Center conditions. III: Parametric and model center problems
The authors consider the Abel equations \[ (*)\;y'=p(x)y^2+ q(x)y^3\quad\text{and}\quad (**)\;y'=p(x)y^2+ \varepsilon q(x)y^3 \] with \(p(x)\), \(q(x)\) polynomials in \(x\) of degrees \(d_1,d_2\), respectively, and \(\varepsilon\) a complex parameter. A center condition for (*) is that \(y(0)=y(1)\) for any solution \(y(x)\) to (*). The equation (**) is said to have a parameter center if, for any \(\varepsilon\) and for any solution \(y(x,\varepsilon)\) to (**), \(y(0, \varepsilon)= y(1,\varepsilon)\). The authors first give another proof, shown before by the same authors, that the parametric center condition implies vanishing of all the moments \(m_k(1)\), where \(m_k(x)= \int^x_0 P^k(t) q(t)dt\), \(P(x)=\int^x_0 p(t)dt\), and then investigate the structure of zeros of \(m_k(x)\) and generalize a canonical representation of \(m_k(x)\). On this basis, the authors study some parametric center problems. For part II see the review above.
scientific article; zbMATH DE number 1506583