Some new results on Darboux integrable differential systems

Abstract We deal with complex planar differential systems having a Darboux first integral H . We present a definition of remarkable values and remarkable curves associated to H and characterize the existence of a polynomial inverse integrating factor for these systems. Furthermore, we study the relation between the characteristic polynomial F and the inverse integrating factors of the system and show the importance of the numerator of the exponential factor of H in the construction of F .

[1]  Jaume Llibre,et al.  Polynomial inverse integrating factors for polynomial vector fields , 2006 .

[2]  Jaume Giné,et al.  On the Integrability of Two-Dimensional Flows , 1999 .

[3]  H. Zoladek.  On algebraic solutions of algebraic Pfaff equations , 1995 .

[4]  Dana Schlomiuk,et al.  The Full Study of Planar Quadratic Differential Systems Possessing a Line of Singularities at Infinity , 2008 .

[5]  Sebastian Walcher,et al.  On the Poincaré Problem , 2000 .

[6]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[7]  J. Walker,et al.  Book Reviews : THEORY OF BIFURCATIONS OF DYNAMIC SYSTEMS ON A PLANE A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier J. Wiley & Sons, New York , New York (1973) , 1976 .

[8]  H. Giacomini,et al.  On the vanishing set of inverse integrating factors , 2000 .

[9]  I. A. García,et al.  Integral invariants and limit sets of planar vector fields , 2005 .

[10]  Jaume Llibre,et al.  Multiplicity of invariant algebraic curves in polynomial vector fields , 2007 .

[11]  J. Llibre,et al.  Polynomial differential systems having a given Darbouxian first integral , 2004 .

[12]  Isaac A. García,et al.  A Survey on the Inverse Integrating Factor , 2009, 0903.0941.

[13]  Antoni Ferragut,et al.  A New Algorithm for Finding Rational First Integrals of Polynomial Vector Fields , 2010 .

[14]  Jaume Llibre,et al.  On the nonexistence, existence and uniqueness of limit cycles , 1996 .

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

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

[17]  Jaume Llibre,et al.  On the remarkable values of the rational first integrals of polynomial vector fields , 2007 .

[18]  Michael F. Singer,et al.  Liouvillian First Integrals of Differential Equations , 1988, ISSAC.

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

[20]  M. J. Prelle,et al.  Elementary first integrals of differential equations , 1981, SYMSAC '81.

[21]  R. Kooij,et al.  Algebraic invariant curves and the integrability of polynomial systems , 1993 .

[22]  Antoni Ferragut,et al.  Local behavior of planar analytic vector fields via integrability , 2012 .

[23]  Jaume Llibre,et al.  Polynomial inverse integrating factors for quadratic differential systems , 2010 .

[24]  Colin Christopher,et al.  LIOUVILLIAN FIRST INTEGRALS OF SECOND ORDER POLYNOMIAL DIFFERENTIAL EQUATIONS , 1999 .

[25]  Jaume Llibre,et al.  Configurations of limit cycles and planar polynomial vector fields , 2004 .