Parametric centers for trigonometric Abel equations
The paper is concerned with the one-dimensional version of the center problem for the trigonometric Abel equation \[ \frac{\mathrm{d}y}{\mathrm{d}\theta}=p( \theta) y^{2}+\varepsilon q( \theta) y^{3},\tag{1} \] where \(\varepsilon\) is a small parameter, \(p\) and \(q\) are polynomials in \(\sin\theta\) and \(\cos\theta,\) and \(y( 2\pi,\varepsilon,c) \) denotes the solution of equation (1) satisfying the initial condition \(y( 0,\varepsilon) =c.\) The author is interested in dependence of solutions on the parameter \(\varepsilon\) and, in particular, in obtaining the formula for the first different from zero derivative of the solution with respect to the parameter \(\varepsilon.\) Using the expansion \[ -\frac{1}{y}=-\frac{1}{c}+\varepsilon Y_{1}( c) +\cdots +\varepsilon^{k}Y_{k}( c) +\cdots, \] one can compute the successive derivatives of \(-1/y( 2\pi) \) easier than those of \(y( 2\pi) .\) A recursive formula for the derivatives is obtained through an algorithm that runs for as long as derivatives vanish. In particular, one has \[ Y_{1}( c) =\int_{0}^{2\pi}\frac{cq( \theta) }{1-cP( \theta) }\mathrm{d}\theta, \] where \(q( \theta) \) and \(P( \theta) \) are given trigonometric polynomials, \(P( \theta) =\int_{0}^{\theta}p( \psi) \mathrm{d}\psi.\) Necessary conditions for the derivative \(Y_{1}( c) \) to vanish are derived by studying iterated integrals over the unit complex disc using the tools of complex analysis. This new promising direction of research relates the discussed problem to many interesting research topics like composition of Laurent polynomials or Cauchy integrals of algebraic functions.
trigonometric Abel differential equation
scientific article; zbMATH DE number 5375865
scientific article