Analytic continuation and fixed points of the Poincaré mapping for a polynomial Abel equation

Abstract

We consider an Abel differential equation $y^{\prime }=p(x)y2+q(x)y3$ with $p(x)$, $q(x)$ polynomials in $x$. For two given points $a$ and $b$ in $\mathbb{C}$, the “Poincaré mapping” of the above equation transforms the values of its solutions at $a$ into their values at $b$. In this paper we study global analytic properties of the Poincaré mapping, in particular, its analytic continuation, its singularities and its fixed points (which correspond to the “periodic solutions” such that $y(a)=y(b)$). On the one hand, we give a general description of singularities of the Poincaré mapping, and of its analytic continuation. On the other hand, we study in detail the structure of the Poincaré mapping for a local model near a simple fixed singularity, where an explicit solution can be written. Yet, the global analytic structure (in particular, the ramification) of the solutions and of the Poincaré mapping in this case is fairly complicated, and, in our view, highly instructive. For a given degree of the coefficients we produce examples with an infinite number of complex “periodic solutions” and analyze their mutual position and branching. Let us recall that Pugh's problem, which is closely related to the classical Hilbert's 16th problem, asks for the existence of a bound to the number of real isolated “periodic solutions”. New findings reported here lead us to propose new insights on the Poincaré mapping. If the “complexity” of the path in the $x$-plane between $a$ and $b$ is a priori bounded, the number of fixed points should be uniformly bounded. We think that, in some sense, this is close to the complex version of Khovansky's fewnomial theory.