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

Teubner

Preface
Der Gedanke, die bahnbrechenden und aufterst fruchtbaren Theorien, die Herr Lebesgue jur den Inhalt von Punktmengen entwickelt hat, auf den Begriff der Ldnge zu iibertragen, liegt sehr nahe: es geniigt eine additive Mengenfunktion zu finden, deren Wert jur jede rektifizierbare Kurve gleich der gewohnlichen Ldnge dieser Kurve ist, und die im ubrigen bei der Bildung von Vereinigungs- und Durchschnittsmengen die Eigenschaften des gewohnlichen Lebesgueschen Maftes besitzt.
— Constantin Caratheodory (1873 - 1950),
,,Uber das lineare Mass von Punktmengen - eine Verallgemeinerung des Langenbegriffs"
Theorems about dimension characteristics had a rather modest place in the classical theory of ordinary differential equations. Among these results is the well-known Liouville theorem about the contraction of volumes under the flow of a vector field with negative divergence (cf. V. V. Nemyckii and V. V. Stepanov, [237]). Another classical result is Hilmy's theorem on the topological dimension of the minimal sets of a dynamical system (G. F. Hilmy, [132]). In the last thirty years when the attractor of a dynamical system became one of the central objects of intensive investigations, the situation has radically changed.
In 1980, investigators of differential equations were greatly impressed by a paper about upper estimates of the Hausdorff dimension of flow and map invariant sets written by A. Douady and J. Oesterle ([81]). The Douady-Oesterle approach, the significance of which can be compared with that of Liouville's theorem, has been developed and modificated in many papers for various types of dimension characteristics of attractors generated by dynamical systems: F. Ledrappier ([168]); P. Constantin, C. Foias, and R. Temam ([65]); R. A. Smith ([29]); A. Eden, C. Foias, and R. Temam ([90]); Zhi-Min Chen ([59]); V. A. Boichenko and G. A. Leonov ([35]); B. Hunt ([138]);

6
After Ya. B. Pesin had worked out ([246]) a general scheine of introducing metric dimension characteristics, this method made it possible to define from a unique point of view various types of outer measures and dimensions, such äs the Hausdorff dimension, the fractal dimension, the Information dimension äs well äs the topological and metric entropies. The Pesin scheme naturally led to the characterization of a class of Caratheodory measures (G. A. Leonov, [18]; G. A. Leonov, K. Geifert, and V. Reitmann, [192]), which are adapted to the specific character of attractors of autonomous differential equations, i.e. to the fact that these attractors consist wholly of trajectories. The neighborhoods of pieces of these trajectories form a covering of the attractor. It serves äs the base for introducing special outer Caratheodory measures. A number of effective tools for estimating these measures were developed within the theory of differential equations in Euclidean space and on manifolds by G. Borg ([41]) and P. Hartman and C. Olech ([126]), when analysing the orbital stability of Solutions. The most important property, used in dimension theory, is the fact that Caratheodory measures are majorants for the associated Hausdorff measures.
Early in the nineties of the last Century the authors of this book observed deep inner connections between the direct method of Lyapunov in stability theory and estimation technics for outer measures in dimension theory. Introducing Lyapunov functions and varying Riemannian metrices (G. A. Leonov, [176, 177]; A. Noack and V. Reitmann, [241]) into upper estimates of dimension characteristics of invariant sets made it possible to generalize and improve (G. A. Leonov and V. A. Boichenko, [190]; G. A. Leonov and S. A. Lyashko, [193]; G. A. Leonov, D. V. Ponomarenko and V. B. Smirnova, [198]; V. A. Boichenko, G. A. Leonov, A. Mirle, and V. Reitmann, [40]) some well-known results of R. A. Smith, P. Constantin, C. Foias, A. Eden, and R. Temam.
On the other hand, Pugh's ciosing lemma ([259]) and theorems about the span-ning of two-dimensional surfaces on a given closed curve, gave the opportunity to apply some theorems about the contraction of Hausdorff measures to global stability investigations of time-continuous dynamical Systems (R. A. Smith, [294]; G. A. Leonov, [176, 177]; M. Y. Li and J. S. Muldowney, [204]). In this book the effectiveness of introducing Lyapunov functions into dimen-sional characteristics is shown for a number of concrete dynamical Systems: the Henon map, the Systems of Lorenz and Rössler äs well äs their gener-alizations for various physical Systems and models (rotation of a rigid body in a resisting medium, convection of liquid in a rotating ellipsoid, interaction between waves in plasma, etc.).
This book provides a systematic presentation of research activities in the dimension theory of differential equations in the direction described above.

7
Let us briefly sketch the contents of the book. In the first chapter some facts on linear algebra Lozinskii matrix-norms, and exterior calculus, necessary for estimation technics of outer measures, are presented. In addition to this, the Yakubovich-Kalman frequency theorem and the Kalman-Szego theorem about the solvability of certain matrix inequalities are formulated and discussed.
Chapter II is devoted to Lyapunov functions and their use both in classical stability theory (Lyapunov stability, orbital stability, and Zhukovskii stability) and in the investigation of various types of global attractors for dynamical systems (global B-attractors, minimal global B-attractors). The theoretical results are applied to the stability investigation of a generalized Lorenz system, and dynamical systems on the flat cylinder. One section of this chapter is concerned with the existence proof of a homoclinic orbit in the Lorenz system (G. A. Leonov, [151, 173]; S. P. Hastings and W. C. Troy, [128]; X. Chen, [60]).
The third chapter is an introduction to dimension theory. It starts with the definition and the basic properties of the topological dimension in the spirit of W. Hurewicz and H. Wallman ([139]). Next the notions of Hausdorff mea-sure, Hausdorff dimension and fractal dimension are introduced. After this the topological entropy of a continuous map is discussed. The last part of the chapter deals with Pesin's scheme of introducing Caratheodory dimension characteristics.
The fourth chapter is the central chapter of the book. It begins with some results of M. L. Cartwright ([55]) and R. A. Smith ([294]) on the estimation of the topological dimension of a minimal set. Then the so called limit theorem about the evolution of Hausdorff measures under the action of smooth maps is presented. This theorem gives the opportunity to include into Hausdorff dimension and Hausdorff measure estimates, certain varying functions which turn over into Lyapunov functions when the theorem is applied to differential equations. The method of varying functions is used in estimations of fractal dimension and topological entropy. Simultaneously, with the introduction of Lyapunov functions in estimates for the Hausdorff measure, the Lozinskii norms were used by J. S. Muldowney [236], for estimating the two-dimensional Riemannian volumes of compact sets shifted along the orbits of differential equations. One of the main goals of Chapter IV is to combine the Lyapunov function and Lozinskii norm approaches (V. A. Boichenko and G. A. Leonov, [38]), in order to solve a number of problems in the qualitative theory of ordinary differential equations, such as the generalization of the Liouville for-mula and the Bendixson criterion. We also concentrate in Chapter IV on the Kaplan-Yorke formula and the Lyapunov dimension formulas for the Lorenz and Henon attractors (G. A. Leonov and S. A. Lyashko [195]; G. A. Leonov,

[188]). Then we discuss the construction of special Caratheodory measures for the estimation of Hausdorff measures connected with flow invariant sets on Riemannian manifolds.
Chapter IV contains also an important result on the estimation of the fractal dimension of an invariant set on an arbitrary finite-dimensional smooth manifold, which goes back to B. Hunt ([138]) and K. Gelfert ([112]). The last topic we are concerned with in this chapter are lower estimates of map or flow invariant sets which are due to G. A. Leonov ([186]) and M. Y. Li and J. S. Muldowney ([205]).
In the appendix, we present mathematical knowledge that is necessary for a complete understanding of the statements and proofs in the main part of the monograph.
A great part of the results due to A. Franz, K. Gelfert, A. Noack and V. Reit-mann, which are presented in this book, stem from investigations which were supported by the German Priority Research Programs "Dynamics: Analysis, efficient simulation and ergodic theory" and "Mathematical methods for time series analysis and digital image processing". These programs were founded by the German Research Foundation (DFG).
The authors of the book are greatly indebted to Margitta Reitmann for her accurate typing of the manuscript in ETgX and to John Sewell for his careful reviewing of the English version of the book.
St. Petersburg and Dresden, December 2004
V. A. Boichenko G. A. Leonov V. Reitmann

Contents
Singular values, exterior calculus and Lozinskii-norms	15
1	Singular values and covering of ellipsoids		15
1.1	Introduction		15
1.2	Definition of singular values		17
1.3	Lemmas on covering of ellipsoids    		19
2	Singular value inequalities		21
2.1	The Fischer-Courant theorem		21
2.2	The Binet-Cauchy theorem    		24
2.3	The inequalities of Horn, Weyl and Fan		28
3	Compound matrices		30
3.1	Multiplicative compound matrices		30
3.2	Additive compound matrices		35
3.3	Applications to stability theory		38
4	Logarithmic matrix norms		40
4.1	Lozinskii's theorem		40
4.2	Generalization of the Liouville equation		45
5	The Yakubovich-Kalman frequency theorem		49
5.1	The frequency theorem for ODE's		49
5.2	The frequency theorem for discrete-time systems		52
6	Frequency-domain estimation of singular values		54
6.1	Linear differential equations		54
6.2	Linear difference equations		58
7	Exterior calculus in linear spaces    		62
7.1	Multiplicative and additive compounds of operators 	62
7.2	Singular values of an operator acting between Euclidean
spaces		72
7.3	Lemmas on covering of ellipsoids in an Euclidean space  .	75
7.4	Singular value inequalities for operators		76

10	CONTENTS
II	Attractors, stability and Lyapunov functions	79
1	Dynamical systems, limit sets and attractors	79
1.1	Dynamical systems in metric spaces	79
1.2	Minimal global attractors    	85
1.3	Time-invariant control systems    	89
2	Dissipativity	91
2.1	Dissipativity in the sense of Levinson	91
2.2	Dissipativity and completeness of the Lorenz system    92
2.3	Lyapunov-type results for dissipativity	96
2.4	Convergence in systems with several equilibrium states   . 100
3	Stability of motion	106
3.1	Lyapunov stability	107
3.2	Orbital stability	115
3.3	Zhukovskii stability	119
4	Existence of a homoclinic orbit in the Lorenz system    	124
4.1	Introduction	124
4.2	Estimates for the shape of global attractors    	125
4.3	The existence of homoclinic orbits	127
5	The generalized Lorenz system	132
5.1	Definition of the system    	132
5.2	Equilibrium states    	133
5.3	Global asymptotic stability    	134
5.4	Dissipativity	136
6	Orbital stability for flows on manifolds	138
6.1	Introduction	138
6.2	Dynamical systems with a local contraction property   . . 139
6.3	The Andronov-Vitt theorem	143
6.4	Various types of variational equations	144
6.5	Asymptotic orbital stability conditions	147
6.6	Estimating the singular values and orbital stability   . .  . 162
6.7	Frequency-domain conditions for orbital stability in
feedback control equations on the cylinder	169
III   Introduction to dimension theory	175
1	Topological dimension   	175
1.1	The inductive topological dimension    	176
1.2	The covering dimension    	182
2	Hausdorff and fractal dimensions    	186
2.1	The Hausdorff measure and the Hausdorff dimension   .  186
2.2	Fractal dimension and lower box dimension  196

CONTENTS	11
2.3	Self-similar sets	202
2.4	Dimension of Cartesian products    	205
Topological entropy	207
3.1	The Bowen-Dinaburg definition	208
3.2	The characterization by open covers	211
3.3	Some properties of the topological entropy	214
4       Dimension-like characteristics	219
4.1	Caratheodory measure, dimension and capacity	219
4.2	Properties of the Caratheodory dimension and
Caratheodory capacity	223
IV Dimension and Lyapunov functions	229
1	Estimation of the topological dimension	229
1.1	Hilmy's theorem    	229
1.2	Minimal sets for almost periodic flows    	230
1.3	The frequency spectrum of almost periodic solutions    .  . 236
1.4	Frequency-domain conditions for upper topological
dimension estimates of orbit closures	242
2	Upper estimates for the Hausdorff dimension	244
2.1	The limit theorem for Hausdorff measures	244
2.2	Corollaries of the limit theorem for Hausdorff measures  . 250
2.3	Application of the limit theorem to the Henon map   . .  . 255
3	The application of the limit theorem to ODE's	260
3.1	An auxiliary result	260
3.2	Estimates of the Hausdorff measure and of Hausdorff
dimension	262
3.3	The generalized Bendixson criterion	266
3.4	On the firiiteness of the number of periodic solutions    . . 267
3.5	Convergence theorems                268
4	Convergence in third-order nonlinear systems	269
4.1	The generalized Lorenz system	269
4.2	Euler's equations describing the rotation of a rigid body
in a resisting medium	275
4.3	A nonlinear system arising from fluid convection in a
rotating ellipsoid	276
4.4	A system describing the interaction of three waves in
plasma	277
5	Estimates of fractal dimension	280
5.1        Maps with a constant Jacobian	280

12	CONTENTS
5.2        Autonomous differential equations which are conserva
tive on the invariant set    	283
6	Estimates of the topological entropy	285
6.1	Ito's generalized entropy estimate for maps	285
6.2	Application to differential equations	290
7	Fractal dimension estimates	292
7.1	The Rossler system	293
7.2	Lorenz equation	295
7.3	Equations of the third order	302
7.4	Equations describing the interaction between waves in
plasma	307
8	Upper Lyapunov dimension    	310
8.1	Definition of local Lyapunov exponents	310
8.2	An upper estimate for the upper Lyapunov dimension of
the attractors of the Lorenz System	313
9	Formulas for the Lyapunov dimension	318
9.1	General results	318
9.2	The Henon map	325
9.3	Lorenz system	328
10	Invariant sets of vector fields	331
10.1	Introduction	331
10.2	Hausdorff dimension bounds for invariant sets of maps
on manifolds	332
10.3	Time-dependent vector fields on manifolds	337
10.4	Convergence for autonomous vector fields	343
11	Use of a tubular Caratheodory structure	346
11.1	The system in normal variation	346
11.2	Tubular Caratheodory structure	350
11.3	Dimension estimates for sets which are negatively
invariant for a flow	352
11.4	Flow invariant sets with an equivariant tangent bundle
splitting	359
11.5	Generalizations of the theorems of Hartman-Olech and
Borg	362
12	The Lyapunov dimension as upper bound	364
12.1	Statement of the results	364
12.2	Proof of Theorem 12.1.1	365
12.3	Global Lyapunov exponents and upper Lyapunov
dimension	372
12.4	Application to the Lorenz system	373