Density of first Poincaré returns, periodic orbits, and Kolmogorov–Sinai entropy

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Abstract

It is known that unstable periodic orbits of a given map give information about the natural measure of a chaotic attractor. In this work we show how these orbits can be used to calculate the density function of the first Poincaré returns. The close relation between periodic orbits and the Poincaré returns allows for estimates of relevant quantities in dynamical systems, as the Kolmogorov–Sinai entropy, in terms of this density function. Since return times can be trivially observed and measured, our approach to calculate this entropy is highly oriented to the treatment of experimental systems. We also develop a method for the numerical computation of unstable periodic orbits.

Original languageEnglish
Pages (from-to)863 -875
Number of pages14
JournalCommunications in Nonlinear Science & Numerical Simulation
Volume16
Issue number2
Early online date25 May 2010
DOIs
Publication statusPublished - Feb 2011

Fingerprint

Periodic Orbits
Orbits
Entropy
Density Function
Unstable
Probability density function
Calculate
Return Time
Chaotic Attractor
Numerical Computation
Dynamical system
Orbit
Dynamical systems
Estimate

Keywords

  • Time returns
  • Periodic orbits
  • Lyapunov exponents
  • Kolmogorov entropy

Cite this

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title = "Density of first Poincar{\'e} returns, periodic orbits, and Kolmogorov–Sinai entropy",
abstract = "It is known that unstable periodic orbits of a given map give information about the natural measure of a chaotic attractor. In this work we show how these orbits can be used to calculate the density function of the first Poincar{\'e} returns. The close relation between periodic orbits and the Poincar{\'e} returns allows for estimates of relevant quantities in dynamical systems, as the Kolmogorov–Sinai entropy, in terms of this density function. Since return times can be trivially observed and measured, our approach to calculate this entropy is highly oriented to the treatment of experimental systems. We also develop a method for the numerical computation of unstable periodic orbits.",
keywords = "Time returns, Periodic orbits, Lyapunov exponents, Kolmogorov entropy",
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T1 - Density of first Poincaré returns, periodic orbits, and Kolmogorov–Sinai entropy

AU - Baptista, Murilo Da Silva

PY - 2011/2

Y1 - 2011/2

N2 - It is known that unstable periodic orbits of a given map give information about the natural measure of a chaotic attractor. In this work we show how these orbits can be used to calculate the density function of the first Poincaré returns. The close relation between periodic orbits and the Poincaré returns allows for estimates of relevant quantities in dynamical systems, as the Kolmogorov–Sinai entropy, in terms of this density function. Since return times can be trivially observed and measured, our approach to calculate this entropy is highly oriented to the treatment of experimental systems. We also develop a method for the numerical computation of unstable periodic orbits.

AB - It is known that unstable periodic orbits of a given map give information about the natural measure of a chaotic attractor. In this work we show how these orbits can be used to calculate the density function of the first Poincaré returns. The close relation between periodic orbits and the Poincaré returns allows for estimates of relevant quantities in dynamical systems, as the Kolmogorov–Sinai entropy, in terms of this density function. Since return times can be trivially observed and measured, our approach to calculate this entropy is highly oriented to the treatment of experimental systems. We also develop a method for the numerical computation of unstable periodic orbits.

KW - Time returns

KW - Periodic orbits

KW - Lyapunov exponents

KW - Kolmogorov entropy

U2 - 10.1016/j.cnsns.2010.05.018

DO - 10.1016/j.cnsns.2010.05.018

M3 - Article

VL - 16

SP - 863

EP - 875

JO - Communications in Nonlinear Science & Numerical Simulation

JF - Communications in Nonlinear Science & Numerical Simulation

SN - 1007-5704

IS - 2

ER -