Search results for "Darboux integral"
showing 3 items of 3 documents
Indefinite integrals involving complete elliptic integrals of the third kind
2017
ABSTRACTA method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete …
Pseudo-Abelian integrals along Darboux cycles
2008
We study polynomial perturbations of integrable, non-Hamiltonian system with first integral of Darboux-type with positive exponents. We assume that the unperturbed system admits a period annulus. The linear part of the Poincare return map is given by pseudo-Abelian integrals. In this paper we investigate analytic properties of these integrals. We prove that iterated variations of these integrals vanish identically. Using this relation we prove that the number of zeros of these integrals is locally uniformly bounded under generic hypothesis. This is a generic analog of the Varchenko-Khovanskii theorem for pseudo-Abelian integrals. Finally, under some arithmetic properties of exponents, the p…
Darboux Linearization and Isochronous Centers with a Rational First Integral
1997
Abstract In this paper we study isochronous centers of polynomial systems. It is known that a center is isochronous if and only if it is linearizable. We introduce the notion of Darboux linearizability of a center and give an effective criterion for verifying Darboux linearizability. If a center is Darboux linearizable, the method produces a linearizing change of coordinates. Most of the known polynomial isochronous centers are Darboux linearizable. Moreover, using this criterion we find a new two-parameter family of cubic isochronous centers and give the linearizing changes of coordinates for centers belonging to that family. We also determine all Hamiltonian cubic systems which are Darbou…