Conditions to the existence of center in planar systems and center for Abel equations

2017/07/10 by Anderson L. A. de Araujo, de Araujo, Anderson L. A., Abílio Lemos +3 · 1 citation
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA

paper · pdf · doi:10.48550/arxiv.1707.02664

arxiv created 2017/07/10 · arxiv updated 2017/07/11

Abstract

Abel equations of the form x'(t)=f(t)x³(t)+g(t)x²(t), t ∈ [-a,a], where a>0 is a constant, f and g are continuous functions, are of interest because of their close relation to planar vector fields. If f and g are odd functions, we prove, in this paper, that the Abel equation has a center at the origin. We also consider a class of polynomial differential equations x = -y+P_n(x,y) and y = x+Q_n(x,y), where P_n and Q_n are homogeneous polynomials of degree n. Using the results obtained for Abel's equation, we obtain a new subclass of systems having a center at the origin.

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