A note on the fractalization of saddle invariant curves in quasiperiodic systems

Jordi-Lluís Figueras; Àlex Haro

doi:10.3934/dcdss.2016043

Article Contents

2016, Volume 9, Issue 4: 1095-1107. Doi: 10.3934/dcdss.2016043

This issue Previous Article Null controllable sets and reachable sets for nonautonomous linear control systems Next Article Normal forms à la Moser for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium

A note on the fractalization of saddle invariant curves in quasiperiodic systems

Jordi-Lluís Figueras^1, and
Àlex Haro^2,

1.
Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala
2.
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona

Received: October 31, 2015

Revised: March 31, 2016

Published: July 2016

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Abstract

The purpose of this paper is to describe a new mechanism of destruction of saddle invariant curves in quasiperiodically forced systems, in which an invariant curve experiments a process of fractalization, that is, the curve gets increasingly wrinkled until it breaks down. The phenomenon resembles the one described for attracting invariant curves in a number of quasiperiodically forced dissipative systems, and that has received the attention in the literature for its connections with the so-called Strange Non-Chaotic Attractors. We present a general conceptual framework that provides a simple unifying mathematical picture for fractalization routes in dissipative and conservative systems.

Keywords:

Breakdown of saddle invariant curves,
quasiperiodic systems.

Mathematics Subject Classification: Primary: 37G35, 37C55; Secondary: 37D45.

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