Two dimensional Fuchsian systems and the Chebyshev property

Abstract

Let $(x(t),y(t))^\top$ be a solution of a Fuchsian system of order two with three singular points. The vector space of functions of the form $P(t)x(t)+Q(t)y(t)$, where $P,Q$ are real polynomials, has a natural filtration of vector spaces, according to the asymptotic behaviour of the functions at infinity. We describe a two-parameter class of Fuchsian systems, for which the corresponding vector spaces obey the Chebyshev property (the maximal number of isolated zeroes of each function is less than the dimension of the vector space). Up to now, only a few particular systems were known to possess such a nonoscillation property. It is remarkable that most of these systems are of the type studied in the present paper. We apply our results in estimating the number of limit cycles that appear after small polynomial perturbations of several quadratic or cubic Hamiltonian systems in the plane.