The Oseledec and Sacker-Sell spectra for almost periodic linear systems: an example

Johnson, Russell

doi:10.1090/s0002-9939-1987-0870782-7

Cited by 9 publications

(2 citation statements)

References 12 publications

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“…As a consequence, thick annuli in the Mather spectrum correspond to nondegenerate intervals in the Sacker-Sell spectrum, while circles to isolated points. See also [13,27] for related results.…”

Section: 2mentioning

confidence: 98%

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

Figueras¹,

Haro²

2016

Discrete &Amp; Continuous Dynamical Systems - S

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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.

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Section: 2mentioning

confidence: 98%

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

Figueras¹,

Haro²

2016

Discrete &Amp; Continuous Dynamical Systems - S

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“…One can check that the subspaces appearing in the Sacker-Sell decomposition are a sum of subspaces appearing in the Oseledets decomposition. It can happen that the (non-uniform) Oseledets decomposition strictly refines the uniform Sacker-Sell decomposition [9], even in cases where the Sacker-Sell decomposition is trivial [8]. Bochi [3] has shown that generically for a matrix cocycle over a minimal base, the two decompositions coincide.…”

Section: Introductionmentioning

confidence: 99%

On irreducibility of oseledets subspaces

2017

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Abstract. For a cocycle of invertible real n-by-n matrices, the Multiplicative Ergodic Theorem gives an Oseledets subspace decomposition of R n ; that is, above each point in the base space, R n is written as a direct sum of equivariant subspaces, one for each Lyapunov exponent of the cocycle. It is natural to ask if these summands may be further decomposed into equivariant subspaces; that is, if the Oseledets subspaces are reducible. We prove a theorem yielding sufficient conditions for irreducibility of the trivial equivariant subspaces R 2 and C 2 for O 2 (R)-valued cocycles and give explicit examples where the conditions are satisfied.

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Application to Differential Equations and Inclusions

Andres

Górniewicz

2003

Topological Fixed Point Principles for Boundary Value Problems

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The Oseledec and Sacker-Sell spectra for almost periodic linear systems: an example

Abstract: ABSTRACT. We give an example illustrating the relation between the Oseledec spectrum (roughly speaking, the set of Lyapunov exponents) and the SackerSell (or continuous) spectrum for Bohr almost periodic linear systems.

Cited by 9 publications

References 12 publications

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

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

On irreducibility of oseledets subspaces

Application to Differential Equations and Inclusions

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