Integrability of planar polynomial differential systems through linear differential equations

Rocky Mountain Journal of Mathematics

Volumn 36, Issue 2, 2006, Pages 457-485

  Giacomini, H ^a   Gine J ^b   Grau, M ^b  

^a UNIVERSITÉ DE TOURS   (France)

^b UNIVERSITY OF LLEIDA   (Spain)

Author keywords

Darboux integrability; Invariant curves; Lirst integral; Planar polynomial system

Indexed keywords

EID: 33746350150     PISSN: 00357596     EISSN: None     Source Type: Journal    
DOI: 10.1216/rmjm/1181069462     Document Type: Article

References (28)

1. M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables (Milton Abramowitz and Irene A. Stegun, eds.), Dover Publ., Inc., New York, 1992 (reprint of the 1972 edition).
2. N.N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or centre type, Trans. Amer. Math. Soc. 100 (1954), 397-413.
3. M. Berthier, D. Cerveau and A. Lins Neto, Sur les feuilletages analytiques réels et le problème du centre, J. Differential Equations 131 (1996), 244-266.
4. M. Carnicer, The Poincaré problem in the nondicritical case, Ann. Math. 140 (1994), 289-294.
5. D. Cerveau and A. Lins Neto, Holomorphic foliations in CP(2) having an invariant algebraic curve, Ann. Inst. Fourier 41 (1991), 883-903.
6. J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, On the integrability of two-dimensional flows, J. Differential Equations 157 (1999), 163-182.
7. J. Chavarriga and M. Grau, Invariant algebraic curves linear in one variable for planar real quadratic systems, Appl. Math. Comput. 138 (2003), 291-308.
8. _, A family of quadratic polynomial differential systems non-integrables Darboux and with algebraic solutions of arbitrarily high degree, Appl. Math. Lett. 16 (2003), 833-837.
9. C. Christopher, Invariant algebraic curves and conditions for a centre, Proc. Roy. Soc. Edinburgh 124 (1994), 1209-1229.
10. _, Liouvillian first integrals of second order polynomial differential systems, Electronic J. Differential Equations 1999 1-7.
11. C. Christopher and J. Llibre, A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals, Proc. Amer. Math. Soc. 130 (2002), 2025-2030 (electronic).
12. G. Darboux, Mémoire sur les équations diffé rentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. 32 (1878), 60-96; 123-144; 151-200.
13. H. Dulac, Détermination et intégration d'une certaine classe d'équations différentielles ayant pour point singulier un centre, Bull. Sciences Math. 32 (1908). 230-252.
14. I.A. García and J. Giné, Generalized cofactors and nonlinear superposition principles, Appl. Math. Lett. 16 (2003), 1137-1141.
15. J.P. Jouanolou, Équations de Pfaff algébriques, Lecture Notes in Math., vol. 708, Springer, Berlin, 1979.
16. A. Lins Neto, Folheaçōesem CP(2) com curva algébrica invariante e singularidades radiais, Instituto de Estudios con Iberoamerica y Portugal, Seminarios Temáticos, Vol. 2, Fasc. III, 1998.
17. _, Some examples for the Poincaré and Painlevé problems, Ann. Scient. Éc. Norm. Sup. 35 (2002), 231-266.
18. V.A. Lunkevich and K.S. Sibirskiǐ, Integrals of a general quadratic differential system in cases of a center, Diff. Equations 18 (1982), 563-568.
19. J. Moulin Ollagnier, About a conjecture on quadratic vector fields, J. Pure Appl. Algebra 165 (2001), 227-234.
20. R. Moussu, Une démonstration d'un théorème de Lyapunov-Poincaré, Astérisque 98-99 (1982), 216-223.
21. H. Poincaré, Sur l'intégration algébrique des équations différentielles du premier ordre et du premier degré Rend. Circ. Mat. Palermo 5 (1891), 161-191.
22. M.J. Prelle and M.F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1983), 215-229.
23. D. Schlomiuk, Algebraic paticular integrals, integrability and the problem of center, Trans. Amer. Math. Soc. 338 (1993), 799-841.
24. _, Algebraic and geometric aspects of the theory of polynomial vector fields, in Bifurcations and periodic orbits of vector fields, (Montreal, PQ, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 408, Kluwer Acad. Publ., Dordrecht, 1993, pp. 429-467.
25. M.F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673-688.
26. Wolfram Research, Inc., http://functions.wolfram.com, 2000-2003.
27. H. Zoła̧dek, The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal. 4 (1994), 79-136.
28. _, Remarks on: "The classification of reversible cubic systems with center," Topol. Methods Nonlinear Anal. 8 (1996), 335-342.