Dimension Theory for Ordinary Differential Equations

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1 Vladimir A. Boichenko, Gennadij A. Leonov, Volker Reitmann Dimension Theory for Ordinary Differential Equations Teubner

2 Contents Singular values, exterior calculus and Lozinskii-norms 15 1 Singular values and covering of ellipsoids Introduction Definition of singular values Lemmas on covering of ellipsoids 19 2 Singular value inequalities The Fischer-Courant theorem The Binet-Cauchy theorem The inequalities of Horn, Weyl and Fan 28 3 Compound matrices Multiplicative compound matrices Additive compound matrices Applications to stability theory 38 4 Logarithmic matrix norms Lozinskii's theorem Generalization of the Liouville equation 45 5 The Yakubovich-Kalman frequency theorem The frequency theorem for ODE's The frequency theorem for discrete-time systems 52 6 Frequency-domain estimation of singular values Linear differential equations Linear difference equations 58 7 Exterior calculus in linear spaces Multiplicative and additive compounds of operators Singular values of an operator acting between Euclidean spaces Lemmas on covering of ellipsoids in an Euclidean space Singular value inequalities for operators 76

3 10 CONTENTS 11 Attractors, stability and Lyapunov functions 79 1 Dynamical systems, limit sets and attractors Dynamical systems in metric spaces Minimal global attractors Time-invariant control systems 89 2 Dissipativity Dissipativity in the sense of Levinson Dissipativity and completeness of the Lorenz system Lyapunov-type results for dissipativity Convergence in systems with several equilibrium states Stability of motion Lyapunov stability Orbital stability Zhukovskii stability Existence of a homoclinic orbit in the Lorenz system Introduction Estimates for the shape of global attractors The existence of homoclinic orbits The generalized Lorenz system Definition of the system Equilibrium states Global asymptotic stability Dissipativity Orbital stability for flows on manifolds Introduction Dynamical systems with a local contraction property The Andronov-Vitt theorem Various types of variational equations Asymptotic orbital stability conditions Estimating the singular values and orbital stability Frequency-domain conditions for orbital stability in feedback control equations on the cylinder 169 III Introduction to dimension theory Topological dimension The inductive topological dimension The covering dimension Hausdorff and fractal dimensions The Hausdorff measure and the Hausdorff dimension Fractal dimension and lower box dimension 196

4 CONTENTS Self-similar sets Dimension of Cartesian products Topological entropy The Bowen-Dinaburg definition The characterization by open covers Some properties of the topological entropy Dimension-like characteristics Caratheodory measure, dimension and capacity Properties of the Caratheodory dimension and Caratheodory capacity 223 IV Dimension and Lyapunov functions Estimation of the topological dimension Hilmy's theorem Minimal sets for almost periodic flows The frequency spectrum of almost periodic solutions Frequency-domain conditions for upper topological dimension estimates of orbit closures Upper estimates for the Hausdorff dimension The limit theorem for Hausdorff measures Corollaries of the limit theorem for Hausdorff measures Application of the limit theorem to the Henon map The application of the limit theorem to ODE's An auxiliary result Estimates of the Hausdorff measure and of Hausdorff dimension The generalized Bendixson criterion On the finiteness of the number of periodic solutions Convergence theorems Convergence in third-order nonlinear systems The generalized Lorenz system Euler's equations describing the rotation of a rigid body in a resisting medium A nonlinear system arising from fluid convection in a rotating ellipsoid A system describing the interaction of three waves in plasma Estimates of fractal dimension Maps with a constant Jacobian 280

5 12 CONTENTS 5.2 Autonomous differential equations which are conservative on the invariant set Estimates of the topological entropy Ito's generalized entropy estimate for maps Application to differential equations Fractal dimension estimates The Rossler system Lorenz equation Equations of the third order Equations describing the interaction between waves in plasma Upper Lyapunov dimension Definition of local Lyapunov exponents An upper estimate for the upper Lyapunov dimension of the attractors of the Lorenz System Formulas for the Lyapunov dimension General results The Henon map Lorenz system Invariant sets of vector fields Introduction Hausdorff dimension bounds for invariant sets of maps on manifolds Time-dependent vector fields on manifolds Convergence for autonomous vector fields Use of a tubular Caratheodory structure The system in normal variation Tubular Caratheodory structure Dimension estimates for sets which are negatively invariant for a flow Flow invariant sets with an equivariant tangent bundle splitting Generalizations of the theorems of Hartman-Olech and Borg The Lyapunov dimension as upper bound Statement of the results Proof of Theorem Global Lyapunov exponents and upper Lyapunov dimension Application to the Lorenz system 373

6 CONTENTS Lower estimates of the dimension of B-attractors Introduction Frequency-domain conditions for lower topological dimension bounds of global #-attractors Lower estimates of the Hausdorff dimension of global B-attractors Lower dimension estimates for global attractors based on the evolution of currents 382 A Some tools 385 A.I Definition of a differentiable manifold 385 A.2 Tangent space, tangent bundle and differential 387 A.3 Tensor products, exterior products and tensor fields A.4 Riemannian manifolds 390 A.5 Covariant derivative 392 A.6 Vector fields 392 A.7 Spaces of vector fields and maps 394 A.8 Parallel transport, geodesies and exponential map A.9 Curvature and torsion 398 A.10 Fiber bundles and distributions 400 A.ll Recurrence and hyperbolicity in dynamical systems A.12 Homology theory 403 A.13 Degree theory 405 A.14 Simple 5-linked parameterized m-boundaries 407 A.15 Geometric measure theory 409 A.16 Totally ordered sets 411 A.17 Almost periodic functions 412 Bibliography 415 Index 435

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