Liouvillian first integrals for the planar Lotka-Volterra system

We complete the classication of all Lotka-Volterra systemsx=x(ax+by+c),y=y(Ax+By+C), having a Liouvillian first integral. In our classification we take into account the first integrals coming from the existence of exponential factors.

[1]  Alain Goriely,et al.  Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations , 1996 .

[2]  S. Labrunie,et al.  On the polynomial first integrals of the ({ital a},{ital b},{ital c}) Lotka{endash}Volterra system , 1996 .

[3]  Michael F. Singer.  Liouvillian first integrals of differential equations , 1992 .

[4]  J. M. Ollagnier.  Liouvillian integration of the Lotka-Volterra system , 2001 .

[5]  Jacques-Arthur Weil.  Constantes et polynomes de darboux en algebre differentielle : application aux systemes differentiels lineaires , 1995 .

[6]  T. Carleman.  Application de la théorie des équations intégrales linéaires aux systèmes d'équations différentielles non linéaires , 1932 .

[7]  Jaume Llibre,et al.  Algebraic aspects of integrability for polynomial systems , 1999 .

[8]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[9]  Jarmo Hietarinta,et al.  Direct methods for the search of the second invariant , 1987 .

[10]  J. M. Ollagnier.  Some remarks about the integration of polynomial planar vector fields , 2002 .

[11]  Roberto F. S. Andrade,et al.  The Lorenz model and the method of Carleman embedding , 1981 .

[12]  A. J. Lotka.  Analytical Note on Certain Rhythmic Relations in Organic Systems , 1920, Proceedings of the National Academy of Sciences.

[13]  J. M. Ollagnier.  About a conjecture on quadratic vector fields , 2001 .

[14]  JAUME LLIBRE,et al.  Darbouxian Integrability of Polynomial Vector Fields with Special Emphasis on the Two-Dimensional Surfaces , 2002, Int. J. Bifurc. Chaos.

[15]  P. Lax.  INTEGRALS OF NONLINEAR EQUATIONS OF EVOLUTION AND SOLITARY WAVES. , 1968 .

[16]  Jaume Llibre,et al.  The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order , 2004, Canadian Journal of Mathematics.

[17]  Dana Schlomiuk,et al.  Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields , 1993 .

[18]  M. A. Almeida,et al.  Lie symmetries and invariants of the Lotka–Volterra system , 1995 .

[19]  O. P. Zandron,et al.  Integrals of motion for three-dimensional non-Hamiltonian dynamical systems , 1991 .

[20]  Frans Cantrijn,et al.  GENERALIZATIONS OF NOETHER'S THEOREM IN CLASSICAL MECHANICS* , 1981 .

[21]  J. Llibre,et al.  Invariant Algebraic Curves and Rational First Integrals for Planar Polynomial Vector Fields , 2001 .

[22]  J. Strelcyn,et al.  Integrals of quadratic ordinary differential equations in R3: The Lotka-Volterra system , 1990 .

[23]  C. Christopher.  Invariant algebraic curves and conditions for a centre , 1994, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[24]  Michael F. Singer,et al.  Elementary first integrals of differential equations , 1983 .

[25]  Jaume Giné,et al.  Darboux integrability and the inverse integrating factor , 2003 .

[26]  Jean Moulin Ollagnier,et al.  Rational integration of the Lotka–Volterra system , 1999 .

[27]  Dana Schlomiuk.  Algebraic particular integrals, integrability and the problem of the center , 1993 .

[28]  M. Poincaré,et al.  Sur Ľintégration algébrique des équations différentielles du premier ordre et du premier degré , 1891 .

[29]  J. Llibre,et al.  Multiplicity of Invariant Algebraic Curves and Darboux Integrability , 2000, math/0009020.

[30]  S. Wojciechowski,et al.  A method of finding integrals for three-dimensional dynamical systems , 1988 .

[31]  Jaume Llibre,et al.  Darboux integrability for 3D Lotka-Volterra systems , 2000 .

[32]  J. Moulin-Ollagnier,et al.  Polynomial first integrals of the Lotka-Volterra system , 1997 .

[33]  J. Llibre,et al.  Integrability and Algebraic Solutions for Planar Polynomial Differential Systems with Emphasis on the Quadratic Systems , 1998 .

[34]  Xiang Zhang,et al.  Darboux integrability of real polynomial vector fields on regular algebraic hypersurfaces , 2002 .

[35]  J. Llibre,et al.  Darbouxian integrability for polynomial vector fields on the 2- dimensional sphere , 2002 .

[36]  Alfred Ramani,et al.  The Painlevé property and singularity analysis of integrable and non-integrable systems , 1989 .

[37]  Tassos Bountis,et al.  On the complete and partial integrability of non-Hamiltonian systems , 1984 .

[38]  J. Jouanolou,et al.  Equations de Pfaff algébriques , 1979 .

[39]  J. Llibre,et al.  Darbouxian first integrals and invariants for real quadratic systems having an invariant conic , 2002 .

[40]  J. Llibre,et al.  Invariant hyperplanes and Darboux integrability for d-dimensional polynomial differential systems , 2000 .

[41]  K United,et al.  INTEGRABILITY VIA INVARIANT ALGEBRAIC CURVES FOR PLANAR POLYNOMIAL DIFFERENTIAL SYSTEMS , 2000 .