Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors

P Grassberger, R Badii, A Politi

Research output: Contribution to journalArticle

372 Citations (Scopus)

Abstract

The analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems. Prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective Liapunov exponents. For hyperbolic attractors, the latter determines the metric entropies and provides one constraint on the partial dimensions. For nonhyperbolic attractors, there are important modifications. We discuss them for the examples of the logistic and Hénon map. We show, in particular, that the generalized dimensions have singularities with noncontinuous derivative, similar to first-order phase transitions in statistical mechanics.
Original languageEnglish
Pages (from-to)135-178
Number of pages44
JournalJournal of Statistical Physics
Volume51
Issue number1-2
DOIs
Publication statusPublished - 1 Apr 1988

Keywords

  • dynamical systems
  • generalized dimensions and entropies
  • Liapunov exponents
  • scaling functions
  • hyperbolicity
  • phase transitions

Cite this

Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors. / Grassberger, P ; Badii, R ; Politi, A .

In: Journal of Statistical Physics, Vol. 51, No. 1-2, 01.04.1988, p. 135-178.

Research output: Contribution to journalArticle

@article{307aa95f42e94d29b727bbd6edcd09b5,
title = "Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors",
abstract = "The analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems. Prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective Liapunov exponents. For hyperbolic attractors, the latter determines the metric entropies and provides one constraint on the partial dimensions. For nonhyperbolic attractors, there are important modifications. We discuss them for the examples of the logistic and H{\'e}non map. We show, in particular, that the generalized dimensions have singularities with noncontinuous derivative, similar to first-order phase transitions in statistical mechanics.",
keywords = "dynamical systems, generalized dimensions and entropies, Liapunov exponents, scaling functions, hyperbolicity, phase transitions",
author = "P Grassberger and R Badii and A Politi",
year = "1988",
month = "4",
day = "1",
doi = "10.1007/BF01015324",
language = "English",
volume = "51",
pages = "135--178",
journal = "Journal of Statistical Physics",
issn = "0022-4715",
publisher = "Springer New York",
number = "1-2",

}

TY - JOUR

T1 - Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors

AU - Grassberger, P

AU - Badii, R

AU - Politi, A

PY - 1988/4/1

Y1 - 1988/4/1

N2 - The analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems. Prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective Liapunov exponents. For hyperbolic attractors, the latter determines the metric entropies and provides one constraint on the partial dimensions. For nonhyperbolic attractors, there are important modifications. We discuss them for the examples of the logistic and Hénon map. We show, in particular, that the generalized dimensions have singularities with noncontinuous derivative, similar to first-order phase transitions in statistical mechanics.

AB - The analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems. Prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective Liapunov exponents. For hyperbolic attractors, the latter determines the metric entropies and provides one constraint on the partial dimensions. For nonhyperbolic attractors, there are important modifications. We discuss them for the examples of the logistic and Hénon map. We show, in particular, that the generalized dimensions have singularities with noncontinuous derivative, similar to first-order phase transitions in statistical mechanics.

KW - dynamical systems

KW - generalized dimensions and entropies

KW - Liapunov exponents

KW - scaling functions

KW - hyperbolicity

KW - phase transitions

U2 - 10.1007/BF01015324

DO - 10.1007/BF01015324

M3 - Article

VL - 51

SP - 135

EP - 178

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -