Algebraic geometry of the center-focus problem for Abel differential equations. (English) Zbl 1362.37055
This paper deals with centers of the Abel’s equation \(y'=p(x)y^3+q(x)y^2\), with polynomial coefficients \(p\), \(q\), from an algebraic geometry point of view.
In fact, by translating some previous results by the second author et al. [Compos. Math. 149, No. 4, 705–728 (2013; Zbl 1273.30024); Proc. Lond. Math. Soc. (3) 99, No. 3, 633–657 (2009; Zbl 1177.30046)] into the language of center equations, the authors show that the center conditions can be expressed in terms of the composition algebra, thus providing an extension of results by M. Briskin et al. [Ann. Math. (2) 172, No. 1, 437–483 (2010; Zbl 1216.34025)].
Further, this paper initiates the study of some second-order approximations of the center equations, called “second Melnikov coefficients”, that along with the vanishing of the moments can be used to characterize centers.
In fact, by translating some previous results by the second author et al. [Compos. Math. 149, No. 4, 705–728 (2013; Zbl 1273.30024); Proc. Lond. Math. Soc. (3) 99, No. 3, 633–657 (2009; Zbl 1177.30046)] into the language of center equations, the authors show that the center conditions can be expressed in terms of the composition algebra, thus providing an extension of results by M. Briskin et al. [Ann. Math. (2) 172, No. 1, 437–483 (2010; Zbl 1216.34025)].
Further, this paper initiates the study of some second-order approximations of the center equations, called “second Melnikov coefficients”, that along with the vanishing of the moments can be used to characterize centers.
Reviewer: Marco Spadini (Firenze)
MSC:
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 37C10 | Dynamics induced by flows and semiflows |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34A26 | Geometric methods in 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 |
Keywords:
Abel differential equation; center-focus problem; planar vector field; composition algebra; moment equations; Melnikov coefficientsReferences:
| [1] | DOI: 10.1017/S0143385799141737 · Zbl 0990.34017 · doi:10.1017/S0143385799141737 |
| [2] | DOI: 10.1007/s13398-012-0112-4 · Zbl 1308.34037 · doi:10.1007/s13398-012-0112-4 |
| [3] | DOI: 10.1090/conm/382/07048 · doi:10.1090/conm/382/07048 |
| [4] | DOI: 10.1093/qmath/46.1.21 · Zbl 0881.20008 · doi:10.1093/qmath/46.1.21 |
| [5] | DOI: 10.1017/S0308210500021971 · Zbl 0618.34026 · doi:10.1017/S0308210500021971 |
| [6] | DOI: 10.1016/j.jde.2013.04.009 · Zbl 1290.34015 · doi:10.1016/j.jde.2013.04.009 |
| [7] | Alwash, Differ. Equ. Appl. 5 pp 1– (2013) |
| [8] | DOI: 10.1016/j.jmaa.2012.09.006 · Zbl 1269.34035 · doi:10.1016/j.jmaa.2012.09.006 |
| [9] | DOI: 10.1007/s12591-011-0089-z · Zbl 1272.34050 · doi:10.1007/s12591-011-0089-z |
| [10] | DOI: 10.1007/s12346-011-0054-9 · Zbl 1264.34055 · doi:10.1007/s12346-011-0054-9 |
| [11] | DOI: 10.1016/j.exmath.2009.02.002 · Zbl 1190.34048 · doi:10.1016/j.exmath.2009.02.002 |
| [12] | Christopher, Limit Cycles of Differential Equations (2007) · Zbl 1359.34001 |
| [13] | Alwash, Electron. J. Differential Equations 2003 pp 1– (2003) |
| [14] | DOI: 10.1017/etds.2012.143 · Zbl 1291.37085 · doi:10.1017/etds.2012.143 |
| [15] | Brudnyi, C. R. Math. Acad. Sci. Soc. R. Can. 31 pp 33– (2009) |
| [16] | DOI: 10.1353/ajm.2006.0012 · Zbl 1104.34022 · doi:10.1353/ajm.2006.0012 |
| [17] | DOI: 10.4007/annals.2010.172.437 · Zbl 1216.34025 · doi:10.4007/annals.2010.172.437 |
| [18] | DOI: 10.1090/S0002-9947-1922-1501189-9 · doi:10.1090/S0002-9947-1922-1501189-9 |
| [19] | DOI: 10.1088/0951-7715/23/7/007 · Zbl 1203.30007 · doi:10.1088/0951-7715/23/7/007 |
| [20] | DOI: 10.1007/BF02916714 · Zbl 1078.30034 · doi:10.1007/BF02916714 |
| [21] | DOI: 10.1090/conm/532/10490 · doi:10.1090/conm/532/10490 |
| [22] | DOI: 10.1112/plms/pdp010 · Zbl 1177.30046 · doi:10.1112/plms/pdp010 |
| [23] | Pakovich, Mosc. Math. J. 13 pp 693– (2013) |
| [24] | DOI: 10.1112/S0010437X12000620 · Zbl 1273.30024 · doi:10.1112/S0010437X12000620 |
| [25] | DOI: 10.1016/j.bulsci.2009.06.003 · Zbl 1205.30025 · doi:10.1016/j.bulsci.2009.06.003 |
| [26] | DOI: 10.1016/j.bulsci.2005.06.001 · Zbl 1089.30039 · doi:10.1016/j.bulsci.2005.06.001 |
| [27] | DOI: 10.1090/S0002-9939-02-06755-2 · Zbl 1008.34024 · doi:10.1090/S0002-9939-02-06755-2 |
| [28] | Nakai, Mosc. Math. J. 10 pp 415– (2010) |
| [29] | DOI: 10.1112/plms/pdq009 · Zbl 1241.20031 · doi:10.1112/plms/pdq009 |
| [30] | DOI: 10.3934/cpaa.2012.11.2023 · Zbl 1280.34038 · doi:10.3934/cpaa.2012.11.2023 |
| [31] | DOI: 10.1090/S0273-0979-02-00946-1 · Zbl 1004.34017 · doi:10.1090/S0273-0979-02-00946-1 |
| [32] | DOI: 10.1007/978-1-4615-7904-5 · doi:10.1007/978-1-4615-7904-5 |
| [33] | Giné, Electron. J. Qual. Theory Differ. Equ. (2014) |
| [34] | Giné, J. Appl. Anal. Comput. 3 pp 133– (2013) |
| [35] | DOI: 10.1112/plms/pds050 · Zbl 1277.34034 · doi:10.1112/plms/pds050 |
| [36] | DOI: 10.2307/2371520 · Zbl 0025.10403 · doi:10.2307/2371520 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
