Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-04-30T15:55:10.917Z Has data issue: false hasContentIssue false
  • English
  • Français

Summing up the Dynamics of Quadratic Hamiltonian Systems With a Center

Published online by Cambridge University Press:  20 November 2018

Janos Pal  and
Dana Schlomiuk
Janos Pal
Affiliation:
Département de mathématiques et de statistique and Centre de recherches mathématiques, Université de Montréal, C.P. 6128, succ., Centre-ville, Montréal, Québec, H3C 3J7
Dana Schlomiuk
Affiliation:
Département de mathématiques et de statistique and Centre de recherches mathématiques, Université de Montréal, C.P. 6128, succ., Centre-ville, Montréal, Québec, H3C 3J7
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this work we study the global geometry of planar quadratic Hamiltonian systems with a center and we sum up the dynamics of these systems in geometrical terms. For this we use the algebro-geometric concept of multiplicity of intersection Ip(P,Q) of two complex projective curves P(x, y, z) = 0, Q(x,y,z) = 0 at a point p of the plane. This is a convenient concept when studying polynomial systems and it could be applied for the analysis of other classes of nonlinear systems.

Keywords

34C58F
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 3 , 01 June 1997 , pp. 582 - 599
Copyright
Copyright © Canadian Mathematical Society 1997

References

[ALGM] Andronov, A.A., Leontovich, E .A., Gordon, I.I. and Maier, A.G., Qualitative theory of second-order dynamic systems, Israel Program for Scientific Translations, John Wiley & Sons, 1973. 524.Google Scholar
[A] Andronova, E.A., Decomposition of the parameter space of a quadratic equation with a singular point of center type and topological structures with limit cycles, Ph.D. Thesis, Gorky, Russia, 1988. Russian, 114.Google Scholar
[AL] Artes, J.C. and Llibre, J., Sistemes quadratics Hamiltonians, (1992), 60, preprint.Google Scholar
[B] Berlinskii, V., On the behaviour of the integral curves of a differential equation, Izv. Vyssh. Uchebn. Zaved. Mat. (2) 15(1960), 3–18.Google Scholar
[BM] Brunella, M. and Miari, M., Topological equivalence of a plane vector field with its principal part defined through Newton polyhedra, J. Differential Equations 85(1990), 338–366.Google Scholar
[D] Darboux, G., Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré, (Mélanges), Bull. Sci. Math. (1878), 60–96. 123–144; 151–200.Google Scholar
[DT] Tsutomu Date, Classification and analysis of two-dimensional real homogeneous quadratic differential equation systems, J. Differential Equations 32(1979), 311–334.Google Scholar
[G] Gione, F., Géométrie projective, Notes de cours, Cours Math. l’UQTR 13(1978), 267.Google Scholar
[GV] Gonz, E.A.ález Velasco, Generic properties of polynomial vector fields at infinity, Trans. Amer.Math. Soc. 143(1969), 201–222.Google Scholar
[J] Jager, P.de., Phase portraits of quadratic systems—Higher order singularities and separatrix cycles, Ph.D. Thesis, Technische Universiteit Delft, May 1989, 139.Google Scholar
[Ki] Kirwan, F., Complex algebraic curves, LondonMathematical Society Student Texts 23, Cambridge Univ. Press, 1992. 264.Google Scholar
[PS] Pal, J. and Schlomiuk, D., Geometric analysis of the bifurcation diagram of the quadratic Hamiltonian systems with a center, CRM Report, Université de Montréal, CRM-2211, 1994. 25.Google Scholar
[P81] Poincaré, H., Mémoire sur les courbes définies par les équations différentielles, J.Math. (3) 7, 375–422. Oeuvres de Henri Poincaré vol. I, Paris, Gauthiers-Villars et Cie, Editeurs, 1951. 3–84.Google Scholar
[P85] Poincaré, H., Sur les courbes définies par les équations différentielles, J. Math. Pures Appl. (4) 1, 167–244. Oeuvres de Henri Poincaré vol. I, Paris, Gauthiers-Villars et Cie, 1951. 95–114.Google Scholar
[P91] Poincaré, H., Sur l’intégration algébrique des équations différentielles, C. R. Acad. Sci. Paris 112(1891), 761–764.Google Scholar
[P97] Poincaré, H., Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré, Rend. Circ. Mat. Palermo 11(1897), 193–239.Google Scholar
[S1] Schlomiuk, D., The “center-space” of plane quadratic systems and its bifurcation diagram, Rapport de recherche du Département de mathématiques et de statisique, D.M.S. No 88-18, Université deMontréal, Octobre 1988, 26.Google Scholar
[S2] Schlomiuk, D., Algebraic particular integrals, integrability and the problem of the center, Trans. Amer.Math. Soc. (2) 338(1993), 799–841.Google Scholar
[V] Vulpe, N.I., Affine-invariant conditions for the topological discrimination of quadratic systems with a center, Translated from Differentsial’nye Uravneniya (3) 19(1983), 371–379.Google Scholar