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

PI：Dmitry Novikov/A.Gabrielov，Purdu UniversityDMS-0200861Abstract%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%D.Novikov和A.Gabrielov提出了研究Fuchs常微分方程解的全局有限性。这项研究的目的是根据Fuchsian系统的数量特征建立这些解的多项式组合的全局振动性的有效上界，并将这些结果应用于无穷小Hilbert的第十六问题。主要研究超几何方程在汇合奇点附近解的振动性。在生物、电子、工程、气象等许多自然系统中，用相应的动力系统的极限环来描述系统的稳定振动。如果动力系统接近于保守系统，则其极限环对应于阿贝尔积分的零点，阿贝尔积分是具有多项式系数的常微分方程解。与代数方程的解不同，即使当系数是低次多项式时，这类函数也可以有许多零。该研究的目的是根据系数的大小给出这些零点个数的有效上界，以及相应的动力系统极限环个数的上界。这可以被认为是著名的希尔伯特第十六问题的一个有效版本。

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.