Dimension Theory for Ordinary Differential Equations

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

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.

Inhaltsverzeichnis:

I Singular values, exterior calculus and Lozinskii-norms.- 1 Singular values and covering of ellipsoids.- 2 Singular value inequalities.- 3 Compound matrices.- 4 Logarithmic matrix norms.- 5 The Yakubovich-Kalman frequency theorem.- 6 Frequency-domain estimation of singular values.- 7 Exterior calculus in linear spaces.- II Attractors, stability and Lyapunov functions.- 1 Dynamical systems, limit sets and attractors.- 2 Dissipativity.- 3 Stability of motion.- 4 Existence of a homoclinic orbit in the Lorenz system.- 5 The generalized Lorenz system.- 6 Orbital stability for flows on manifolds.- III Introduction to dimension theory.- 1 Topological dimension.- 2 Hausdorff and fractal dimensions.- 3 Topological entropy.- 4 Dimension-like characteristics.- IV Dimension and Lyapunov functions.- 1 Estimation of the topological dimension.- 2 Upper estimates for the Hausdorff dimension.- 3 The application of the limit theorem to ODE's.- 4 Convergence in third-order nonlinear systems.- 5 Est imates of fractal dimension.- 6 Estimates of the topological entropy.- 7 Fractal dimension estimates.- 8 Upper Lyapunov dimension.- 9 Formulas for the Lyapunov dimension.- 10 Invariant sets of vector fields.- 11 Use of a tubular Carathéodory structure.- 12 The Lyapunov dimension as upper bound.- 13 Lower estimates of the dimension of B-attractors.- A Some tools.- A.1 Definition of a differentiable manifold.- A.2 Tangent space, tangent bundle and differential.- A.3 Tensor products, exterior products and tensor fields.- A.4 Riemannian manifolds.- A.5 Covariant derivative.- A.6 Vector fields.- A.7 Spaces of vector fields and maps.- A.8 Parallel transport, geodesics and exponential map.- A.9 Curvature and torsion.- A.10 Fiber bundles and distributions.- A.11 Recurrence and hyperbolicity indynamical systems.- A.12 Homology theory.- A.13 Degree theory.- A.15 Geometric measure theory.- A.16 Totally ordered sets.- A.17 Almost periodic functions.

Rezension:

"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