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Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems
1. | Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida |
2. | Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia |
References:
[1] |
A. Algaba, E. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 395-420.
doi: 10.1088/0951-7715/22/2/009. |
[2] |
A. Algaba, C. García and M. Reyes, Rational integrability of two-dimensional quasi-homogeneous polynomial differential systems, Nonlinear Anal., 73 (2010), 1318-1327.
doi: 10.1016/j.na.2010.04.059. |
[3] |
J. C. Artés, J. Llibre and N. Vulpe, Quadratic systems with a polynomial first integral: A complete classification in the coefficient space $\mathbbR^{12}$, J. Differential Equations, 246 (2009), 3535-3558.
doi: 10.1016/j.jde.2008.12.010. |
[4] |
J. Chavarriga, I. A. García and J. Giné, On integrability of differential equations defined by the sum of homogeneous vector fields with degenerate infinity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 711-722.
doi: 10.1142/S0218127401002390. |
[5] |
J. Chavarriga, H. Giacomini and J. Giné, Polynomial inverse integrating factors, Ann. Differential Equations, 16 (2000), 320-329. |
[6] |
S. D. Furta, On non-integrability of general systems of differential equations, Z. Angew Math. Phys., 47 (1996), 112-131.
doi: 10.1007/BF00917577. |
[7] |
I. A. García, On the integrability of quasihomogeneous and related planar vector fields, Int. J. Bifurcation and Chaos, 13 (2003), 995-1002.
doi: 10.1142/S021812740300700X. |
[8] |
B. García, J. Llibre and J. S. Pérez del Río, Quasi-homogeneous planar polynomial differential systems and their integrability, preprint, (2012). |
[9] |
A. Goriely, Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys., 37 (1996), 1871-1893.
doi: 10.1063/1.531484. |
[10] |
J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar homogeneous polynomial differential systems, preprint (2012). |
[11] |
J. Giné and J. Llibre, On the planar integrable differential systems, Z. Angew. Math. Phys., 62 (2011), 567-574.
doi: 10.1007/s00033-011-0116-5. |
[12] |
J. Giné and X. Santallusia, Essential variables in the integrability problem of planar vector fields, Phys. Lett. A, 375 (2011), 291-297.
doi: 10.1016/j.physleta.2010.11.026. |
[13] |
E. Isaacson and H. B. Keller, "Analysis of Numerical Methods," Dover Publications, Inc., New York, 1994. |
[14] |
W. Li, J. Llibre and X. Zhang, Planar analytic vector fields with generalized rational first integrals, Bull. Sci. Math., 125 (2001), 341-361.
doi: 10.1016/S0007-4497(01)01083-1. |
[15] |
W. Li, J. Llibre and X. Zhang, Local first integrals of differential systems and diffeomorphisms, Z. Angew. Math. Phys., 54 (2003), 235-255.
doi: 10.1007/s000330300003. |
[16] |
J. Llibre, C. Pantazi and S. Walcher, First integrals of local analytic differential systems, Bull. Sci. Math., 136 (2012), 342-359.
doi: 10.1016/j.bulsci.2011.10.003. |
[17] |
J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280.
doi: 10.1088/0951-7715/15/4/313. |
[18] |
J. Moulin-Ollagnier, Polynomial first integrals of the Lotka-Volterra system, Bull. Sci. Math., 121 (1997), 463-476. |
[19] |
J. Moulin-Ollagnier, Rational integration of the Lotka-Volterra system, Bull. Sci. Math., 123 (1999), 437-466.
doi: 10.1016/S0007-4497(99)00111-6. |
[20] |
H. Poincaré, Sur l'intégration algébrique des équations differentiels, C. R. Acad. Sci., 112 (1891), 761-764. |
[21] |
H. Poincaré, Sur l'intégration algébrique des équations differentiels du 1er ordre et du 1er degré, Rend. Circ. Mat. Palermo, 5 (1891), 161-191. |
[22] |
H. Poincaré, Sur l'intégration algébrique des équations differentiels du 1er ordre et du 1er degré, Rend. Circ. Mat. Palermo, 11 (1897), 193-239. |
[23] |
A. Tsygvintsev, On the existence of polynomial first integrals of quadratic homogeneous systems of ordinary differential equations, J. Phys. A: Math. Gen., 34 (2001), 2185-2193.
doi: 10.1088/0305-4470/34/11/311. |
[24] |
H. Yoshida, Necessary conditions for existence of algebraic first integrals I and II, Celestial Mech., 31 (1983), 363-379, 381-399. |
show all references
References:
[1] |
A. Algaba, E. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 395-420.
doi: 10.1088/0951-7715/22/2/009. |
[2] |
A. Algaba, C. García and M. Reyes, Rational integrability of two-dimensional quasi-homogeneous polynomial differential systems, Nonlinear Anal., 73 (2010), 1318-1327.
doi: 10.1016/j.na.2010.04.059. |
[3] |
J. C. Artés, J. Llibre and N. Vulpe, Quadratic systems with a polynomial first integral: A complete classification in the coefficient space $\mathbbR^{12}$, J. Differential Equations, 246 (2009), 3535-3558.
doi: 10.1016/j.jde.2008.12.010. |
[4] |
J. Chavarriga, I. A. García and J. Giné, On integrability of differential equations defined by the sum of homogeneous vector fields with degenerate infinity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 711-722.
doi: 10.1142/S0218127401002390. |
[5] |
J. Chavarriga, H. Giacomini and J. Giné, Polynomial inverse integrating factors, Ann. Differential Equations, 16 (2000), 320-329. |
[6] |
S. D. Furta, On non-integrability of general systems of differential equations, Z. Angew Math. Phys., 47 (1996), 112-131.
doi: 10.1007/BF00917577. |
[7] |
I. A. García, On the integrability of quasihomogeneous and related planar vector fields, Int. J. Bifurcation and Chaos, 13 (2003), 995-1002.
doi: 10.1142/S021812740300700X. |
[8] |
B. García, J. Llibre and J. S. Pérez del Río, Quasi-homogeneous planar polynomial differential systems and their integrability, preprint, (2012). |
[9] |
A. Goriely, Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys., 37 (1996), 1871-1893.
doi: 10.1063/1.531484. |
[10] |
J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar homogeneous polynomial differential systems, preprint (2012). |
[11] |
J. Giné and J. Llibre, On the planar integrable differential systems, Z. Angew. Math. Phys., 62 (2011), 567-574.
doi: 10.1007/s00033-011-0116-5. |
[12] |
J. Giné and X. Santallusia, Essential variables in the integrability problem of planar vector fields, Phys. Lett. A, 375 (2011), 291-297.
doi: 10.1016/j.physleta.2010.11.026. |
[13] |
E. Isaacson and H. B. Keller, "Analysis of Numerical Methods," Dover Publications, Inc., New York, 1994. |
[14] |
W. Li, J. Llibre and X. Zhang, Planar analytic vector fields with generalized rational first integrals, Bull. Sci. Math., 125 (2001), 341-361.
doi: 10.1016/S0007-4497(01)01083-1. |
[15] |
W. Li, J. Llibre and X. Zhang, Local first integrals of differential systems and diffeomorphisms, Z. Angew. Math. Phys., 54 (2003), 235-255.
doi: 10.1007/s000330300003. |
[16] |
J. Llibre, C. Pantazi and S. Walcher, First integrals of local analytic differential systems, Bull. Sci. Math., 136 (2012), 342-359.
doi: 10.1016/j.bulsci.2011.10.003. |
[17] |
J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280.
doi: 10.1088/0951-7715/15/4/313. |
[18] |
J. Moulin-Ollagnier, Polynomial first integrals of the Lotka-Volterra system, Bull. Sci. Math., 121 (1997), 463-476. |
[19] |
J. Moulin-Ollagnier, Rational integration of the Lotka-Volterra system, Bull. Sci. Math., 123 (1999), 437-466.
doi: 10.1016/S0007-4497(99)00111-6. |
[20] |
H. Poincaré, Sur l'intégration algébrique des équations differentiels, C. R. Acad. Sci., 112 (1891), 761-764. |
[21] |
H. Poincaré, Sur l'intégration algébrique des équations differentiels du 1er ordre et du 1er degré, Rend. Circ. Mat. Palermo, 5 (1891), 161-191. |
[22] |
H. Poincaré, Sur l'intégration algébrique des équations differentiels du 1er ordre et du 1er degré, Rend. Circ. Mat. Palermo, 11 (1897), 193-239. |
[23] |
A. Tsygvintsev, On the existence of polynomial first integrals of quadratic homogeneous systems of ordinary differential equations, J. Phys. A: Math. Gen., 34 (2001), 2185-2193.
doi: 10.1088/0305-4470/34/11/311. |
[24] |
H. Yoshida, Necessary conditions for existence of algebraic first integrals I and II, Celestial Mech., 31 (1983), 363-379, 381-399. |
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