Differential Equation
| Dimension
Theory for Ordinary Differential Equation |
Contents
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, singular values of an operator and covering lemmas
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 of a minimal set
2. Upper estimates for the Hausdorff dimension of negatively invariant sets
3. The application of the limit theorem to ODE's
4. Convergence in third-order nonlinear systems arising from physical models
5. Estimates of fractal dimension
6. Estimates of the topological entropy
7. Fractal dimension estimates for invariant sets and attractors of concrete systems
8. Upper Lyapunov dimension
9. Formulas for the Lyapunov dimension of the Hénon and Lorenz systems
10. Hausdorff dimension estimates for invariant sets of vector fields
11. Hausdorff dimension estimates by use of a tubular Carathéodory structure and their
application to stability theory
12. The Lyapunov dimension as upper bound of the fractal dimension
13. Lower estimates of the dimension of global B-attractors
Appendix: Some tools
Bibliography
Index
