On the geometric structure of the class of planar quadratic differential systems. (English) Zbl 1043.34035
This is a well-written article which uses a geometric approach in the study of real quadratic planar differential systems, i.e., systems of two first-order quadratic ordinary differential equations with real coefficients that have a vector field. The interest is in the global theory. Probably the most famous open problem related to this study is Hilbert’s 16th problem which seeks to determine the maximum number of limit cycles which appear for such systems and their possible relative positions in the plane. The principal aim of this work is to seek geometric clarification on results and methods such as the isocline method and the role of rotation parameters amongst other things.
Reviewer: F. M. Mahomed (Johannesburg)
MSC:
| 34C14 | Symmetries, invariants of ordinary differential equations |
| 34C08 | Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) |
| 37C10 | Dynamics induced by flows and semiflows |
References:
| [1] | Camacho, C.; Schlomiuk, D., Complex foliations arising from polynomial differential equations, Bifurcations and Periodic Orbits of Vector Fields (Montreal, 1992), 1-18 (1993), Dordrecht: Kluwer Acad. Publ., Dordrecht · Zbl 0808.32034 |
| [2] | Duff, G. F., Limit cycles and rotated vector fields, Annals of Math., 57, 2, 15-31 (1953) · Zbl 0050.09103 · doi:10.2307/1969724 |
| [3] | F. Dumortier,Local study of planar vector fields: singularities and their unfoldings, in Structure in Dynamics (H.W. Broer, F. Dumortier, S.T. van Strien, F. Takens, eds), Studies in mathematical Physics, North Holland vol.2 (1991), 161-241. · Zbl 0746.58002 |
| [4] | Dumortier, F.; Roussarie, R.; Rousseau, C., Hilbert’s 16th problem for quadratic vector fields, J. Differential. Equations, 110, no. 1, 86-133 (1994) · Zbl 0802.34028 · doi:10.1006/jdeq.1994.1061 |
| [5] | Dumortier, F.; Roussarie, R.; Rousseau, C., Elementary graphics of cyclicity 1 and 2, Nonlinearity, 7, no. 3, 1001-1043 (1994) · Zbl 0855.58043 · doi:10.1088/0951-7715/7/3/013 |
| [6] | J. Écalle,Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Math., Hermann, Paris (1992), 337 pp · Zbl 1241.34003 |
| [7] | Erugin, N. P., On certain questions of stability of motion and the qualitative theory of differential equations in the large, Akad. Nauk SSSR. Prikl. Mat. Meh., 14, 459-512 (1950) · Zbl 0039.09502 |
| [8] | Erugin, N. P., A qualitative investigation of the integral curves of a system of differential equations, Akad. Nauk SSSR. Prikl. Mat. Meh., 14, 659-664 (1950) · Zbl 0039.09601 |
| [9] | W. Fulton,Algebraic curves, W.A. Benjamin (1969), 269 pp. · Zbl 0181.23901 |
| [10] | Gaiko, V. A., On an application of two isoclines method to investigation of two-dimensional dynamic systems, Advances in Synergetics, 2, 104-109 (1994) |
| [11] | Il’Yashenko, Yu. S., Finiteness theorems for limit cycles (1991), Providence, RI: Amer. Math. Soc., Providence, RI · Zbl 0743.34036 |
| [12] | Kirwan, F., Complex algebraic curves (1992), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 0744.14018 |
| [13] | J. Llibre and D. Schlomiuk,The Geometry of Quadratic Differential Systems with a Weak Focus of third order, to appear in the C.R. of the C.R.M. of Barcelona, 47 pp. · Zbl 1058.34034 |
| [14] | Pal, J.; Schlomiuk, D., On the Geometry in the Neighbourhood of Infinity of Quadratic Differential Systems with a Weak Focus, Qualitative Theory of Dynamical Systems, 2, 1-43 (2001) · Zbl 0989.34018 · doi:10.1007/BF02969379 |
| [15] | R.J. Walker,Algebraic curves, Dover Publications, Inc., New York (1962), 201 pp. · Zbl 0103.38202 |
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.
