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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

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 & Continuous Dynamical Systems - A, 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.  doi: 10.1088/0951-7715/22/2/009.  Google Scholar

[2]

A. Algaba, C. García and M. Reyes, Rational integrability of two-dimensional quasi-homogeneous polynomial differential systems,, Nonlinear Anal., 73 (2010), 1318.  doi: 10.1016/j.na.2010.04.059.  Google Scholar

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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.  doi: 10.1016/j.jde.2008.12.010.  Google Scholar

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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.  doi: 10.1142/S0218127401002390.  Google Scholar

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J. Chavarriga, H. Giacomini and J. Giné, Polynomial inverse integrating factors,, Ann. Differential Equations, 16 (2000), 320.   Google Scholar

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S. D. Furta, On non-integrability of general systems of differential equations,, Z. Angew Math. Phys., 47 (1996), 112.  doi: 10.1007/BF00917577.  Google Scholar

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I. A. García, On the integrability of quasihomogeneous and related planar vector fields,, Int. J. Bifurcation and Chaos, 13 (2003), 995.  doi: 10.1142/S021812740300700X.  Google Scholar

[8]

B. García, J. Llibre and J. S. Pérez del Río, Quasi-homogeneous planar polynomial differential systems and their integrability,, preprint, (2012).   Google Scholar

[9]

A. Goriely, Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations,, J. Math. Phys., 37 (1996), 1871.  doi: 10.1063/1.531484.  Google Scholar

[10]

J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar homogeneous polynomial differential systems,, preprint (2012)., (2012).   Google Scholar

[11]

J. Giné and J. Llibre, On the planar integrable differential systems,, Z. Angew. Math. Phys., 62 (2011), 567.  doi: 10.1007/s00033-011-0116-5.  Google Scholar

[12]

J. Giné and X. Santallusia, Essential variables in the integrability problem of planar vector fields,, Phys. Lett. A, 375 (2011), 291.  doi: 10.1016/j.physleta.2010.11.026.  Google Scholar

[13]

E. Isaacson and H. B. Keller, "Analysis of Numerical Methods,", Dover Publications, (1994).   Google Scholar

[14]

W. Li, J. Llibre and X. Zhang, Planar analytic vector fields with generalized rational first integrals,, Bull. Sci. Math., 125 (2001), 341.  doi: 10.1016/S0007-4497(01)01083-1.  Google Scholar

[15]

W. Li, J. Llibre and X. Zhang, Local first integrals of differential systems and diffeomorphisms,, Z. Angew. Math. Phys., 54 (2003), 235.  doi: 10.1007/s000330300003.  Google Scholar

[16]

J. Llibre, C. Pantazi and S. Walcher, First integrals of local analytic differential systems,, Bull. Sci. Math., 136 (2012), 342.  doi: 10.1016/j.bulsci.2011.10.003.  Google Scholar

[17]

J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems,, Nonlinearity, 15 (2002), 1269.  doi: 10.1088/0951-7715/15/4/313.  Google Scholar

[18]

J. Moulin-Ollagnier, Polynomial first integrals of the Lotka-Volterra system,, Bull. Sci. Math., 121 (1997), 463.   Google Scholar

[19]

J. Moulin-Ollagnier, Rational integration of the Lotka-Volterra system,, Bull. Sci. Math., 123 (1999), 437.  doi: 10.1016/S0007-4497(99)00111-6.  Google Scholar

[20]

H. Poincaré, Sur l'intégration algébrique des équations differentiels,, C. R. Acad. Sci., 112 (1891), 761.   Google Scholar

[21]

H. Poincaré, Sur l'intégration algébrique des équations differentiels du 1er ordre et du 1er degré,, Rend. Circ. Mat. Palermo, 5 (1891), 161.   Google Scholar

[22]

H. Poincaré, Sur l'intégration algébrique des équations differentiels du 1er ordre et du 1er degré,, Rend. Circ. Mat. Palermo, 11 (1897), 193.   Google Scholar

[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.  doi: 10.1088/0305-4470/34/11/311.  Google Scholar

[24]

H. Yoshida, Necessary conditions for existence of algebraic first integrals I and II,, Celestial Mech., 31 (1983), 363.   Google Scholar

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.  doi: 10.1088/0951-7715/22/2/009.  Google Scholar

[2]

A. Algaba, C. García and M. Reyes, Rational integrability of two-dimensional quasi-homogeneous polynomial differential systems,, Nonlinear Anal., 73 (2010), 1318.  doi: 10.1016/j.na.2010.04.059.  Google Scholar

[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.  doi: 10.1016/j.jde.2008.12.010.  Google Scholar

[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.  doi: 10.1142/S0218127401002390.  Google Scholar

[5]

J. Chavarriga, H. Giacomini and J. Giné, Polynomial inverse integrating factors,, Ann. Differential Equations, 16 (2000), 320.   Google Scholar

[6]

S. D. Furta, On non-integrability of general systems of differential equations,, Z. Angew Math. Phys., 47 (1996), 112.  doi: 10.1007/BF00917577.  Google Scholar

[7]

I. A. García, On the integrability of quasihomogeneous and related planar vector fields,, Int. J. Bifurcation and Chaos, 13 (2003), 995.  doi: 10.1142/S021812740300700X.  Google Scholar

[8]

B. García, J. Llibre and J. S. Pérez del Río, Quasi-homogeneous planar polynomial differential systems and their integrability,, preprint, (2012).   Google Scholar

[9]

A. Goriely, Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations,, J. Math. Phys., 37 (1996), 1871.  doi: 10.1063/1.531484.  Google Scholar

[10]

J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar homogeneous polynomial differential systems,, preprint (2012)., (2012).   Google Scholar

[11]

J. Giné and J. Llibre, On the planar integrable differential systems,, Z. Angew. Math. Phys., 62 (2011), 567.  doi: 10.1007/s00033-011-0116-5.  Google Scholar

[12]

J. Giné and X. Santallusia, Essential variables in the integrability problem of planar vector fields,, Phys. Lett. A, 375 (2011), 291.  doi: 10.1016/j.physleta.2010.11.026.  Google Scholar

[13]

E. Isaacson and H. B. Keller, "Analysis of Numerical Methods,", Dover Publications, (1994).   Google Scholar

[14]

W. Li, J. Llibre and X. Zhang, Planar analytic vector fields with generalized rational first integrals,, Bull. Sci. Math., 125 (2001), 341.  doi: 10.1016/S0007-4497(01)01083-1.  Google Scholar

[15]

W. Li, J. Llibre and X. Zhang, Local first integrals of differential systems and diffeomorphisms,, Z. Angew. Math. Phys., 54 (2003), 235.  doi: 10.1007/s000330300003.  Google Scholar

[16]

J. Llibre, C. Pantazi and S. Walcher, First integrals of local analytic differential systems,, Bull. Sci. Math., 136 (2012), 342.  doi: 10.1016/j.bulsci.2011.10.003.  Google Scholar

[17]

J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems,, Nonlinearity, 15 (2002), 1269.  doi: 10.1088/0951-7715/15/4/313.  Google Scholar

[18]

J. Moulin-Ollagnier, Polynomial first integrals of the Lotka-Volterra system,, Bull. Sci. Math., 121 (1997), 463.   Google Scholar

[19]

J. Moulin-Ollagnier, Rational integration of the Lotka-Volterra system,, Bull. Sci. Math., 123 (1999), 437.  doi: 10.1016/S0007-4497(99)00111-6.  Google Scholar

[20]

H. Poincaré, Sur l'intégration algébrique des équations differentiels,, C. R. Acad. Sci., 112 (1891), 761.   Google Scholar

[21]

H. Poincaré, Sur l'intégration algébrique des équations differentiels du 1er ordre et du 1er degré,, Rend. Circ. Mat. Palermo, 5 (1891), 161.   Google Scholar

[22]

H. Poincaré, Sur l'intégration algébrique des équations differentiels du 1er ordre et du 1er degré,, Rend. Circ. Mat. Palermo, 11 (1897), 193.   Google Scholar

[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.  doi: 10.1088/0305-4470/34/11/311.  Google Scholar

[24]

H. Yoshida, Necessary conditions for existence of algebraic first integrals I and II,, Celestial Mech., 31 (1983), 363.   Google Scholar

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