##
**Dimension theory for ordinary differential equations.**
*(English)*
Zbl 1094.34002

Teubner-Texte zur Mathematik 141. Wiesbaden: Teubner (ISBN 3-519-00437-2/pbk). 441 p. (2005).

The field covered by this book is really wider than that indicated by its title: in addition to various dimensional characteristics of invariant sets both for flows (dynamical systems with continuous time) and cascades (dynamical systems with discrete time), the authors study various stability properties of such systems as well as the global structure of their trajectories.

They give a survey of numerous modern methods for the global study of nonlinear differential equations and dynamical systems. The text provides an intruduction to the fundamental ideas as well as to more recent developments.

A definite advantage of the book is that the results are applied to several systems which are very important for applications. For example, the famous Lorenz system is studied in detail: the authors establish the existence of homoclinic trajectories, characterize the shape of the global attractor, and estimate its fractal and Lyapunov dimensions.

In the study of dimensional characteristics, the authors develop far-reaching generalizations of the famous method of Lyapunov functions and combine this method with various analytical approaches.

Let us briefly describe the contents of the four chapters of the book.

Chapter 1, Singular values, exterior calculus, and Lozinskii norms, gives an introduction to some basic technical tools employed in the book. It must be emphasized that a lot of attention is paid to Lozinskii norms of matrices (these norms may take negative values and help to get very effective estimates for singular values of Cauchy matrices for linear systems). In addition, various frequency theorems in stability theory are discussed. Sections: 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.

Chapter 2, Attractors, stability, and Lyapunov functions. This chapter covers a wide field of stability theory, from the classical Lyapunov stability to the up-to-date theory of attractors (for example, a rather complete classification of attractors is given, including global attractors, global \({\mathcal B}\)-attractors, Milnor attractors, etc). Sections: 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.

Chapter 3, Introduction to dimension theory. In this chapter, the authors introduce topological, Hausdorff, and fractal dimensions. In addition, they describe Pesin’s scheme of introducing Carathéodory dimension characteristics. Sections: 1. Topological dimension. 2. Hausdorff and fractal dimensions. 3. Topological entropy. 4. Dimension-like characteristics.

Chapter 4, Dimension and Lyapunov functions, is the central part of the book. The methods described use various types of Lyapunov functions and their generalizations to study the evolution of Hausdorff measures along trajectories of dynamical systems. In this approach, the application of Lozinskii norms plays a very important role. Far-reaching generalizations of the Liouville formula and Bendixson criterion are obtained. It must be emphasized that the methods developed by the authors allow them to obtain both upper and lower estimates for dimension characteristics. Sections: 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 ODEs. 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 \({\mathcal B}\)-attractors.

The book also contains an appendix which gives an introduction to some basic notions (Riemannian manifolds, vector fields, curvature and torsion, fiber bundles and distributions, homologies and degrees, etc).

Concluding, one may say that the introductory parts of the book are suitable for graduate students, and in the more advanced sections even experts in the field will certainly discover novelties.

They give a survey of numerous modern methods for the global study of nonlinear differential equations and dynamical systems. The text provides an intruduction to the fundamental ideas as well as to more recent developments.

A definite advantage of the book is that the results are applied to several systems which are very important for applications. For example, the famous Lorenz system is studied in detail: the authors establish the existence of homoclinic trajectories, characterize the shape of the global attractor, and estimate its fractal and Lyapunov dimensions.

In the study of dimensional characteristics, the authors develop far-reaching generalizations of the famous method of Lyapunov functions and combine this method with various analytical approaches.

Let us briefly describe the contents of the four chapters of the book.

Chapter 1, Singular values, exterior calculus, and Lozinskii norms, gives an introduction to some basic technical tools employed in the book. It must be emphasized that a lot of attention is paid to Lozinskii norms of matrices (these norms may take negative values and help to get very effective estimates for singular values of Cauchy matrices for linear systems). In addition, various frequency theorems in stability theory are discussed. Sections: 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.

Chapter 2, Attractors, stability, and Lyapunov functions. This chapter covers a wide field of stability theory, from the classical Lyapunov stability to the up-to-date theory of attractors (for example, a rather complete classification of attractors is given, including global attractors, global \({\mathcal B}\)-attractors, Milnor attractors, etc). Sections: 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.

Chapter 3, Introduction to dimension theory. In this chapter, the authors introduce topological, Hausdorff, and fractal dimensions. In addition, they describe Pesin’s scheme of introducing Carathéodory dimension characteristics. Sections: 1. Topological dimension. 2. Hausdorff and fractal dimensions. 3. Topological entropy. 4. Dimension-like characteristics.

Chapter 4, Dimension and Lyapunov functions, is the central part of the book. The methods described use various types of Lyapunov functions and their generalizations to study the evolution of Hausdorff measures along trajectories of dynamical systems. In this approach, the application of Lozinskii norms plays a very important role. Far-reaching generalizations of the Liouville formula and Bendixson criterion are obtained. It must be emphasized that the methods developed by the authors allow them to obtain both upper and lower estimates for dimension characteristics. Sections: 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 ODEs. 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 \({\mathcal B}\)-attractors.

The book also contains an appendix which gives an introduction to some basic notions (Riemannian manifolds, vector fields, curvature and torsion, fiber bundles and distributions, homologies and degrees, etc).

Concluding, one may say that the introductory parts of the book are suitable for graduate students, and in the more advanced sections even experts in the field will certainly discover novelties.

Reviewer: Sergei Yu. Pilyugin (St. Petersburg)

### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37C45 | Dimension theory of smooth dynamical systems |

34C45 | Invariant manifolds for ordinary differential equations |

34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |

34D20 | Stability of solutions to ordinary differential equations |