Abelian integrals and Hilbert’s 16th problem. Hilbert’s 16th problem asks for H(n) - the maximal number of limit cycles (isolated periodic orbits) the family of 2D polynomial vector fields of degree n

Description

Abelian integrals and Hilbert’s 16th problem. Hilbert’s 16th problem asks for H(n) - the maximal number of limit cycles (isolated periodic orbits) the family of 2D polynomial vector fields of degree n can display. The restricted version of this problem asks for Z(n) - the number of limit cycles that can bifurcate from a perturbation of a Hamiltonian system. The aim of this project is to significantly improve our knowledge about the solution to Hilbert’s 16th problem and its infinitesimal version by proving upper and lower bounds for important families of planar polynomial vector fields. We will use a combination of tools from dynamical systems, validated numerics, and formal proofs to put the findings on a truly solid foundation.. Scheme: Discovery Projects. Field: 4904 - Pure Mathematics. Lead: Prof Warwick Tucker