Restricted Versions of the Hilbert 16th Problem and Related Topics in the Theory of Analytic Foliations

Cornell University, Ithaca NY

Investigators

Abstract

The Hilbert 16th problem, part 2, is: "What may be said about the number and location of limit cycles of a planar polynomial vector field?" Traditionally this question is interpreted as a problem of finding an upper bound of the number of limit cycles of a polynomial vector field as a function of the degree of the polynomials. Even for the degree two, the upper bound is not yet found, and its existence is not yet proved. It makes sense to consider "restricted versions" of the Hilbert 16th problem. Namely, the set or all polynomial vector fields is replaced by a particular subset of by a similar class. The examples are Lienard and Abel equations (the latter ones are polynomial in the phase variable with coefficients 1-periodic in time). Even for these equations the problem of the number of limit cycles stays open. It is solved by the PI with an extra restriction: Abel equations, and Lienard ones with the polynomial of odd degree, are considered together with an upper bound for the magnitudes of the coefficients; the upper estimate on the number of limit cycles depends on this upper bound of the magnitudes. This is the result from the prior NSF support. In the current project we try to get rid of this latter restriction, and to give an estimate of the number of limit cycles that depends on the degrees of the polynomials in the right hand side only; for Abel equations the coefficients should be trigonometric polynomials of given degree. We hope to use mighty tools of the theory of complex analytic foliations, growth and zeros theorems for holomorphic functions and methods developed under the prior NSF support. Another important problem to be studied is the infinitesimal Hilbert 16th problem. It requires to estimate the number of limit cycles generated by a small perturbation from the ovals of the Hamiltonian polynomial vector field; closed orbits of the latter field form continuous families. This problem is reduced to the estimate of the number of zeros of an Abelian integral, that is, an integral of a polynomial 1-form over the ovals of a real polynomial in the plane; the estimate should be given in terms of the degrees of the polynomial Hamiltonian function and of the integrand. This problem was investigated by the author since 69; later on by Yakovenko, D.Novikov, Horosov, Gavrilov, Petrov, Khovanski, Varchenko and others. Some progress was obtained by Glutsuk and the author for the restricted version of the problem when the Hamiltonian polynomial is taken of a special type and of arbitrary degree. One of the goals of this project is to get an explicit upper estimate that is expected to be an exponential of a polynomial of the degree of the Hamiltonian function, provided that the integrand has a smaller degree. The theory of dynamical systems is the realm of determinism, on one hand, and of chaos, on the other hand. Vector fields in the phase space of dimension higher than two form the realm of chaos. This was understood in 1960's, and henceforth, this realm is the subject of the top interest for mathematicians, computer scientists and physicists. On the other hand, the classical subject of planar differential equations which may be called "realm of order" attracted the interest of researchers during more that one hundred years, beginning with Poincare and Hilbert. Hilbert's 16th problem is the main one in this domain. It persists the efforts of mathematicians during 100 years, and it is clear now that simplified "restricted" versions of the problem should be attacked first. The project suggests some concrete ways of this attack based on new ideas and the progress from the prior NSF support.