Invariant manifolds for delay endomorphisms

Rovella Alvaro; Vilamajó Francesc; Romero Neptalí

doi:10.3934/dcds.2001.7.35

Article Contents

2001, Volume 7, Issue 1: 35-50. Doi: 10.3934/dcds.2001.7.35

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Invariant manifolds for delay endomorphisms

1.
Universidad de La República, Facultad de Ciencias. Centro de Matemática, Iguá 4225. Montevideo 11400
2.
Universidad Politéecnica de Cataluña, Departament de Matemática Aplicada 2, Escola Técnica Superior d'Enginyers Industrials. Colom 11, 08222. Terrasa, Barcelona
3.
Universidad Centro Occidental Lisandro Alvarado, Departamento de Matemática, Decanato de Ciencias. Apartado Postal 400. Barquisimeto

Received: September 30, 1999

Revised: August 31, 2000

Published: December 2000

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Abstract

Let $F_\mu(x_1,\cdots,x_k)=(x_2,\cdots,x_k,-x_1^2+\mu x_1)$. For any $G$ in a $C^2$ neighborhood $\mathcal{U}$ of the family $F_\mu$, the point at $\infty$ is an attractor (with basin denoted by $B_\infty$), and there exists a repelling fixed point in the boundary of $B_\infty$. This gives the initial step to the study of the whole boundary of $B_\infty$ and the changes it suffers: for perturbations of $F_\mu$ with $\mu$ small, the boundary of $B_\infty$ is an invariant codimension one manifold, while for large values of $\mu$ the basin $B_\infty$ is dense and its complementary set an expanding Cantor set. The techniques developed will be applied to delay endomorphisms.

Keywords:

Invariant manifolds,
Expanding Cantor sets.

Mathematics Subject Classification: 37C05, 37D10, 37D20.

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