On evaluation of the topological degree of the Poincare map in some singular situations

TL;DR

This paper develops a method to evaluate the topological degree of the Poincare map for a model of a narrow lagoon under periodic forcing, providing conditions for T-periodic solutions despite singularities.

Contribution

It introduces a novel approach to evaluate the Poincare map's topological degree in singular situations, enabling the analysis of periodic solutions in complex models.

Findings

Derived conditions for parameter values guaranteeing T-periodic solutions.

Addressed the challenge of singular vector fields in the evaluation process.

Extended the applicability of topological degree methods to singular dynamical systems.

Abstract

In the paper we develop a method to evaluate the topological degree of the Poincare map of the mathematical model of narrow lagoon subject to a T-periodic forcing. Using the method developved we arrive to the conditions for the parameteres that guarantee the existence of T-periodic solutions in a given region. The difficulty towards implementing this plan is caused by the fact that the direct employing of M.A. Krasnoselski-A.I. Perov irreversibility approach leads to a singular vector field.