Spectral analysis and an area-preserving extension of a piecewise linear intermittent map

Tomoshige Miyaguchi, Yoji Aizawa

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

We investigate the spectral properties of a one-dimensional piecewise linear intermittent map, which has not only a marginal fixed point but also a singular structure suppressing injections of the orbits into neighborhoods of the marginal fixed point. We explicitly derive generalized eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map for classes of observables and piecewise constant initial densities, and it is found that the Frobenius-Perron operator has two simple real eigenvalues 1 and λd (-1,0) and a continuous spectrum on the real line [0,1]. From these spectral properties, we also found that this system exhibits a power law decay of correlations. This analytical result is found to be in a good agreement with numerical simulations. Moreover, the system can be extended to an area-preserving invertible map defined on the unit square. This extended system is similar to the baker transformation, but does not satisfy hyperbolicity. A relation between this area-preserving map and a billiard system is also discussed.

Original languageEnglish
Article number066201
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume75
Issue number6
DOIs
Publication statusPublished - 2007 Jun 4

Fingerprint

Piecewise Linear Map
Spectral Analysis
Frobenius-Perron Operator
preserving
spectrum analysis
Spectral Properties
Fixed point
Generalized Eigenvalue
Decay of Correlations
eigenvalues
Eigenvalues and Eigenfunctions
Extended Systems
Continuous Spectrum
Hyperbolicity
Billiards
Real Line
operators
Invertible
Injection
continuous spectra

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Condensed Matter Physics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Spectral analysis and an area-preserving extension of a piecewise linear intermittent map. / Miyaguchi, Tomoshige; Aizawa, Yoji.

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 75, No. 6, 066201, 04.06.2007.

Research output: Contribution to journalArticle

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