Article

Inverse Problems for multiple invariant curves

01/2007; DOI:10.1017/S0308210506000400
Source: OAI

ABSTRACT Planar polynomial vector fields which admit invariant algebraic curves, Darboux integrating factors or Darboux first integrals are of special interest. In the present paper we solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors, under generic assumptions regarding the (multiple) invariant algebraic curves involved. In particular we prove, in this generic scenario, that the existence of a Darboux integrating factor implies Darboux integrability. Furthermore we construct examples where the genericity assumption does not hold and indicate that the situation is different for these.

0 0
·
0 Bookmarks
·
61 Views
• Source
Article: Liouvillian first integrals of second order polynomial differential equations
[hide abstract]
ABSTRACT: We consider polynomial dierential systems in the plane with Liouvillian rst in- tegrals. It is shown that all such systems have Darbouxian integrating factors, and that thesearchforsuch integralscanbereduced toasearch fortheinvariantalgebraiccurves of the system and their 'degenerate' counterparts.
Electronic Journal of Differential Equations. 01/1999;
• Article: Algebraic aspects of integrability for polynomial systems
[hide abstract]
ABSTRACT: We present an introductory survey to the Darboux integrability theory of planar complex and real polynomial differential systems. Our presentation contains some improvements to the classical theory.
Qualitative Theory of Dynamical Systems 01/1999;