Liouvillian first integrals for the planar Lotka-Volterra system

Rendiconti del Circolo Matematico di Palermo

Volumn 52, Issue 3, 2003, Pages 389-418

  Cairó, Laurent ^a   Giacomini, Hector ^b   Llibre, Jaume ^c  

^a UNIVERSITÉ D'ORLÉANS   (France)

^b UNIVERSITÉ DE TOURS   (France)

^c UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 80053121201     PISSN: 0009725X     EISSN: 19734409     Source Type: Journal    
DOI: 10.1007/BF02872763     Document Type: Article

References (47)

1. Almeida M. A., Magalhães M. E., Moreira I. C., Lie symmetries and invariants of the Lotka-Volterra system, J. Math. Phys., 36 (1995), 1854-1867.
2. Andrade R. F. S., Rauh A., The Lorenz model and the method of Carleman embedding, Phys. Lett. A, 82 (1981), 276-278.
3. Bountis T., Grammaticos B., Ramani A., On the complete and partial integrability of non-Hamiltonian systems, Phys. Rep., 180 (1989), 159.
4. Brenig L., Goriely A., A. Quasimonomial transformations and integrability. Partially integrable evolution equations in physics, (Les Houches, 1989), 571-572, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 310, Kluwer Acad. Publ., Dordrecht, 1990.
5. Cairó L., Feix M. R., Llibre J., Integrability and invariant algebraic curves for planar polynomial differential systems with emphasis on the quadratic systems, Resenhas de Universidade de Sao Paulo, 4 (1999), 127-161.
6. Cairó L., Llibre J., Darboux integrability for the 3-dimensional Lotka-Volterra systems, J. Phys. A: Math. and Gen., 33 (2000), 2395-2406.
7. Cairó L., Llibre J., Darboux first integrals and invariants for real quadratic systems having an invariant conic, J. Phys. A: Math. and Gen., 35 (2002), 589-608.
8. Cantrijn F., Sarlet W., Generalizations of Noether's theorem in classical mechanics, SIAM Rev., 23 (1981), 467-494.
9. Carleman T., Application de la théorie des équations intégrales linéaires aux systémes d'équations différentielles non linéaires, Acta Mathematica, 59 (1932), 63.
10. Chavarriga J., Giacomini H., Giné J., Llibre J., Darboux integrability and the inverse integrating factor, to appear in J. of Differential Equations.
11. Chavarriga J., Llibre J., Invariant algebraic curves and rational first integrals for planar polynomial vector fields, J. Differential Equations, 169 (2001), 1-16.
12. Chavarriga J., Llibre J., Sotomayor J., Algebraic solutions for polynomial vector fields with emphasis in the quadratic case, Expositions Math., 15 (1997), 161-173.
13. Christopher C., Invariant algebraic curves and conditions for a center, Proc. Roy. Soc. Edinburgh, 124A (1994), 1209-1229.
14. Christopher C. J., Llibre J., Algebraic aspects of integrability for polynomial systems, Qualitative Theory of Dynamical Systems, 1 (1999), 71-95.
15. Christopher C., Llibre J., Integrability via invariant algebraic curves for planar polynomial differential systems, Annals of Differential Equations, 16 (2000), 5-19.
16. Christopher C., Llibre J., Pereira J. V., Multiplicity of invariant algebraic curves and Darboux integrability, preprint, 2002.
17. 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.
18. Forgas P., Giacomini H., Llibre J., Viano M., On the Liouville integrability of Bogomolnyi equations for Einstein-Yang-Mills-dilation theory, preprint.
19. Giacomini H. J., Repetto C. E., Zandron O. P., Integrals of motion of three-dimensional non-Hamiltonian dynamical systems, J. Phys. A, 24 (1991), 4567-4574.
20. Goriely A., Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys., 37 (1996), 1871-1893.
21. Gradshteyn I. S., Ryzhik I. M., Table of integrals, series, and products, Translated from the Russian. Sixth edition. Academic Press, Inc., San Diego, CA, 2000.
22. 3: the Lotka-Volterra system, Physica A, 163 (1990), 683-722.
23. Gutierrez C., Llibre J., Darboux integrability for polynomial vector fields on the 2-dimensional sphere, Extracta Mathematicae, 17 (2002), 289-301.
24. Hietarinta J., Direct methods for the search of the second invariant, Phys. Rep., 147 (1987), 87-154.
25. Jouanolou J. P., Equations de Pfaff algébriques, in "Lectures Notes in Mathematics,"708, Springer-Verlag, New York/Berlin, 1979.
26. Labrunie S., On the polynomial first integrals of the (a, b, c) Lotka-Volterra system, J. Math. Phys., 37 (1996), 5539-5550.
27. Lax P. D., Integrals of Nonlinear Equations of Evolution and Solitary Waves, Commun. Pure Appl. Math., 21 (1968), 467-490.
28. Llibre J., Rodríguez G., Invariant hyperplanes and Darbox integrability for d-dimensional polynomial differential systems, Bull. Sci. Math., 124 (2000), 1-21.
29. Llibre J., Rodríguez G., Darboux integrability of polynomial vector fields on 2-dimensional surfaces, to appear in J. of Bifurcation and Chaos, 12 (2002).
30. Llibre J., Schlomiuk D., The geometry of differential quadratic systems with a weak focus of third order, to appear in Canadian J. of Math.
31. Llibre J., Zhang X., Darboux integrability of real polynomial vector fields on regular algebraic hypersurfaces, Rendiconti del circolo matematico di Palermo, Serie II, LI (2002), 109-126.
32. Lotka A. J., Analytical note on certain rhythmic relations in organic systems, Proc. Nat. Acad. Sci., 6 (1920), 410.
33. Moulin Ollagnier J., Polynomial first integrals of the Lotka-Volterra system, Bull. Sci. math., 121 (1997), 463-476.
34. Moulin Ollagnier J., Rational integration of the Lotka-Volterra system, Bull. Sci. math., 123 (1999), 437-466.
35. Moulin Ollagnier J., Liouvillian Integration of the Lotka-Volterra system, Qualitative Theory of Dynamical Systems, 2 (2001), 307-358.
36. Moulin Ollagnier J., Some remarks about the integration of polynomial planar vector fields, Qualitative Theory of Dynamical Systems, 3 (2002), 19-28.
37. Moulin Ollagnier J., About a conjecture on quadratic vector fields, Journal of Pure and Applied Algebra, 165 (2001), 227-234.
38. Olver P. J., Applications of Lie groups to differential equations, Springer, New York, 1986.
39. Poincaré H., Sur l'intégration des équations différentielles du premier ordre et du premier degré I and II, Rendiconti del Circolo Matematico di Palermo, 5 (1891), 161-191; 11 (1897), 193-239.
40. Prelle M. J., Singer M. F., Elementary first integrals of differential equations, Trans. Amer. Math. Soc., 279 (1983), 613-636.
41. Schlomiuk D., Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc., 338 (1993), 799-841.
42. Schlomiuk D., Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields, in Bifurcations and Periodic Orbits of Vector Fields, D. Schlomiuk (ed.), 1993, 429-467.
43. Singer M. F., Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc., 333 (1992), 673-688.
44. Strelcyn J. M., Wojciechowski S., A method of finding integrals for three-dimensional dynamical systems, Phys. Lett. A, 133 (1988), 207-212.
45. Volterra V., Leçons sur la théorie mathématique de la lutte pour la vie, Gauthier-Villars, Paris, 1931.
46. Yablonskii A. I., Algebraic integrals of a differential-equation system, Differential Equations, 6 (1970), 1326-1333.
47. Weil J. A., Constantes et polynômes de Darboux en algèbre différentielle: applications aux systèmes différentiels linéaires, Ph. D., école Polytecnique, 1995.