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

点击翻译按钮获取中文摘要

PI: Dmitry Novikov / A. Gabrielov, Purdue UniversityDMS-0200861Abstract%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%D. Novikov and A. Gabrielov propose to investigate the globalfiniteness properties of solutions of Fuchsian systems of ordinarydifferential equations. The goal of this research is to establisheffective upper bounds on the global oscillation of polynomialcombinations of these solutions in terms of quantitativecharacteristics of the Fuchsian systems, and to apply these resultsto the infinitesimal Hilbert's sixteenth problem. The main interestis the oscillation of solutions near confluent singular points ofthe hypergeometric equation.Stable oscillations of many natural systems in biology, electronics,engineering, meteorology, etc., are described by limit cycles in thecorresponding dynamical systems. If the dynamical system is closeto a conservative one, its limit cycles correspond to zeros ofan Abelian integral, a solution of an ordinary differentialequation with polynomial coefficients. Unlike solutions ofalgebraic equations, such functions can have many zeros evenwhen the coefficients are low-degree polynomials. The goal of theproposed research is to give an effective upper bound on the number ofthese zeros in terms of the magnitude of the coefficients, and thecorresponding upper bound on the number of limit cycles of thedynamical system. This can be considered as an effective versionof the famous Hilbert's sixteenth problem.