Chapter 5 Integrability of polynomial differential systems

Handbook of Differential Equations: Ordinary Differential Equations

Volumn 1, Issue , 2004, Pages 437-532

  Llibre, Jaume ^a  

^a UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

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EID: 70350376754     PISSN: 18745725     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-5725(00)80007-3     Document Type: Chapter

References (111)

1. Russian. Armukhumedov M.I. On the construction of a differential equation having given curves as limit cycles. Izv. Vyss. Ucebn. Zavcd. Mat. 1 44 (1963) 3-6
2. Andradc R.F.S., and Rauh A. The Lorenz model and the method of Carleman embedding. Phys. Lett. A 82 (1981) 276-278
3. Éditions Mir, Moscou. Arnold V. Equations Différentielles Ordinaires (1974)
4. Bulihrcu K., and Jimenez V. A characterization of the (o-limit sets of planar continuous dynamical systems. J. Differential Filiations 145 (1998) 469-488
5. Bamon R. Quadratic vector fields in the plane have a finite number of limit cycles. Int. Hautes Etudes Sci. Publ. Math. 64 (1986) 111-142
6. 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
7. Bountis T.C., Ramani A., Grammatieos B., and Dorizzi B. On the complete and partial integrability of non-Hamiltonian systems. Phys. A 128 (1984) 268-288
8. Brenig L., and Goriely A. Quasimonomial transformations and integrability. Partially Integrable Evolution Equations in Physics (Les Houches, 1989). NATO Adv. Sci. Inst. Sen C Math. Phys. Sci. Vol. 310 (1990), Kluwer Academic, Dordrecht 571-572
9. Cairo L., Feix M.R., and Llibre J. Integrability and algebraic solutions for planar polynomial differential systems with emphasis on the quadratic systems. Resenhas de Universidade de Sao Paulo 4 (2000) 127-161
10. Cairo L., Giacomini H., and Llibre J. Liouvillian first integrals for the planar Lot ka-Vol terra system. Rend. Circ. Mat. Palermo 52 (2003) 389-418
11. Cairo L., and Llibre J. Darbouxian first integrals and invariants for real quadratic systems having an invariant conic. J. Phys. A 35 (2002) 589-608
12. (£. Trans. Amer. Math. Soc. 349 (1997) 2211-2228
13. Cantrijn F., and Sarlet W. Generalizations of Noether's theorem in classical mechanics. SI AM Rev. 23 (1980) 467-494
14. Carleman T. Application de la theorie des equations integrates lineaires aux systemes d'equations dif- ferentielles non lineaires. Acta Math. 59 (1932) 63
15. Carnicer M.M. The Poincare problem in the nondicritical case. Ann. Math. 140 (1994) 289-294
16. Cerveau D., and Lins Neto A. Holomorphic foliations in CP(2) having an invariant algebraic curve. Ann. Inst. Fourier 41 (1991) 883-903
17. Chavarriga J. Integrable systems in the plane with a center type linear part. Appl. Math. 22 (1994) 285-309
18. Chavarriga J., Giacomini H., and Gine J. An improvement to Darboux integrability theorem for systems having a center. Appl. Math. Lett. 12 (1999) 85-89
19. Chavarriga J., Giacomini H., Gine J., and Llibre J. On the integrability of two-dimensional flows. J. Dif- ferential Equations 157 (1999) 163-182
20. Chavarriga J., Giacomini H., Gine J., and Llibre J. Darboux integrability and the inverse integrating factor. J. Differential Equations 194 (2003) 116-139
21. Chavarriga J., Giacomini H., and Llibre J. Uniqueness of algebraic limit cycles for quadratic systems. J. Math. Anal. Appl. 261 (2001) 85-99
22. Chavarriga J., and Llibre J. On the algebraic limit cycles of quadratic systems. Proceedings of the IV Catalan Days of Applied Mathematics (Tarragona, 1998) (1998), Univ. Rovira Virgili, Tarragona 17-24
23. Chavarriga J., and Llibre J. Invariant algebraic curves and rational first integrals for planar polynomial vector fields. J. Differential Equations 169 (2001) 1-16
24. Chavarriga J., Llibre J., and Moulin Ollagnier J. On a result of Darboux. J. Comput. Math. 4 (2001) 197-210
25. J. Chavarriga, J. Llibre and J. Sorolla. Algebraic limit cycles of quadratic systems. J. Differential Equations, to appear.
26. Chavarriga J., Llibre J., and Sotomayor J. Algebraic solutions for polynomial systems with emphasis in the quadratic case. Exposition. Math. 15 (1997) 161-173
27. Preprint. Christopher C. Polynomial systems with invariant algebraic curves (1991), University of Wales, Aberys-twyth
28. Christopher C.J. Invariant algebraic curves and conditions for a center. Proc. Roy. Soc. Edinburgh A 124 (1994) 1209-1229
29. Preprint. Christopher C. Polynomial systems with invariant algebraic curves (1993)
30. Christopher C., and Llibre J. Algebraic aspects of integrability for polynomial systems. Qualitative Theor. Planar Differential liquations 1 (1999) 71-95
31. Christopher C., and Llibre J. Integrahility via invariant algebraic curves for planar polynomial differential systems. Ann. Differential Equations 16 (2000) 5-19
32. Christopher C., and Llibre J. A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily Jiigh degree without rational first integrals. Proc. Amer. Math. Soc. 130 (2002) 2025-2030
33. Christopher C., Llibre J., Panta/i C., and Zhang X. Darboux integrahility and invariant algebraic curves for planar polynomial systems. J. Phys. A 35 (2002) 2457-2476
34. Preprint. Christopher C., Llibre J., and Pereira J.V. Multiplicity of invariant algebraic curves in polynomial vector fields (2003)
35. C. Christopher, J. Llibre, G. Swirszcz. Invariant algebraic curves of large degree for quadratic systems. J. Math. Anal. Appl., to appear.
36. Coppel W.A. Some quadratic systems with at most one limit cycle. Dynam. Report. 2 (1989) 61-88
37. Co/ma D., and §uba A. Partial integrals and the first focal value in the problem of centre. Nonlinear Differential liquations Appl. 2 (1995) 21-34
38. 2eme Serie. Darboux G. Memoire sur fes equations differentielles algebriques du premier ordre et du premier degre (Melanges). 123-144. 151-200. Bull. Sci. Math. 2 (1878) 60-96
39. Ecalle J. Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac (1992), Hermann
40. Evdokimenco R.M. Construction of algebraic paths and the qualitative investigation in the large of the properties of integral curves of a system of differential equations. Differential Equations 6 (1970) 1349-1358
41. Evdokimenco R.M. Behavior of integnd curves of a dynamic system. Differential Equations 9 (1974) 1095-1103
42. Evdokimenco R.M. Investigation in the large of a dynamic system with a given integral curve. Differential Equations 15 (1979) 215-221
43. Filiptsov V.F. Algebraic limit cycles. Differential Equations 9 (1973) 983-986
44. Frommer M. Uber das Auftreten von Wirbeln unit Strudeln (geschlossener und spiraliger Integralkurven) in tier Umgebung rationaler Unbestimmheitsstellen. Math. Ann. 109 (1934) 395-424
45. Reprinted. Fulton W. Algebraic Curves, Mathematics Lecture Note Series, Benjamin-Cummings (1969). In: Wesley A. (Ed) (1989)
46. Furta S.D. On non-integrability of general systems of differential equations. Z. Angew. Math. Phys. 47 (1996) 112-131
47. Giacomini H., Llibre J., and Viano M. On the nonexistence, existence and uniqueness of limit cycles. Nonlinearity 9 (1996) 501-516
48. Giacomini H., Llibre J., and Viano M. On the shape of limit cycles that bifurcate from Hamiltonian centers. Nonlinear Anal. 41 (2000) 523-537
49. Giacomini H., Llibre J., and Viano M. On the shape of limit cycles that bifurcate from non-Hamiltonian centers. Nonlinear Anal. 43 (2001) 837-859
50. Giacomini H., Llibre J., and Viano M. Arbitrary order bifurcations for perturbed Hamiltonian planar systems via the reciprocal of an integrating factor. Nonlinear Anal. 48 (2002) 117-136
51. Giacomini H.J., Repetto C.E., and Zandron O.P. Integrals of motion of three-dimensional non-Hamiltonian dynamical systems. J. Phys. A 24 (1991) 4567-4574
52. Goriely A. Integrability and Non-lntegrability of Dynamical Systems. Advanced Series in Nonlinear Dynamics (2001), World Scientific
53. Grammaticos B., Moulin Ollagnier J., Ramani A., Strelcyn J.M., and Wojciechowski S. Integrals of quadratic ordinary differential equations in R: the Lotka-Volterra system. Physica A 163 (1990) 683-722
54. Gutierrez C., and Llibre J. Darboux integrability for polynomial vector fields on the 2-dimensional sphere. Extracta Math. 17 (2002) 289-301
55. Hietarinta J. Direct methods for the search of the second invariant. Phys. Rep. 147 (1987) 87-154
56. Hilbert D. Mathematische Prohleme. Lecture, Second Internat. Congr. Math. (1900)
57. Hilbert D. Nachr. Gcs. Wiss. Gottingen Math. Phys. KL. (1900) 253-297
58. Hilbert D. English transl: Bull. Amcr. Math. Soc. 8 (1902) 437-479
59. Ilyashenko Yu. hiniteness Theorems for Limit Cycles. Translations of Math. Monographs. Amer. Math. Soc. (1991)
60. Jouunolou J.P. Equations de PfaffAlgebriques. Lectures Notes in Mathematics (1979), Springer-Verlag, New York/Berlin
61. (in Dutch). Kapteyn W. On the midpoints of integral curves of differential equations of the first degree. Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland (1911) 1446-1457
62. Kapteyn W. New investigations on the midpoints of integrals of differential equations of the first degree. Nederl. Akad. Wetensch. Verslag Afd. Natuurk 20 (1912) 1354-1365
63. (in Dutch). Kapteyn W. New investigations on the midpoints of integrals of differential equations of the first degree. Nederl. Akad. Wetensch. Verslag Afd. Natuurk 21 (1912) 27-33
64. Kooij R., and Christopher C. Algebraic invariant curves and the integrahilitx of polynomial systems. Appl. Math. Lett. 6 (1993) 51-53
65. Lax P.D. Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21 (1968) 467-490
66. Li W., Llibre J., Nicolau M., and Zhang X. On the differentiability of first integrals of two-dimensional flows. Proc. Amer. Math. Soc. 130 (2002) 2079-2088
67. Li W., Llibre J., and Zhang X. Planar analytic vector fields with generalized rational first integrals. Bull. Sci. Math. 125 (2001) 341-361
68. Lins Neto A. Some examples for the Poincare and Painleve problems. Ann. Sci. Ecole Norm. Sup. 35 (2002) 231-266
69. Llibre J., Perez del Rio J.S., and Rodrfguez J.A. Phase portraits of a new class of integrable quadratic- vector fields. Dynam. Contin. Discrete Impuls. Systems 7 (2000) 595-616
70. Llibre J., and Rodrfguez G. Invariant hyperplanes and Darbox integrability for d-dimensional polynomial differential systems. Bull. Sci. Math. 124 (2000) 1-21
71. Llibre J., and Rodrfguez G. Darboux integrability of polynomial vector fields on 2-dimensional surfaces. J. Bifurcation Chaos 12 (2002) 2821-2833
72. Llibre J., and Rodrfguez G. Configurations of limit cycles and planar polynomial vector fields. J. Differ- ential Equations 198 (2004) 374-380
73. J. Llibre and D. Schlomiuk. The geometry of differential quadratic systems with a weak focus of third order. Canadian J. Math., to appear.
74. Serie II LI. Llibre J., and Zhang X. Darboux integrability of real polynomial vector fields on regular algebraic hypersurfaces. Rend. Circ. Mat. Palermo (2002) 109-126
75. Lotka A.J. Analytical note on certain rhythmic relations in organic systems. Proc. Nat. Acad. Sci. 6 (1920) 410
76. Man Y.-K., and Maccallum M.A.H. A rational approach to the Prelle-Singer algorithm. J. Symbolic Comput. 24 (1997) 31-43
77. Markus L. Global structure of ordinary differential equations in the plane. Trans. Amer. Math. Soc. 76 (1954) 127-148
78. Moulin Ollagnier J. Polynomial first integrals of the Lotka-Volterra system. Bull. Sci. Math. 121 (1997) 463-476
79. Moulin Ollagnier J. Rational integration of the Lotka-Volterra system. Bull. Sci. Math. 123 (1999) 437-466
80. Moulin Ollagnier J. Liouvillian Integration of the Lotka-Volterra system. Qualitative Theor. Dynamical Systems 2 (2001) 307-358
81. Moulin Ollagnier J. About a conjecture on quadratic vector fields. J. Pure Appl. Algebra 165 (2001) 227-234
82. Moulin Ollagnier J., Nowicki A., and Strelcyn J.M. On the non-existence of constants of derivations: the proof of a theorem of Jouanolou and its development. Bull. Sci. Math. 119 (1995) 195-233
83. Nemytskii V.V., and Stepanov V.V. Qualitative Theory of Differential Equations (1989), Dover, New York
84. Neumann D.A. Classification of continuous flows on 2-manifolds. Proc. Amer. Math. Soc. 48 (1975) 73-81
85. Odani K. The limit cycle of the van der Pol equation is not algebraic. J. Differential Equations 115 (1995) 146-152
86. Olvcr P.J. Applications of Lie Groups to Differential Equations (1986), Springer, New York
87. Pearson J.M., Lloyd N.G., and Christopher C.J. Algorithmic derivation of centre conditions. SIAM Rev. 38 (1996) 619-636
88. Lima, Peru. Pereira J.V. Integrabilidade de equacoes diferenciais no piano complexo. Monografias del IMCA (2002)
89. Poincare H. Sur /'integration des equations differentielles du premier ordre et du premier degre I and /. Rend. Circ. Mat. Palermo 5 (1891) 161-191
90. Poincare H. Sur /'integration des equations differentielles du premier ordre et du premier degre I and /. Rend. Circ. Mat. Palermo 11 (1897) 193-239
91. Prelle M.J., and Singer M.K. Elementary first integrals of differential equations. Trans. Amer. Math. Soc. 279 (1983) 613-636
92. j. Acta Math.Sinica 8 (1958) 23-35
93. Rousseau C., and Schlomiuk D. Cubic vector fields symmetric with respect to a center. J. Differential Equations 123 (1995) 388-436
94. Schecter S., and Singer M.F. A class of vectorfields on S that are topologically equivalent to polynomial vectorfields. J. Differential Equations 57 (1985) 406-435
95. Schlomiuk D. Elementary first integrals and algebraic invariant curves of differential equations. Exposi- tion. Math. 11 (1993) 433-454
96. Schlomiuk D. Algebraic particular integrals, integrability and the problem of the center. Trans. Amer. Math. Soc. 338 (1993) 799-841
97. Schlomiuk D. Algebraic and geometric aspects of the theory of polynomial vector fields. In: Schlomiuk D. (Ed). Bifurcations and Periodic Orbits of Vector Fields (1993) 429-467
98. Shubc A.S. Partial integrals, integrability, and the center problem. Differential Equations 32 (1996) 884-892
99. Singer M.F. Liouvillian first integrals of differential equations. Trans. Amer. Math. Soc. 333 (1992) 673-688
100. Sotomayor J. Licoes de Equacoes Diferenciais Ordindrias. IMPA, Rio de Janerio (1979)
101. Strelcyn J.M., and Wojciechowski S. A method of finding integrals for three-dimensional dynamical systems. Phys. Lett. A 133 (1988) 207-212
102. Sverdlove R. Inverse problems for dynamical systems. J. Differential Equations 42 (1981) 72-105
103. (in Russian). Valeeva R.T. Construction of a differential equation with given limit cycles and critical points. Volz Math. Sb. 5 (1966) 83-85
104. Volterra V. Lemons sur la theorie Mathematique de la lutte pour la vie (1931), Gauthier-Villars, Paris
105. Walcher S. On the Poincare problem. J. Differential Equations 166 (2000) 51-78
106. Weil J.A. Constcmtes et polynomes de Darboux en algebre differentielle: applications aux systemes dif- ferentiels lineaires. Ph.D., Ecole Polytecnique (1995)
107. (in Russian). Yablonskii A.I. Limit cycles of a certain differential equations. Differential Equations 2 (1966) 335-344
108. Yanqian Y. Theory of Limit Cycles, Translations of Math. Monographs. Amer. Math. Soc., Prov- idence (1986)
109. Zhifcn Z., Tongren D., Wenzao H., and Zhenxi D. Qualitative Theory of Differential Equa- tions. Translations of Math. Monographs. Amer. Math. Soc, Providence (1992)
110. Preprint. Zotadek H. The solution of the center-focus problem (1992)
111. Zotadek H. On algebraic solutions of algebraic Pfaff equations. Studia Math. 114 (1995) 117-126