Dimension Theory for Ordinary Differential Equations

ISBN-13: 9783519004370 / Angielski / Miękka / 2005 / 443 str.

Vladimir A. Boichenko; Genadij A. Leonov; Volker Reitmann

Opis

Der Gedanke, die bahnbrechenden und iiufJerst fruchtbaren Theorien, die Herr Lebesgue fur den Inhalt von Punktmengen entwickelt hat, auf den Begriff der Liinge zu ubertragen, liegt sehr nahe: es genugt eine additive Mengenfunktion zu jinden, deren Wert fur jede rektijizierbare K urve gleich der gewohnlichen Liinge dieser K urve ist, und die im ubrigen bei der Bildung von Vereinigungs- und Durchschnittsmengen die Eigenschaften des gewohnlichen Lebesgueschen MafJes besitzt. - Constantin CaratModory (1873 - 1950), "Uber das lineare Mass von Punktmengen - eine Verallgemeinerung des Uingenbegriffs" Theorems about dimension characteristics had a rather modest place in the classical theory of ordinary differential equations. Among these results is the well-known Liouville theorem about the contraction of volumes under the flow of a vector field with negative divergence (d. V. V. Nemyckii and V. V. Stepanov, 237]). Another classical result is Hilmy's theorem on the topological dimension of the minimal sets of a dynamical system (G. F. Hilmy, 132]). In the last thirty years when the attractor of a dynamical system be came one of the central objects of intensive investigations, the situation has radically changed."

Recenzje

"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

Spis treści

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

O autorze

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

Blurb

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.