Basin of attraction of triangular maps with applications

TL;DR

This paper investigates the dynamics within the basin of attraction of invariant fibers in planar triangular maps, providing insights into their limit behavior and applying findings to various mathematical systems.

Contribution

It introduces a detailed analysis of the limit dynamics in triangular maps with invariant fibers, extending understanding to several classes of planar systems and difference equations.

Findings

Characterization of limit dynamics in the basin of attraction

Application to quasi-homogeneous and difference equations

Identification of conditions for fixed and periodic points

Abstract

We consider some planar triangular maps. These maps preserve certain fibration of the plane. We assume that there exists an invariant attracting fiber and we study the limit dynamics of those points in the basin of attraction of this invariant fiber, assuming that either it contains a global attractor, or it is filled by fixed or 2-periodic points. Finally, we apply our results to a variety of examples, from particular cases of triangular systems to some planar quasi-homogeneous maps, and some multiplicative and additive difference equations, as well.