The Lyapunov dimension and its estimation via the Leonov method

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The Lyapunov dimension and its estimation via the Leonov method. / Kuznetsov, N. V.

In: Physics Letters, Section A: General, Atomic and Solid State Physics, Vol. 380, No. 25-26, 03.06.2016, p. 2142-2149.

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Kuznetsov, N. V. / The Lyapunov dimension and its estimation via the Leonov method. In: Physics Letters, Section A: General, Atomic and Solid State Physics. 2016 ; Vol. 380, No. 25-26. pp. 2142-2149.

BibTeX

@article{cb4b8d86a13943bea0f42de050dd8c8e,

title = "The Lyapunov dimension and its estimation via the Leonov method",

abstract = "Along with widely used numerical methods for estimating and computing the Lyapunov dimension there is an effective analytical approach, proposed by G.A. Leonov in 1991. The Leonov method is based on the direct Lyapunov method with special Lyapunov-like functions. The advantage of the method is that it allows one to estimate the Lyapunov dimension of invariant sets without localization of the set in the phase space and, in many cases, to get effectively an exact Lyapunov dimension formula. In this work the invariance of the Lyapunov dimension with respect to diffeomorphisms and its connection with the Leonov method are discussed. For discrete-time dynamical systems an analog of Leonov method is suggested. In a simple but rigorous way, here it is presented the connection between the Leonov method and the key related works: Kaplan and Yorke (the concept of the Lyapunov dimension, 1979), Douady and Oesterl{\'e} (upper bounds of the Hausdorff dimension via the Lyapunov dimension of maps, 1980), Constantin, Eden, Foia{\c s}, and Temam (upper bounds of the Hausdorff dimension via the Lyapunov exponents and Lyapunov dimension of dynamical systems, 1985-90), and the numerical calculation of the Lyapunov exponents and dimension.",

keywords = "Attractors of dynamical systems, Finite-time Lyapunov exponents, Hausdorff dimension, Invariance with respect to diffeomorphisms, Leonov method, Lyapunov dimension Kaplan-Yorke formula",

author = "Kuznetsov, {N. V.}",

note = "Publisher Copyright: {\textcopyright} 2016 Elsevier B.V. Allrightsreserved.s.",

year = "2016",

month = jun,

day = "3",

doi = "10.1016/j.physleta.2016.04.036",

language = "English",

volume = "380",

pages = "2142--2149",

journal = "Physics Letters A",

issn = "0375-9601",

publisher = "Elsevier",

number = "25-26",

}

RIS

TY - JOUR

T1 - The Lyapunov dimension and its estimation via the Leonov method

AU - Kuznetsov, N. V.

PY - 2016/6/3

Y1 - 2016/6/3

N2 - Along with widely used numerical methods for estimating and computing the Lyapunov dimension there is an effective analytical approach, proposed by G.A. Leonov in 1991. The Leonov method is based on the direct Lyapunov method with special Lyapunov-like functions. The advantage of the method is that it allows one to estimate the Lyapunov dimension of invariant sets without localization of the set in the phase space and, in many cases, to get effectively an exact Lyapunov dimension formula. In this work the invariance of the Lyapunov dimension with respect to diffeomorphisms and its connection with the Leonov method are discussed. For discrete-time dynamical systems an analog of Leonov method is suggested. In a simple but rigorous way, here it is presented the connection between the Leonov method and the key related works: Kaplan and Yorke (the concept of the Lyapunov dimension, 1979), Douady and Oesterlé (upper bounds of the Hausdorff dimension via the Lyapunov dimension of maps, 1980), Constantin, Eden, Foiaş, and Temam (upper bounds of the Hausdorff dimension via the Lyapunov exponents and Lyapunov dimension of dynamical systems, 1985-90), and the numerical calculation of the Lyapunov exponents and dimension.

AB - Along with widely used numerical methods for estimating and computing the Lyapunov dimension there is an effective analytical approach, proposed by G.A. Leonov in 1991. The Leonov method is based on the direct Lyapunov method with special Lyapunov-like functions. The advantage of the method is that it allows one to estimate the Lyapunov dimension of invariant sets without localization of the set in the phase space and, in many cases, to get effectively an exact Lyapunov dimension formula. In this work the invariance of the Lyapunov dimension with respect to diffeomorphisms and its connection with the Leonov method are discussed. For discrete-time dynamical systems an analog of Leonov method is suggested. In a simple but rigorous way, here it is presented the connection between the Leonov method and the key related works: Kaplan and Yorke (the concept of the Lyapunov dimension, 1979), Douady and Oesterlé (upper bounds of the Hausdorff dimension via the Lyapunov dimension of maps, 1980), Constantin, Eden, Foiaş, and Temam (upper bounds of the Hausdorff dimension via the Lyapunov exponents and Lyapunov dimension of dynamical systems, 1985-90), and the numerical calculation of the Lyapunov exponents and dimension.

KW - Attractors of dynamical systems

KW - Finite-time Lyapunov exponents

KW - Hausdorff dimension

KW - Invariance with respect to diffeomorphisms

KW - Leonov method

KW - Lyapunov dimension Kaplan-Yorke formula

UR - http://www.scopus.com/inward/record.url?scp=84964802952&partnerID=8YFLogxK

U2 - 10.1016/j.physleta.2016.04.036

DO - 10.1016/j.physleta.2016.04.036

M3 - Article

AN - SCOPUS:84964802952

VL - 380

SP - 2142

EP - 2149

JO - Physics Letters A

JF - Physics Letters A

SN - 0375-9601

IS - 25-26

ER -

ID: 95260066