The Hilbert 16-th problem and an estimate for cyclicity of an elementary polycycle

TL;DR

This paper investigates the Hilbert 16th problem by providing an estimate for the maximum number of limit cycles near elementary polycycles in planar vector fields, advancing understanding of cyclicity in dynamical systems.

Contribution

It offers a new estimate for the cyclicity of elementary polycycles, contributing to the broader effort to bound limit cycles in polynomial vector fields.

Findings

Derived an upper bound for the cyclicity of elementary polycycles.

Connected local cyclicity estimates to the global Hilbert 16th problem.

Provided methods to estimate limit cycles near polycycles in planar systems.

Abstract

Hilbert-Arnold (HA) problem, motivated by Hilbert 16-th problem, is to prove that for a generic k-parameter family of smooth vector fields {\dot x=v(x,\eps)}_{\eps\in B^k} on the 2-dimensional sphere S^2 has uniformly bounded number of limit cycles (isolated periodic solutions), denoted by LC(\eps), over the parameter \eps, i.e. max_{\eps \in B^k} LC(\eps) <= K < \infty for some K. The HA problem can be reduced to so-called Local Hilbert-Arnold (LHA) problem. Suppose that a generic k-parameter family {\dot x=v(x,\eps)}_{\eps \in B^k}, x\in S^2 for some parameter \eps^*\in B^k has a polycycle (separatrix polygon) gamma consisting of equilibrium points as vertices and connecting separatrices as sides. LHA problem is to estimate B(k)--- the maximal number of limit cycles that can be born in a neighbourhood of gamma for a field \dot x=v(x,\eps), where \eps is close to \eps^*.