TY - JOUR
T1 - Chebyshev property of complete elliptic integrals and its application to abelian integrals
AU - Gasull, Armengol
AU - Li, Weigu
AU - Llibre, Jaume
AU - Zhang, Zhifen
PY - 2002/1/1
Y1 - 2002/1/1
N2 - 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) = ∫oπ/2 dθ/√1-k2 sin2 θ and E(k) = ∫oπ/2 √1 - k2 sin2 θdθ 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.
AB - 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) = ∫oπ/2 dθ/√1-k2 sin2 θ and E(k) = ∫oπ/2 √1 - k2 sin2 θdθ 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.
U2 - 10.2140/pjm.2002.202.341
DO - 10.2140/pjm.2002.202.341
M3 - Article
SN - 0030-8730
VL - 202
SP - 341
EP - 361
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
IS - 2
ER -