Effective Non-oscillation of Solutions of Fuchsian Systems of Differential Equations and Abelian Integrals

Purdue University, West Lafayette IN

Investigators

Abstract

PI: Dmitry Novikov / A. Gabrielov, Purdue University DMS-0200861 Abstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D. Novikov and A. Gabrielov propose to investigate the global finiteness properties of solutions of Fuchsian systems of ordinary differential equations. The goal of this research is to establish effective upper bounds on the global oscillation of polynomial combinations of these solutions in terms of quantitative characteristics of the Fuchsian systems, and to apply these results to the infinitesimal Hilbert's sixteenth problem. The main interest is the oscillation of solutions near confluent singular points of the hypergeometric equation. Stable oscillations of many natural systems in biology, electronics, engineering, meteorology, etc., are described by limit cycles in the corresponding dynamical systems. If the dynamical system is close to a conservative one, its limit cycles correspond to zeros of an Abelian integral, a solution of an ordinary differential equation with polynomial coefficients. Unlike solutions of algebraic equations, such functions can have many zeros even when the coefficients are low-degree polynomials. The goal of the proposed research is to give an effective upper bound on the number of these zeros in terms of the magnitude of the coefficients, and the corresponding upper bound on the number of limit cycles of the dynamical system. This can be considered as an effective version of the famous Hilbert's sixteenth problem.