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The geometry of quadratic polynomial differential systems with a weak focus and an invariant straight line. (English) Zbl 1208.34035

Summary: Planar quadratic differential systems occur in many areas of applied mathematics. Although more than a thousand papers were written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert’s 16th problem, are still open for this family. In this article, we conduct a global study of the class \(QW^{I}\) of all real quadratic differential systems which have a weak focus and invariant straight lines of total multiplicity of at least two. This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real projective space of the parameters of this form. The bifurcation diagram yields 73 phase portraits for systems in \(QW^{I}\) plus 26 additional phase portraits with the center at its border points. Algebraic invariants are used to construct the bifurcation set. We show that all systems in \(QW^{I}\) necessarily have their weak focus of order one and invariant straight lines of total multiplicity exactly two. The phase portraits are represented on the Poincaré disk. The bifurcation set is algebraic and all points in this set are points of bifurcation of singularities. We prove that there is no phase portrait with limit cycles in this class but that there is a total of five phase portraits with graphics, four having the invariant line as a regular orbit and one phase portrait with an infinity of graphics which are all homoclinic loops inside a heteroclinic graphic with two singularities, both at infinity.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C45 Invariant manifolds for ordinary differential equations
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[1] DOI: 10.1006/jdeq.1994.1004 · Zbl 0791.34048 · doi:10.1006/jdeq.1994.1004
[2] DOI: 10.1216/rmjm/1181072378 · Zbl 0814.34020 · doi:10.1216/rmjm/1181072378
[3] DOI: 10.1006/jdeq.1996.0127 · Zbl 0858.34042 · doi:10.1006/jdeq.1996.0127
[4] DOI: 10.5565/PUBLMAT_41197_02 · Zbl 0880.34031 · doi:10.5565/PUBLMAT_41197_02
[5] Artés J. C., Memoires Amer. Math. Soc. 134
[6] Artés J. C., Resenhas 6 pp 85–
[7] DOI: 10.1142/S0218127406016720 · Zbl 1124.34014 · doi:10.1142/S0218127406016720
[8] DOI: 10.1007/BF03032094 · Zbl 1196.34014 · doi:10.1007/BF03032094
[9] DOI: 10.1142/S021812740802032X · Zbl 1206.34051 · doi:10.1142/S021812740802032X
[10] Artés J. C., Electron. J. Diff. Eqs. 82 pp 1–
[11] DOI: 10.1016/j.jde.2008.12.010 · Zbl 1170.34003 · doi:10.1016/j.jde.2008.12.010
[12] DOI: 10.1016/S0362-546X(96)00088-0 · Zbl 0886.34026 · doi:10.1016/S0362-546X(96)00088-0
[13] DOI: 10.1016/j.na.2006.04.021 · Zbl 1189.34065 · doi:10.1016/j.na.2006.04.021
[14] DOI: 10.1016/j.jde.2006.07.022 · Zbl 1117.34035 · doi:10.1016/j.jde.2006.07.022
[15] DOI: 10.2140/pjm.2007.229.63 · Zbl 1160.34003 · doi:10.2140/pjm.2007.229.63
[16] DOI: 10.1016/0022-0396(87)90133-1 · Zbl 0616.34028 · doi:10.1016/0022-0396(87)90133-1
[17] DOI: 10.5565/PUBLMAT_32288_08 · Zbl 0674.34027 · doi:10.5565/PUBLMAT_32288_08
[18] DOI: 10.1142/S0218127409023299 · Zbl 1167.34324 · doi:10.1142/S0218127409023299
[19] DOI: 10.1016/j.na.2010.04.004 · Zbl 1203.34063 · doi:10.1016/j.na.2010.04.004
[20] DOI: 10.1016/0022-0396(66)90070-2 · Zbl 0143.11903 · doi:10.1016/0022-0396(66)90070-2
[21] DOI: 10.1007/978-3-322-96657-5_3 · doi:10.1007/978-3-322-96657-5_3
[22] Darboux G., Bull. Sci. Math. 124 pp 151–
[23] Dulac H., Bull. Soc. Math. France 51 pp 45–
[24] Dumortier F., J. Diff. Eqs. 110 pp 66–
[25] Dumortier F., Qualitative Theory of Planar Differential Systems (2006) · Zbl 1110.34002
[26] Fulton W., Algebraic Curves. An Introduction to Algebraic Geometry (1969) · Zbl 0181.23901
[27] DOI: 10.1007/BF02874780 · Zbl 1150.34008 · doi:10.1007/BF02874780
[28] DOI: 10.1216/RMJ-1986-16-4-751 · Zbl 0609.34040 · doi:10.1216/RMJ-1986-16-4-751
[29] DOI: 10.1090/S0002-9947-1969-0252788-8 · doi:10.1090/S0002-9947-1969-0252788-8
[30] DOI: 10.1007/978-1-4757-3849-0 · doi:10.1007/978-1-4757-3849-0
[31] DOI: 10.1090/S0002-9904-1902-00923-3 · JFM 33.0976.07 · doi:10.1090/S0002-9904-1902-00923-3
[32] DOI: 10.1016/S0007-4497(98)80080-8 · Zbl 0920.34037 · doi:10.1016/S0007-4497(98)80080-8
[33] Ilyanshenko Y., Trans. of Math. Monographs 94, in: Finiteness Theorem for Limit Cycles (1991)
[34] Kapteyn W., Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland (Dutch) 19 pp 1446–
[35] Kapteyn W., Nederl. Akad. Wetensch. Verslag Afd. Natuurk. (Dutch) 20 pp 1354–
[36] Kapteyn W., Nederl. Akad. Wetensch. Verslag Afd. Natuurk. (Dutch) 21 pp 27–
[37] Li C., Chinese Ann. Math. Ser. B 7 pp 174–
[38] DOI: 10.1016/j.jde.2003.10.008 · Zbl 1055.34061 · doi:10.1016/j.jde.2003.10.008
[39] DOI: 10.4153/CJM-2004-015-2 · Zbl 1058.34034 · doi:10.4153/CJM-2004-015-2
[40] DOI: 10.1016/j.na.2008.07.012 · Zbl 1171.34035 · doi:10.1016/j.na.2008.07.012
[41] DOI: 10.1016/j.na.2008.07.012 · Zbl 1171.34035 · doi:10.1016/j.na.2008.07.012
[42] Lunkevitch V. A., Diff. Eqs. 18 pp 786–
[43] DOI: 10.4153/CJM-1997-027-0 · Zbl 0879.34038 · doi:10.4153/CJM-1997-027-0
[44] Pingguang Z., Acta Math. Sinica 44 pp 37–
[45] Pingguang Z., Ann. Diff. Eqs. 17 pp 287–
[46] Pingguang Z., Qualit. Th. Dyn. Phase Portraits 3 pp 1–
[47] Poincaré H., J. Maths. Pures Appl. 7 pp 375–
[48] DOI: 10.1007/BF02584827 · Zbl 0628.34032 · doi:10.1007/BF02584827
[49] DOI: 10.1007/BFb0083072 · doi:10.1007/BFb0083072
[50] DOI: 10.1007/BF02969335 · Zbl 1043.34035 · doi:10.1007/BF02969335
[51] DOI: 10.1090/S0002-9947-1993-1106193-6 · doi:10.1090/S0002-9947-1993-1106193-6
[52] DOI: 10.1007/978-94-015-8238-4_10 · doi:10.1007/978-94-015-8238-4_10
[53] DOI: 10.1007/BF02969379 · Zbl 0989.34018 · doi:10.1007/BF02969379
[54] DOI: 10.1007/978-94-007-1025-2_13 · doi:10.1007/978-94-007-1025-2_13
[55] DOI: 10.1007/BF02968134 · Zbl 1101.34016 · doi:10.1007/BF02968134
[56] DOI: 10.1016/j.jde.2004.11.001 · Zbl 1090.34024 · doi:10.1016/j.jde.2004.11.001
[57] DOI: 10.1016/j.na.2006.11.028 · Zbl 1136.34037 · doi:10.1016/j.na.2006.11.028
[58] DOI: 10.1216/RMJ-2008-38-6-2015 · Zbl 1175.34037 · doi:10.1216/RMJ-2008-38-6-2015
[59] DOI: 10.1007/s10884-008-9117-2 · Zbl 1168.34024 · doi:10.1007/s10884-008-9117-2
[60] Schlomiuk D., Buletinul A.Ş. a R.M., Matematica 56 pp 27–
[61] Shi S., J. Diff. Eqs. 41 pp 301–
[62] Shi S., J. Diff. Eqs. 52 pp 52–
[63] Vulpe N. I., Diff. Eqs. 19 pp 273–
[64] Ye Q. Y., Trans. Mathematical Monographs 66, in: Theory of Limit Cycles (1984)
[65] DOI: 10.1006/jdeq.1994.1049 · Zbl 0797.34044 · doi:10.1006/jdeq.1994.1049
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