential systemsinvariant hyperbolaaffine invariant polynomialsgroup action |
| spellingShingle | Regilene D. S. Oliveira Alex C. Rezende Nicolae Vulpe Family of quadratic differential systems with invariant hyperbolas:a complete classification in the space R^12 Electronic Journal of Differential Equations Quadratic differential systems invariant hyperbola affine invariant polynomials group action |
| title | Family of quadratic differential systems with invariant hyperbolas:a complete classification in the space R^12 |
| title_full | Family of quadratic differential systems with invariant hyperbolas:a complete classification in the space R^12 |
| title_fullStr | Family of quadratic differential systems with invariant hyperbolas:a complete classification in the space R^12 |
| title_full_unstemmed | Family of quadratic differential systems with invariant hyperbolas:a complete classification in the space R^12 |
| title_short | Family of quadratic differential systems with invariant hyperbolas:a complete classification in the space R^12 |
| title_sort | family of quadratic differential systems with invariant hyperbolas a complete classification in the space r 12 |
| topic | Quadratic differential systems invariant hyperbola affine invariant polynomials group action |
| url | http://ejde.math.txstate.edu/Volumes/2016/162/abstr.html |
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