The displacement map associated to polynomial perturbations of some nongeneric Hamiltonians

TL;DR

This paper generalizes the description of the Principal Poincaré Pontryagin Function for certain Hamiltonians, showing it can be expressed as an iterated integral involving a logarithmic function with specific properties.

Contribution

It extends previous results by characterizing the Principal Poincaré Pontryagin Function for a broader class of Hamiltonians with real points at infinity.

Findings

Principal Poincaré Pontryagin Function is an iterated integral of length at most 2.

The integral involves a logarithmic function with a single ramification point at infinity.

Generalization applies to Hamiltonians with real points at infinity under certain conditions.

Abstract

It is known that the Principal Poincar\'e Pontryagin Function is generically an Abelian integral. In non generic cases it is an iterated integral. In previous papers one of the authors gives a precise description of the Principal Poincar\'e Pontryagin Function, an iterated integral af length at most 2, involving a logarithmic function with only one ramification at a point at infinity. We show here that this property can be generalized to Hamiltonians having real points at infinity and satisfying some properties.