Generalized nonlinear superposition principles for polynomial planar vector fields

Journal of Lie Theory

Volumn 15, Issue 1, 2005, Pages 89-104

  García, Isaac A ^a   Giacomini, Hector ^b   Giné, Jaume ^a  

^a UNIVERSITY OF LLEIDA   (Spain)

^b UNIVERSITÉ DE TOURS   (France)

Author keywords

Darboux first integral; Liouvillian first integral; Nonlinear differential equations; Nonlinear superposition principle; Polynomial planar vector fields

Indexed keywords

EID: 12744276070     PISSN: 09495932     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (18)

1. Cariñena, J. F., J. Grabowski, and G. Marmo, "Lie-Scheffers systems: a geometric approach," Napoli Series on Physics and Astrophysics. Bibliopolis, Naples, 2000.
2. Chavarriga, J., H. Giacomini, J. Giné, and J. Llibrc, Local analytic integrability for nilpotent centers, Ergodic Theory Dynam. Systems 23 (2003), 417-428.
3. Christopher, C. and J. Llibre, Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. Differential Equations 16 (2000), 5-19.
4. Christopher, C., and N. G. Lloyd, On the paper of Jin and Wang concerning the conditions for a centre in certain cubic systems, Bull. London Math. Soc. 22 (1990), 5-12.
5. Darboux, G., Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. 2ème série, 2 (1878), 60-96; 123-144; 151-200.
6. Garcia, I. A., and J. Giné, Generalized cofactors and nonlinear superposition principles, Appl. Math. Lett. 16 (2003), 1137-1141.
7. _, Non-algebraic Invariant Curves for Polynomial Planar Vector Fields, Discrete Contin. Dyn. Syst. 10 (2004), 755-768.
8. Jin, X., and D. Wang, On the conditions of Kukles for the existence of a centre, Bull. London Math. Soc. 22 (1990), 1-4.
9. Kukles, I. S., Sur quelques cas de distinction entre un foyer et un centre, Dokl. Akad. Nauk. SSSR 42 (1944), 208-211.
10. Jouanolou, J. P., "Équations de Pfaff algébriques, " Lecture Notes in Mathematics 708, Springer-Verlag, 1979.
11. Lie, S. and Scheffers, G., "Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen," Teubner-Verlag, Leipzig, 1893.
12. Odani, K., The limit cycle of the van der Pol equation is not algebraic, J. Differential Equations 115 (1995), 146-152.
13. _, The integration of polynomial Liénard systems by elementary functions, Differential Equations Dynam. Systems 5 (1997), 347-354.
14. λn =C, Ann. Fac. Sc. Univ., Toulouse (1896), 1-37; reprinted in Œuvres, tome 2, 546-582.
15. Seidenberg, A., Reduction of singularities of the differential equation A dy = B dx, Amer. J. Math. 90 (1968), 248-269.
16. Stephani, H., "Differential equations. Their solutions using symmetries," Cambridge University Press, Cambridge, 1989.
17. Walcher, S., On the Poincaré problem, J. Differential Equations 166 (2000), 51-78.
18. Wilson, J. C., Algebraic periodic solutions of Liénard equations, Contributions to Differential Equations 3 (1964), 1-20.