TY - JOUR

T1 - Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation

AU - Batenkov, Dmitry

AU - Binyamini, Gal

N1 - Publisher Copyright:
© 2015 Elsevier Inc.

PY - 2015/12/5

Y1 - 2015/12/5

N2 - Given two polynomials P, q we consider the following question: "how large can the index of the first non-zero moment mk=∫abPkq be, assuming the sequence is not identically zero?" The answer K to this question is known as the moment Bautin index, and we provide the first general upper bound: K≤2+deg q +3(deg P-1)2. The proof is based on qualitative analysis of linear ODEs, applied to Cauchy-type integrals of certain algebraic functions.The moment Bautin index plays an important role in the study of bifurcations of periodic solution in the polynomial Abel equation y'=py2+εqy3 for p, q polynomials and ε≪1. In particular, our result implies that for p satisfying a well-known generic condition, the number of periodic solutions near the zero solution does not exceed 5+deg q + 3deg2 p. This is the first such bound depending solely on the degrees of the Abel equation.

AB - Given two polynomials P, q we consider the following question: "how large can the index of the first non-zero moment mk=∫abPkq be, assuming the sequence is not identically zero?" The answer K to this question is known as the moment Bautin index, and we provide the first general upper bound: K≤2+deg q +3(deg P-1)2. The proof is based on qualitative analysis of linear ODEs, applied to Cauchy-type integrals of certain algebraic functions.The moment Bautin index plays an important role in the study of bifurcations of periodic solution in the polynomial Abel equation y'=py2+εqy3 for p, q polynomials and ε≪1. In particular, our result implies that for p satisfying a well-known generic condition, the number of periodic solutions near the zero solution does not exceed 5+deg q + 3deg2 p. This is the first such bound depending solely on the degrees of the Abel equation.

UR - http://www.scopus.com/inward/record.url?scp=84941806411&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2015.07.009

DO - 10.1016/j.jde.2015.07.009

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AN - SCOPUS:84941806411

SN - 0022-0396

VL - 259

SP - 5769

EP - 5781

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 11

ER -