Attractors of one-dimensional maps generated by b oundary value problems

Abstract

There exist boundary value problems for which the investigation is reduced to a study of one-dimensional maps given implicitly by the relationship xn+1 + ax n = F (xn + bx n+1). We discuss some problems connected with investigation of such one-dimensional maps, when F is a polynomial. The second part of Hilbert's 16th problem deals with polynomial differential equations in the plane (see (1), (2), (3)). What may be said about the number and location of limit cycles of a planar polynomial vector field of degree k? This Hilbert's problem remains up to now unsolved even for quadratic polynomials. At the same time, it inspired significant progress, in particular, in the geometric theory of planar differential equations, in bifurcation theory, in the theory of normal forms and foliations. As Ilyashenko wrote in (2), traditionally, Hilbert's ques- tion is split into three, each one requiring a stronger answer. Problem 1. Is it true that a planar polynomial vector field