Dimension Theory for Ordinary Differential Equations

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 Estimates 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.