Simplified Versions of Hilbert 16th Problem and Related Topics in Complex Dynamics and Analytic Foliations

Cornell University, Ithaca NY

Investigators

Abstract

The project deals with numerous topics in the theory of planar differential equations related to Hilbert's 16th problem and continues two previous projects. It considers both real and complex equations. There are two major achievements in the work over the previous project that will be developed in the current one. First, rather unexpectedly, the Kupka-Smale (KS) property was proved for polynomial automorphisms both of real and complex planes. The proof is based on "persistence theorems" and "Petrovski-Landis (PL) strategy". Generally speaking, persistence theorems claim the possibility of global extension of some geometric properties of polynomial dynamical systems over the whole parameter space. PL strategy makes use of persistence theorems to prove or disprove such geometric properties. New persistence theorems for heteroclinic points of polynomial automorphisms of a complex space are expected. Genericity of Kupka-Smale property for such automorphisms is suggested as a consequence. The main tool would be the PL strategy. Another achievement is an upper estimate by Glutsyuk and the PI of the number of zeros of Abelian integrals both in real and complex domains. This estimate is the best amidst other estimates of this kind due to Yakovenko and his students. On the other hand, it provides an approach to the complete solution of the restricted version of the Infinitesimal Hilbert 16th Problem. Together with the PL strategy, this gives an approach to the Infinitesimal Hilbert 16th Problem itself: give an upper bound of the number of real zeros of an integral of a polynomial one-form over the ovals of another polynomial in the plane. The project suggests numerous problems on the persistence properties for polynomial dynamical systems, simultaneous uniformization and topological properties of polynomial foliations. Study of the relations between these branches of the theory is an important part of the project. Moreover, new simultaneous uniformization theorems, together with new generic properties of polynomial and analytic foliation of the complex space are expected. Theory of dynamical systems is split into two parts: multidimensional systems (realm of chaos); two-dimensional systems (realm of order). Hilbert 16th problem is a central one in the theory of two-dimensional systems. The problem itself persists the efforts of mathematicians during more than a hundred years. Centennial history of investigations related to Hilbert 16th problem is reviewed in a survey article by the PI published in the Bulletin of the AMS in 2002. The survey contains, in particular, many results of the previous NSF projects, as well as problems that are subject to the current project. Note that two-dimensional dynamical systems provide models for various problems in physics, engineering and biology (predator-prey models). Understanding of real two-dimensional dynamics is therefore a subject of general scientific interest. On the other hand, study of complex extensions of real dynamical systems provides important new information about real systems and is interesting in itself.