Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems

Héctor Barge; José M. R. Sanjurjo

doi:10.3934/dcds.2021204

Article Contents

2022, Volume 42, Issue 6: 2585-2601. Doi: 10.3934/dcds.2021204

This issue Previous Article On the one dimensional cubic NLS in a critical space Next Article Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space

Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems

Héctor Barge^1, , and
José M. R. Sanjurjo^2,

1.
E.T.S. Ingenieros Informáticos, Universidad Politécnica de Madrid, Madrid 28660, Spain
2.
Facultad de Ciencias Matemáticas and Instituto de Matemática Interdisciplinar (IMI), Universidad Complutense de Madrid, Madrid 28040, Spain

^* Corresponding author: Héctor Barge
^* Corresponding author: Héctor Barge

Affectionally dedicated to María Jesús Chasco on the ocassion of her 65th birthday

Received: July 2021

Revised: November 2021

Early access: January 2022

Published: June 2022

The authors are supported by Spanish Ministerio de Ciencia e Innovación grant PGC2018-098321-B-I00

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Abstract

In this paper we study generalized Poincaré-Andronov-Hopf bifurcations of discrete dynamical systems. We prove a general result for attractors in $ n $-dimensional manifolds satisfying some suitable conditions. This result allows us to obtain sharper Hopf bifurcation theorems for fixed points in the general case and other attractors in low dimensional manifolds. Topological techniques based on the notion of concentricity of manifolds play a substantial role in the paper.

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Mathematics Subject Classification: Primary: 37G35, 37B35; Secondary: 57N99.

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Figure 1. Construction of the dyadic solenoid as an attractor of a homeomorphism of $ \mathbb{R}^3 $

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Figure 2. Construction of the Whitehead continuum as an attractor of a homeomorphism of $ \mathbb{R}^3 $

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