Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-29T02:51:22.414Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytic integrability of quadratic–linear polynomial differential systems

Published online by Cambridge University Press:  04 November 2009

JAUME LLIBRE  and
CLÀUDIA VALLS
JAUME LLIBRE
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain (email: [email protected])
CLÀUDIA VALLS
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais 1049-001, Lisboa, Portugal (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

For the quadratic–linear polynomial differential systems with a finite singular point, we classify the ones which have a global analytic first integral, and provide the explicit expression of their first integrals.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 1 , February 2011 , pp. 245 - 258
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Anosov, D. V. and Arnold, V. I.. Dynamical Systems I. Springer, Berlin, 1988.CrossRefGoogle Scholar
[2]Bautin, N. N.. On the number of limit cycles which appear with the variation of the coefficients from an equilibrium position of focus or center type. Math. USSR Sb. 100 (1954), 397–413.Google Scholar
[3]Cairó, L. and Llibre, J.. Phase portraits of planar semi-homogeneous systems I. Nonlinear Anal. 29 (1997), 783–811.CrossRefGoogle Scholar
[4]Chavarriga, J., Garcia, B., Llibre, J., Perez del Rio, J. S. and Rodriguez, J. A.. Polynomial first integrals of quadratic vector fields. J. Differential Equations 230 (2006), 393–421.CrossRefGoogle Scholar
[5]Chavarriga, J., Giacomini, H., Giné, J. and Llibre, J.. On the integrability of two-dimensional flows. J. Differential Equations 157 (1999), 163–182.CrossRefGoogle Scholar
[6]Furta, S. D.. On non-integrability of general systems of differential equations. Z. Angew. Math. Phys. 47 (1996), 112–131.CrossRefGoogle Scholar
[7]Kapteyn, W.. On the midpoints of integral curves of differential equations of the first degree. Verh. Kon. Ned. Akad. Wet. Afd. Natuurk. 19 (1911), 1446–1457 (in Dutch).Google Scholar
[8]Kapteyn, W.. New investigations on the midpoints of integrals of differential equations of the first degree. Verh. Kon. Ned. Akad. Wet. Afd. Natuurk. 20 (1912), 1354–1365; 21, 27–33 (in Dutch).Google Scholar
[9]Liapunov, A. M.. Problème général de la stabilité du mouvement (Annals of Mathematics Studies, 17). Princeton University Press, Princeton, NJ, 1949.Google Scholar
[10]Li, W., Llibre, J. and Zhang, X.. Local first integrals of differential systems and diffeomorphisms. Z. Angew. Math. Phys. 54 (2003), 1–21.CrossRefGoogle Scholar
[11]Llibre, J. and Valls, C.. Global analytic first integrals for the real planar Lotka–Volterra systems. J. Math. Phys. 48 (2007), 1–13.CrossRefGoogle Scholar
[12]Joyal, P. and Rousseau, C.. Saddle quantities and applications. J. Differential Equations 78 (1989), 374–399.CrossRefGoogle Scholar
[13]Moussu, R.. Symétrie et forme normale des centres et foyers dégénérés. Ergod. Th. & Dynam. Sys. 2 (1982), 241–251.CrossRefGoogle Scholar
[14]Poincaré, H.. Mémoire sur les courbes définies par les équations différentielles (Oeuvreus de Henri Poincaré, I). Gauthiers–Villars, Paris, 1951, pp. 95–114.Google Scholar
[15]Reyn, J. W.. A bibliography of the qualitative theory of quadratic systems of differential equations in the plane, Delf University of Technology, 1997, http://ta.twi.tudelft.nl/DV/Staff/J.W.Reyn.html.CrossRefGoogle Scholar
[16]Ye, Y.et al. Theory of Limit Cycles (Translations of Mathematical Monographs, 66). American Mathematical Society, Providence, RI, 1984.Google Scholar
[17]Ye, Y.. Qualitative Theory of Polynomial Differential Systems. Shanghai Scientific & Technical Publishers, Shanghai, 1995 (in Chinese).Google Scholar