| I |
Singular values, exterior calculus and Lozinskii-norms |
| 1 |
Singular values, exterior calculus 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 of the fractal dimension |
| 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 in dynamical systems |
| A.12 |
Homology theory |
| A.13 |
Degree theory |
| A.14 |
Simple d-linked parameterized m-boundaries |
| A.15 |
Geometric measure theory |
| A.16 |
Totally ordered sets |
| A.17 |
Almost periodic functions |
| |
Bibliography |
| |
Index |
