Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems

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October 2013, 33(10): 4531-4547. doi: 10.3934/dcds.2013.33.4531

Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems

Jaume Giné ^1, , Maite Grau ^1, and Jaume Llibre ^2,

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

Received November 2012 Revised January 2013 Published April 2013

Abstract
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In this paper we find necessary and sufficient conditions in order that a planar quasi--homogeneous polynomial differential system has a polynomial or a rational first integral. We also prove that any planar quasi--homogeneous polynomial differential system can be transformed into a differential system of the form $\dot{u} \, = \, u f(v)$, $\dot{v} \, = \, g(v)$ with $f(v)$ and $g(v)$ polynomials, and vice versa.

Keywords: polynomial first integral, rational first integral., Quasi--homogeneous polynomial differential equations, integrability problem.

Mathematics Subject Classification: Primary: 34C05, 34A34, 34C2.

Citation: Jaume Giné, Maite Grau, Jaume Llibre. Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4531-4547. doi: 10.3934/dcds.2013.33.4531

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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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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