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Dimension Theory for Ordinary Differential Equations
(Englisch)
Teubner-Texte zur Mathematik 141
Vladimir A. Boichenko & Genadij A. Leonov & Volker Reitmann

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Dimension Theory for Ordinary Differential Equations

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Produktbeschreibung

Moderne Methoden für die Dimensionstheorie gewöhnlicher Differentialgleichungen: Fraktale Dimension und dynamische Systeme


Dr. Vladimir A. Boichenko, Barrikada Company, St. Petersburg
Prof. Dr. Gennadij A. Leonov, St. Petersburg State University
Dr. Volker Reitmann, MPI for the Physics of Complex Systems, Dresden
This book is devoted to the estimation of dimension-like characteristics (Hausdorff dimension, fractal dimension, Lyapunov dimension, topological entropy) for attractors
(mainly global B-attractors) of ordinary differential equations, time-discrete systems and dynamical systems on finite-dimensional manifolds. The contraction under flows of
parameter-dependent outer measures is shown by introducing varying Lyapunov functions or metric tensors in the calculation of singular values. For the attractors of the Henon and Lorenz systems, exact formulae for the Lyapunov dimension are derived.
|The book is concerned with upper bounds for the Hausdorff and Fractal dimensions of flow invariant compact sets in Euclidean space and on Riemannian manifolds and the application of such bounds to global stability investigations of equilibrium points. The dimension estimates are formulated in terms of the eigenvalues of the symmetric part of the linearized vector field by including Lyapunov functions into the contraction conditions for outer Hausdorff measures. Various types of local, global and uniform Lyapunov exponents are introduced. On the base of such exponents the Lyapunov dimension of a set is defined and the Kaplan-Yorke formula is discussed. Upper estimates for the topological entropy are derived using Lyapunov functions and adapted Lozinskii norms.
Basic facts from matrix theory - Attractors, stability and Lyapunov functions - Introduction to dimension theory - Dimension and Lyapunov functions - Dimension estimates for invariant sets of vector fields on manifolds
"Concluding, one may say that the introductory parts of the book are suitable for graduate students, and in the advanced sections even experts in the field will certainly discover novelties."
Zentralblatt Mathematik, 20/2006
Modern Topics in Applied Analysis and Dynamical Systems
This book is devoted to the estimation of dimension-like characteristics (Hausdorff dimension, fractal dimension, Lyapunov dimension, topological entropy) for attractors
(mainly global B-attractors) of ordinary differential equations, time-discrete systems and dynamical systems on finite-dimensional manifolds. The contraction under flows of
parameter-dependent outer measures is shown by introducing varying Lyapunov functions or metric tensors in the calculation of singular values. For the attractors of the Henon and Lorenz systems, exact formulae for the Lyapunov dimension are derived.

Basic facts from matrix theory - Attractors, stability and Lyapunov functions - Introduction to dimension theory - Dimension and Lyapunov functions - Dimension estimates for invariant sets of vector fields on manifolds
Dr. Vladimir A. Boichenko, Barrikada Company, St. Petersburg
Prof. Dr. Gennadij A. Leonov, St. Petersburg State University
Dr. Volker Reitmann, MPI for the Physics of Complex Systems, Dresden

Über den Autor



Dr. Vladimir A. Boichenko, Barrikada Company, St. Petersburg
Prof. Dr. Gennadij A. Leonov, St. Petersburg State University
Dr. Volker Reitmann, MPI for the Physics of Complex Systems, Dresden


Inhaltsverzeichnis



Basic facts from matrix theory - Attractors, stability and Lyapunov functions - Introduction to dimension theory - Dimension and Lyapunov functions - Dimension estimates for invariant sets of vector fields on manifolds


Klappentext



This book is devoted to the estimation of dimension-like characteristics (Hausdorff dimension, fractal dimension, Lyapunov dimension, topological entropy) for attractors

(mainly global B-attractors) of ordinary differential equations, time-discrete systems and dynamical systems on finite-dimensional manifolds. The contraction under flows of

parameter-dependent outer measures is shown by introducing varying Lyapunov functions or metric tensors in the calculation of singular values. For the attractors of the Henon and Lorenz systems, exact formulae for the Lyapunov dimension are derived.




The book is concerned with upper bounds for the Hausdorff and Fractal dimensions of flow invariant compact sets in Euclidean space and on Riemannian manifolds and the application of such bounds to global stability investigations of equilibrium points. The dimension estimates are formulated in terms of the eigenvalues of the symmetric part of the linearized vector field by including Lyapunov functions into the contraction conditions for outer Hausdorff measures. Various types of local, global and uniform Lyapunov exponents are introduced. On the base of such exponents the Lyapunov dimension of a set is defined and the Kaplan-Yorke formula is discussed. Upper estimates for the topological entropy are derived using Lyapunov functions and adapted Lozinskii norms.

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