Chebyshev property of complete elliptic integrals and its application to abelian integrals

Armengol Gasull, Weigu Li, Jaume Llibre and Zhifen Zhang

Vol. 202 (2002), No. 2, 341–361

DOI: 10.2140/pjm.2002.202.341

Abstract

This paper has two parts. In the first one we study the maximum number of zeros of a function of the form f(k)K(k) + g(k)E(k), where k ∈ (−1,1), f and g are polynomials, and K(k) = ∫ ₀^π∕2 and E(k) = ∫ ₀^π∕2d𝜃 are the complete normal elliptic integrals of the first and second kinds, respectively. In the second part we apply the first one to obtain an upper bound for the number of limit cycles which appear from a small polynomial perturbation of the planar isochronous differential equation ż = iz + z³, where z = x + iy ∈ ℂ.

Milestones

Received: 9 March 2000

Revised: 13 June 2000

Published: 1 February 2002