Emergence, Complexity and Computation

Volume 38

Series Editors

Ivan Zelinka, Technical University of Ostrava, Ostrava, Czech Republic

Andrew Adamatzky, University of the West of England, Bristol, UK

Guanrong Chen, City University of Hong Kong, Hong Kong, China

Editorial Board

Ajith Abraham, MirLabs, USA

Ana Lucia, Universidade Federal do Rio Grande do Sul, Porto Alegre, Rio Grandedo Sul, Brazil

Juan C. Burguillo, University of Vigo, Spain

Sergej Čelikovský, Academy of Sciences of the Czech Republic, Czech Republic

Mohammed Chadli, University of Jules Verne, France

Emilio Corchado, University of Salamanca, Spain

Donald Davendra, Technical University of Ostrava, Czech Republic

Andrew Ilachinski, Center for Naval Analyses, USA

Jouni Lampinen, University of Vaasa, Finland

Martin Middendorf, University of Leipzig, Germany

Edward Ott, University of Maryland, USA

Linqiang Pan, Huazhong University of Science and Technology, Wuhan, China

Gheorghe Păun, Romanian Academy, Bucharest, Romania

Hendrik Richter, HTWK Leipzig University of Applied Sciences, Germany

Juan A. Rodriguez-Aguilar , IIIA-CSIC, Spain

Otto Rössler, Institute of Physical and Theoretical Chemistry, Tübingen, Germany

Vaclav Snasel, Technical University of Ostrava, Czech Republic

Ivo Vondrák, Technical University of Ostrava, Czech Republic

Hector Zenil, Karolinska Institute, Sweden

The Emergence, Complexity and Computation (ECC) series publishes newdevelopments, advancements and selected topics in the fields of complexity,computation and emergence. The series focuses on all aspects of reality-basedcomputation approaches from an interdisciplinary point of view especially fromapplied sciences, biology, physics, or chemistry. It presents new ideas andinterdisciplinary insight on the mutual intersection of subareas of computation,complexity and emergence and its impact and limits to any computing based onphysical limits (thermodynamic and quantum limits, Bremermann’s limit, SethLloyd limits…) as well as algorithmic limits (Gödel’s proof and its impact oncalculation, algorithmic complexity, the Chaitin’s Omega number and Kolmogorovcomplexity, non-traditional calculations like Turing machine process and itsconsequences,…) and limitations arising in artificial intelligence. The topics are(but not limited to) membrane computing, DNA computing, immune computing,quantum computing, swarm computing, analogic computing, chaos computing andcomputing on the edge of chaos, computational aspects of dynamics of complexsystems (systems with self-organization, multiagent systems, cellular automata,artificial life,…), emergence of complex systems and its computational aspects, andagent based computation. The main aim of this series is to discuss the abovementioned topics from an interdisciplinary point of view and present new ideascoming from mutual intersection of classical as well as modern methods ofcomputation. Within the scope of the series are monographs, lecture notes, selectedcontributions from specialized conferences and workshops, special contributionfrom international experts.

More information about this series at http://www.springer.com/series/10624

Nikolay Kuznetsov • Volker Reitmann

Attractor DimensionEstimates for DynamicalSystems: Theoryand ComputationDedicated to Gennady Leonov

123

Nikolay KuznetsovDepartment of Applied CyberneticsFaculty of Mathematics and MechanicsSt. Petersburg State UniversitySt. Petersburg, Russia

Volker ReitmannDepartment of Applied CyberneticsFaculty of Mathematics and MechanicsSt. Petersburg State UniversitySt. Petersburg, Russia

Faculty of Information TechnologyUniversity of JyväskyläJyväskylä, Finland

ISSN 2194-7287 ISSN 2194-7295 (electronic)Emergence, Complexity and ComputationISBN 978-3-030-50986-6 ISBN 978-3-030-50987-3 (eBook)https://doi.org/10.1007/978-3-030-50987-3

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer NatureSwitzerland AG 2021This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whetherthe whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse ofillustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, andtransmission or information storage and retrieval, electronic adaptation, computer software, or by similaror dissimilar methodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

In this book, we continue the investigations of global attractors and invariant setsfor dynamical systems by means of Lyapunov functions and adapted metrics. Theeffectiveness of such approaches for the approximation and localization of attractorsfor different classes of dynamical systems was already shown in Abramovich et al.[1, 2] and in [30, 32]. In particular, Lyapunov functions and adapted metrics wereconstructed for global stability problems and the existence of homoclinic orbits inthe Lorenz system using frequency-domain methods and reduction principles.

In 1980, investigators of differential equations and general dynamical systemswere greatly impressed by a paper about upper estimates of the Hausdorff dimen-sion of flow and map invariant sets written by Douady and Oesterlé [12]. TheDouady–Oesterlé approach, the significance of which can be compared with that ofLiouville’s theorem, has been developed and modificated in many papers for var-ious types of dimension characteristics of attractors generated by dynamical sys-tems: Ledrappier [22], Constantin et al. [11], Smith [42], Eden et al. [13], Chen [9],Hunt [17], Boichenko and Leonov [4].

After Ya. B. Pesin had worked out [39] a general scheme of introducing metricdimension characteristics, this method made it possible to define from a uniquepoint of view various types of outer measures and dimensions, such as theHausdorff dimension, the fractal dimension, the information dimension as well asthe topological and metric entropies. The Pesin scheme naturally led to the char-acterization of a class of Carathéodory measures [25, 27], which are adapted to thespecific character of attractors of autonomous differential equations, i.e. to the factthat these attractors consist wholly of trajectories. The neighborhoods of piecesof these trajectories form a covering of the attractor. It serves as the base forintroducing the special outer Carathéodory measures. A number of effective toolsfor estimating these measures were developed within the theory of differentialequations in Euclidean space and well-known results by Borg [8], Hartman andOlech [15], when analysing the orbital stability of solutions. The most importantproperty, used in dimension theory, is the fact that the Carathéodory measures aremajorants for the associated Hausdorff measures.

v

Early in the nineties of the last century, G. A. Leonov and his co-workersobserved deep inner connections between the mentioned direct method ofLyapunov in stability theory and estimation technics for outer measures indimension theory. Introducing the Lyapunov functions and varying Riemannianmetrices (Leonov [24], Noack and Reitmann [38]) into upper estimates of dimen-sion characteristics of invariant sets made it possible to generalize and improve[4, 6, 28, 29] some well-known results of R. A. Smith, P. Constantin, C. Foias,A. Eden, and R. Temam.

On the other hand, Pugh’s closing lemma [40] and theorems about the spanningof two-dimensional surfaces on a given closed curve gave the opportunity to applysome theorems about the contraction of Hausdorff measures to global stabilityinvestigations of time-continuous dynamical systems (Smith [42], Leonov [24], Liand Muldowney [35]). In this book, the effectiveness of introducing the Lyapunovfunctions into dimensional characteristics is shown for a number of concretedynamical systems: the Hénon map, the systems of Lorenz and Rössler as well astheir generalizations for various physical systems and models (rotation of a rigidbody in a resisting medium, convection of liquid in a rotating ellipsoid, interactionbetween waves in plasma, etc.).

Additionally to the derivation of upper Hausdorff dimension estimates, exactformulas for the Lyapunov dimensions for Lorenz type systems were shown[26, 28]. Many of these results were presented in [7].

In the following decade, the modified Douady–Oesterlé approach was also usedfor new classes of attractors [33].

It was also possible to get different versions of the Douady–Oesterlé theorem forpiecewise continuous maps and differential equations [37, 41]. For cocycles gen-erated by non-autonomous systems, the upper Hausdorff dimension estimates arederived in [31, 34]. Some of these results are included in the present book whichprovides a systematic presentation of research activities in the dimension theory ofdynamical systems in finite-dimensional Euclidean spaces and manifolds. Let usbriefly sketch the contents of the book.

In Part I, we consider the basic facts from attractor theory, exterior products anddimension theory. Chapter 1 is devoted to the investigation of various types ofglobal attractors of dynamical systems in general metric spaces (global B-attractors,minimal global B-attractors and others). The theoretical results are applied to thegeneralized Lorenz system and dynamical systems on the flat cylinder. One sectionis concerned with the existence proof of a homoclinic orbit in the Lorenz system(Leonov [23], Hastings and Troy [16], Chen [10]).

In Chap. 2, some facts on singular values of matrices, the exterior calculus forspaces and matrices and the Lozinskii matrix norm, necessary for estimationtechniques of outer measures, are presented. In addition to this, the Yakubovich–Kalman frequency theorem and the Kalman–Szegö theorem about the solvability ofcertain matrix inequalities are formulated and used for the estimation of singularvalues.

vi Preface

Chapter 3 is an introduction to dimension theory. It starts with the definition andthe basic properties of the topological dimension in the spirit of Hurewicz andWallman [18]. Next, the notions of Hausdorff measure, Hausdorff dimension andfractal dimension are introduced. After this, the topological entropy of a continuousmap is discussed. The last part of the chapter deals with Pesin’s scheme of intro-ducing the Carathéodory dimension characteristics.

In Part II, we investigate dimension properties of dynamical systems inEuclidean spaces. It includes estimates of topological dimension of the Hausdorffand fractal dimensions for invariant sets of concrete physical systems and estimatesof the Lyapunov dimension.

Chapter 4 is concerned with the investigation of dimension properties of almostperiodic flows [3]. We thank M. M. Anikushin for helping us to prepare the firstversion of Chap. 4.

Chapter 5 begins with the so-called limit theorem about the evolution ofHausdorff measures under the action of smooth maps in Euclidean spaces. Thistheorem gives the opportunity to include into Hausdorff dimension and Hausdorffmeasure estimates, certain varying functions which turn over into Lyapunovfunctions when the theorem is applied to differential equations. The method ofvarying functions is used in estimations of fractal dimension and topologicalentropy. Simultaneously, with the introduction of Lyapunov functions in estimatesfor the Hausdorff measure, the logarithmic norms were used by Muldowney [36],for estimating the two-dimensional Riemannian volumes of compact sets shiftedalong the orbits of differential equations. One of the main goals of Chap. 5 is tocombine the Lyapunov function and logarithmic norm approaches (Boichenko andLeonov [5]), in order to solve a number of problems in the qualitative theory ofordinary differential equations, such as the generalization of the Liouville formulaand the Bendixson criterion.

Chapter 6 is devoted to finite-dimensional dynamical systems in Euclidean spaceand its aim is to explain, in a simple but rigorous way, the connection between thekey works in the area: Kaplan and Yorke (the concept of Lyapunov dimension [19]1979), Douady and Oesterlé (estimation of Hausdorff dimension via the Lyapunovdimension of maps [12]), Constantin, Eden, Foias, and Temam (estimation ofHausdorff dimension via the Lyapunov exponents and Lyapunov dimension ofdynamical systems [13]), Leonov (estimation of the Lyapunov dimension via thedirect Lyapunov method [20, 24]), and numerical methods for the computation ofLyapunov exponents and Lyapunov dimension [21]. We also concentrate in thischapter on the Kaplan–Yorke formula and the Lyapunov dimension formulas forthe Lorenz and Hénon attractors (Leonov [26]).

In Part III, we consider dimension properties for dynamical systems on mani-folds. Chapter 7 gives a presentation of the exterior calculus in general linearspaces. It contains also some results about orbital stability for vector fields onmanifolds.

Chapter 8 is devoted to dimension estimates of invariant sets and attractors ofdynamical systems on Riemannian manifolds. The Douady–Oesterlé theorem forthe upper Hausdorff dimension estimates for invariant sets of smooth dynamical

Preface vii

systems on Riemannian manifolds is proved. Chapter 8 contains also an importantresult on the estimation of the fractal dimension of an invariant set on an arbitraryfinite-dimensional smooth manifold by the upper Lyapunov dimension, which goesback to Hunt [17], Gelfert [14]. Then we discuss the construction of the specialCarathéodory measures for the estimation of Hausdorff measures connected withflow invariant sets on Riemannian manifolds.

In Chap. 9, we derive dimension and entropy estimates for invariant sets andglobal B-attractors of cocycles. A version of the Douady–Oesterlé theorem (Leonovet al. [31]) is proved for local cocycles in a Euclidean space and for cocycles onRiemannian manifolds. As examples, we consider cocycles, generated by theRössler system with variable coefficients. We also introduce time-discrete cocycleson fibered spaces and define the topological entropy of such cocycles. We thankA. O. Romanov for helping us to prepare this chapter.

In Chap. 10, we derive some versions of the Douady–Oesterlé theorem forsystems with singularities. In the first part of this chapter, we consider a specialclass of non-injective maps, for which we introduce a factor describing the “degreeof non-injectivity” (Boichenko et al. [6]). This factor can be included in thedimension estimates of Chap. 8 in order to weaken the condition to the singularvalue function. In the second part of Chap. 10, we derive the upper Hausdorffdimension estimates for invariant sets of a class of not necessarily invertible andpiecewise smooth maps on manifolds with controllable preimages of thenon-differentiability sets in terms of the singular values of the derivative of thesmoothly extended map (Reitmann and Schnabel [41], Neunhäuserer [37]) Theseestimates generalize some Douady–Oesterlé type results for differentiable maps in aEuclidean space, derived in Chaps. 5 and 8.

In the last section of Chap. 10, we discuss some classes of functionals which areuseful for the estimation of topological and metric dimensions.

St. Petersburg, Russia Nikolay KuznetsovVolker Reitmann

References

1. Abramovich, S., Koryakin, Yu., Leonov, G., Reitmann, V.: Frequency-domain conditions foroscillations in discrete systems. I., Oscillations in the sense of Yakubovich in discrete sys-tems. Wiss. Zeitschr. Techn. Univ. Dresden. 25(5/6), 1153–1163 (1977) (German)

2. Abramovich, S., Koryakin, Yu., Leonov, G., Reitmann, V.: Frequency-domain conditions foroscillations in discrete systems. II., Oscillations in discrete phase systems. Wiss. Zeitschr.Techn. Univ. Dresden. 26(1), 115–122 (1977) (German)

3. Anikushin, M.M.: Dimension theory approach to the complexity of almost periodic trajec-tories. Intern. J. Evol. Equ. 10(3–4), 215–232 (2017)

4. Boichenko, V.A., Leonov, G.A.: Lyapunov’s direct method in the estimation of the Hausdorffdimension of attractors. Acta Appl. Math. 26, 1–60 (1992)

viii Preface

5. Boichenko, V.A., Leonov, G.A.: Lyapunov functions, Lozinskii norms, and the Hausdorffmeasure in the qualitative theory of differential equations. Amer. Math. Soc. Transl. 193(2),1–26 (1999)

6. Boichenko, V.A., Leonov, G.A., Franz, A., Reitmann,V.: Hausdorff and fractal dimensionestimates of invariant sets of non-injective maps. Zeitschrift für Analysis und ihreAnwendungen (ZAA). 17(1), 207–223 (1998)

7. Boichenko, V.A., Leonov, G.A., Reitmann, V.: Dimension Theory for Ordinary DifferentialEquations. Teubner, Stuttgart (2005)

8. Borg, G.: A condition for existence of orbitally stable solutions of dynamical systems. Kungl.Tekn. Högsk. Handl. Stockholm. 153, 3–12 (1960)

9. Chen, Zhi-Min.: A note on Kaplan-Yorke-type estimates on the fractal dimension of chaoticattractors. Chaos, Solitons & Fractals 3, 575–582 (1993)

10. Chen, X.: Lorenz equations, part I: existence and nonexistence of homoclinic orbits.SIAM J. Math. Anal. 27(4), 1057–1069 (1996)

11. Constantin, P., Foias, C., Temam, R.: Attractors representing turbulent flows. Amer. Math.Soc. Memoirs., Providence, Rhode Island. 53(314), (1985)

12. Douady, A., Oesterlé, J.: Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris, Ser.A. 290, 1135–1138 (1980)

13. Eden, A., Foias, C., Temam, R.: Local and global Lyapunov exponents. J. Dynam. Diff. Equ.3, 133–177 (1991) [Preprint No. 8804, The Institute for Applied Mathematics and ScientificComputing, Indiana University, 1988]

14. Gelfert, K.: Maximum local Lyapunov dimension bounds the box dimension. Direct proof forinvariant sets on Riemannian manifolds. Zeitschrift für Analysis und ihre Anwendungen(ZAA). 22(3), 553–568 (2003)

15. Hartman, P., Olech, C.: On global asymptotic stability of solutions of ordinary differentialequations. Trans. Amer. Math. Soc. 104, 154–178 (1962)

16. Hastings, S.P., Troy, W.C.: A shooting approach to chaos in the Lorenz equations. J. Diff.Equ. 127(1), 41–53 (1996)

17. Hunt, B.: Maximum local Lyapunov dimension bounds the box dimension of chaoticattractors. Nonlinearity. 9, 845–852 (1996)

18. Hurewicz, W., Wallman, H.: Dimension Theory. Princeton Univ. Press, Princeton (1948)19. Kaplan, J.L., Yorke, J.A.: Chaotic behavior of multidimensional difference equations. In:

Functional Differential Equations and Approximations of Fixed Points, 204–227, Springer,Berlin (1979)

20. Kuznetsov, N.V.: The Lyapunov dimension and its estimation via the Leonov method.Physics Letters A, 380(25–26), 2142–2149 (2016)

21. Kuznetsov, N.V., Leonov, G.A., Mokaev, T.N., Prasad, A., Shrimali, M.D.: Finite-timeLyapunov dimension and hidden attractor of the Rabinovich system. Nonlinear Dyn. 92 (2),267–285 (2018)

22. Ledrappier, F.: Some relations between dimension and Lyapunov exponents. Commun. Math.Phys. 81, 229–238 (1981)

23. Leonov, G.A.: On the estimation of the bifurcation parameter values of the Lorenz system.Uspekhi Mat. Nauk. 43(3), 189–200 (1988) (Russian); English transl. Russian Math. Surveys.43(3), 216–217 (1988)

24. Leonov, G.A.: Estimation of the Hausdorff dimension of attractors of dynamical systems.Diff. Urav. 27(5), 767–771 (1991) (Russian); English transl. Diff. Equations, 27, 520–524(1991)

25. Leonov, G.A.: Construction of a special outer Carathéodory measure for the estimation of theHausdorff dimension of attractors. Vestn. S. Peterburg Gos. Univ. 1(22), 24–31 (1995)(Russian); English transl. Vestn. St. Petersburg Univ. Math. Ser. 1, 28(4), 24–30 (1995)

26. Leonov, G.A.: Lyapunov dimensions formulas for Hénon and Lorenz attractors. Alg. & Anal.13, 155–170 (2001) (Russian); English transl. St. Petersburg Math. J. 13(3), 453–464 (2002)

Preface ix

27. Leonov, G.A., Gelfert, K., Reitmann, V.: Hausdorff dimension estimates by use of a tubularCarathéodory structure and their application to stability theory. Nonlinear Dyn. Syst. Theory,1(2), 169–192 (2001)

28. Leonov, G.A., Lyashko, S.: Eden’s hypothesis for a Lorenz system. Vestn. S. Peterburg Gos.Univ., Matematika. 26(3), 15–18 (1993) (Russian); English transl. Vestn. St. Petersburg Univ.Math. Ser. 1, 26(3), 14–16 (1993)

29. Leonov, G.A., Ponomarenko, D.V., Smirnova, V.B.: Frequency-Domain Methods forNonlinear Analysis. World Scientific, Singapore-New Jersey-London-Hong Kong (1996)

30. Leonov, G. A., Reitmann, V.: Localization of Attractors for Nonlinear Systems.Teubner-Texte zur Mathematik, Bd. 97, B. G. Teubner Verlagsgesellschaft, Leipzig, (1987)(German)

31. Leonov, G.A., Reitmann, V., Slepuchin, A.S.: Upper estimates for the Hausdorff dimension ofnegatively invariant sets of local cocycles. Dokl. Akad. Nauk, T. 439, No. 6 (2011) (Russian);English transl. Dokl. Mathematics. 84(1), 551–554 (2011)

32. Leonov, G.A., Reitmann, V., Smirnova, V.B.: Non-local Methods for Pendulum-likeFeedback Systems. Teubner-Texte zur Mathematik, Bd. 132, B. G. Teubner Stuttgart- Leipzig(1992)

33. Leonov, G.A., Kuznetsov, N.V., Mokaev T.N.: Homoclinic orbits, and self-excited andhidden attractors in a Lorenz-like system describing convective fluid motion. Eur. Phys.J. Special Topics. 224(8), 1421–1458 (2015)

34. Maltseva, A.A., Reitmann, V.: Existence and dimension properties of a global B-pullbackattractor for a cocycle generated by a discrete control system. J. Diff. Equ. 53(13), 1703–1714(2017)

35. Li, M.Y., Muldowney, J.S.: On Bendixson’s criterion. J. Diff. Equ. 106(1), 27–39 (1993)36. Muldowney, J.S.: Compound matrices and ordinary differential equations. Rocky

Mountain J. Math. 20, 857–871 (1990)37. Neunhäuserer, J.: A Douady-Oesterlé type estimate for the Hausdorff dimension of invariant

sets of piecewise smooth maps. Preprint, University of Technology Dresden (2000)38. Noack, A., Reitmann, V.: Hausdorff dimension estimates for invariant sets of time-dependent

vector fields. Zeitschrift für Analysis und ihre Anwendungen (ZAA). 15(2), 457–473 (1996)39. Pesin, Ya. B.: Dimension type characteristics for invariant sets of dynamical systems. Uspekhi

Mat. Nauk. 43(4), 95–128 (1988) (Russian); English transl. Russian Math. Surveys. 43(4),111–151 (1988)

40. Pugh, C.C.: An improved closing lemma and a general density theorem. Amer. J. Math. 89,1010–1021 (1967)

41. Reitmann, V., Schnabel, U.: Hausdorff dimension estimates for invariant sets of piecewisesmooth maps. ZAMM 80(9), 623–632 (2000)

42. Smith, R.A.: Some applications of Hausdorff dimension inequalities for ordinary differentialequations. Proc. Roy. Soc. Edinburgh. 104A, 235–259 (1986)

x Preface

Acknowledgements

While still working on this book, our coauthor Prof. G. A. Leonov, correspondingmember of the Russian Academy of Science, died in 2018. This work is dedicatedto his memory, with our deepest and most sincere admiration, gratitude, and love.He was an excellent mathematician with a sharp view on problems and a wonderfulcolleague and friend. He will stay forever in our mind.

The preparation of this book was carried out in 2017–2019 at the St. PetersburgState University, at the Institute for Problems in Mechanical Engineering of theRussian Academy of Science, and at the University of Jyväskylä within theframework of the Russian Science Foundation projects 14-21-00041 and19-41-02002.

One of the authors (V.R.) was supported in 2017–2018 by the Johann GottfriedHerder Programme of the German Academic Exchange Service (DAAD).

The authors of the book are greatly indebted to Margitta Reitmann for heraccurate typing of the manuscript in LATEX.

St. Petersburg, Russia Nikolay KuznetsovFebruary 2020 Volker Reitmann

xi

Contents

Part I Basic Elements of Attractor and Dimension Theories

1 Attractors and Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Dynamical Systems, Limit Sets and Attractors . . . . . . . . . . . . . 3

1.1.1 Dynamical Systems in Metric Spaces . . . . . . . . . . . . . 31.1.2 Minimal Global Attractors . . . . . . . . . . . . . . . . . . . . . 10

1.2 Dissipativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.1 Dissipativity in the Sense of Levinson . . . . . . . . . . . . . 151.2.2 Dissipativity and Completeness of The Lorenz

System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.3 Lyapunov-Type Results for Dissipativity . . . . . . . . . . . 20

1.3 Existence of a Homoclinic Orbit in the Lorenz System . . . . . . . 251.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.3.2 Estimates for the Shape of Global Attractors . . . . . . . . 251.3.3 The Existence of Homoclinic Orbits . . . . . . . . . . . . . . 28

1.4 The Generalized Lorenz System . . . . . . . . . . . . . . . . . . . . . . . 331.4.1 Definition of the System . . . . . . . . . . . . . . . . . . . . . . . 331.4.2 Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.4.3 Global Asymptotic Stability . . . . . . . . . . . . . . . . . . . . 351.4.4 Dissipativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2 Singular Values, Exterior Calculus and Logarithmic Norms . . . . . 412.1 Singular Values and Covering of Ellipsoids . . . . . . . . . . . . . . . 41

2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.1.2 Definition of Singular Values . . . . . . . . . . . . . . . . . . . 432.1.3 Lemmas on Covering of Ellipsoids . . . . . . . . . . . . . . . 45

xiii

2.2 Singular Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2.1 The Fischer-Courant Theorem . . . . . . . . . . . . . . . . . . . 472.2.2 The Binet–Cauchy Theorem . . . . . . . . . . . . . . . . . . . . 482.2.3 The Inequalities of Horn, Weyl and Fan . . . . . . . . . . . 50

2.3 Compound Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.3.1 Multiplicative Compound Matrices . . . . . . . . . . . . . . . 522.3.2 Additive Compound Matrices . . . . . . . . . . . . . . . . . . . 562.3.3 Applications to Stability Theory . . . . . . . . . . . . . . . . . 58

2.4 Logarithmic Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.4.1 Lozinskii’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 602.4.2 Generalization of the Liouville Equation . . . . . . . . . . . 662.4.3 Applications to Orbital Stability . . . . . . . . . . . . . . . . . 69

2.5 The Yakubovich-Kalman Frequency Theorem . . . . . . . . . . . . . 732.5.1 The Frequency Theorem for ODE’s . . . . . . . . . . . . . . . 732.5.2 The Frequency Theorem for Discrete-Time

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.6 Frequency-Domain Estimation of Singular Values . . . . . . . . . . 77

2.6.1 Linear Differential Equations . . . . . . . . . . . . . . . . . . . . 772.6.2 Linear Difference Equations . . . . . . . . . . . . . . . . . . . . 81

2.7 Convergence in Systems with Several Equilibrium States . . . . . 842.7.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.7.2 Convergence in the Lorenz System . . . . . . . . . . . . . . . 88

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3 Introduction to Dimension Theory . . . . . . . . . . . . . . . . . . . . . . . . . 953.1 Topological Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.1.1 The Inductive Topological Dimension . . . . . . . . . . . . . 963.1.2 The Covering Dimension . . . . . . . . . . . . . . . . . . . . . . 101

3.2 Hausdorff and Fractal Dimensions . . . . . . . . . . . . . . . . . . . . . . 1053.2.1 The Hausdorff Measure and the Hausdorff

Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053.2.2 Fractal Dimension and Lower Box Dimension . . . . . . . 1143.2.3 Self-similar Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.2.4 Dimension of Cartesian Products . . . . . . . . . . . . . . . . . 122

3.3 Topological Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253.3.1 The Bowen-Dinaburg Definition . . . . . . . . . . . . . . . . . 1253.3.2 The Characterization by Open Covers . . . . . . . . . . . . . 1283.3.3 Some Properties of the Topological Entropy . . . . . . . . 131

3.4 Dimension-Like Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 1363.4.1 Carathéodory Measure, Dimension and Capacity . . . . . 1363.4.2 Properties of the Carathéodory Dimension

and Carathéodory Capacity . . . . . . . . . . . . . . . . . . . . . 139References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

xiv Contents

Part II Dimension Estimates for Almost Periodic Flows andDynamical Systems in Euclidean Spaces

4 Dimensional Aspects of Almost Periodic Dynamics . . . . . . . . . . . . . 1494.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1494.2 Topological Dimension of Compact Groups . . . . . . . . . . . . . . . 1504.3 Frequency Module and Cartwright’s Theorem on Almost

Periodic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534.4 Minimal Sets in Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . 1574.5 Almost Periodic Solutions of Almost Periodic ODEs . . . . . . . . 1584.6 Frequency Spectrum of Almost Periodic Solutions

for DDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664.7 Fractal Dimensions of Almost Periodic Trajectories

and The Liouville Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . 1714.8 Fractal Dimensions of Forced Almost Periodic Regimes

in Chua’s Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5 Dimension and Entropy Estimates for Dynamical Systems . . . . . . . 1915.1 Upper Estimates for the Hausdorff Dimension of Negatively

Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.1.1 The Limit Theorem for Hausdorff Measures . . . . . . . . . 1915.1.2 Corollaries of the Limit Theorem for Hausdorff

Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1965.1.3 Application of the Limit Theorem to the Hénon

Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2015.2 The Application of the Limit Theorem to ODE’s . . . . . . . . . . . 206

5.2.1 An Auxiliary Result . . . . . . . . . . . . . . . . . . . . . . . . . . 2065.2.2 Estimates of the Hausdorff Measure and of Hausdorff

Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2085.2.3 The Generalized Bendixson Criterion . . . . . . . . . . . . . 2125.2.4 On the Finiteness of the Number of Periodic

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2135.2.5 Convergence Theorems . . . . . . . . . . . . . . . . . . . . . . . . 214

5.3 Convergence in Third-Order Nonlinear Systems Arisingfrom Physical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2155.3.1 The Generalized Lorenz System . . . . . . . . . . . . . . . . . 2155.3.2 Euler’s Equations Describing the Rotation

of a Rigid Body in a Resisting Medium . . . . . . . . . . . 2205.3.3 A Nonlinear System Arising from Fluid Convection

in a Rotating Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . 2215.3.4 A System Describing the Interaction of Three Waves

in Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Contents xv

5.4 Estimates of Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . 2255.4.1 Maps with a Constant Jacobian . . . . . . . . . . . . . . . . . . 2255.4.2 Autonomous Differential Equations Which

are Conservative on the Invariant Set . . . . . . . . . . . . . 2295.5 Fractal Dimension Estimates for Invariant Sets

and Attractors of Concrete Systems . . . . . . . . . . . . . . . . . . . . . 2305.5.1 The Rössler System . . . . . . . . . . . . . . . . . . . . . . . . . . 2305.5.2 Lorenz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2325.5.3 Equations of the Third Order . . . . . . . . . . . . . . . . . . . 2395.5.4 Equations Describing the Interaction Between

Waves in Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2445.6 Estimates of the Topological Entropy . . . . . . . . . . . . . . . . . . . . 247

5.6.1 Ito’s Generalized Entropy Estimate for Maps . . . . . . . . 2475.6.2 Application to Differential Equations . . . . . . . . . . . . . . 252

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

6 Lyapunov Dimension for Dynamical Systems in EuclideanSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2576.1 Singular Value Function and Invariant Sets of Maps

of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2576.2 Lyapunov Dimension of Maps . . . . . . . . . . . . . . . . . . . . . . . . . 2636.3 Lyapunov Dimensions of a Dynamical System . . . . . . . . . . . . . 265

6.3.1 Lyapunov Exponents: Various Definitions . . . . . . . . . . 2696.3.2 Kaplan-Yorke Formula of the Lyapunov

Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2726.4 Analytical Estimates of the Lyapunov Dimension

and its Invariance with Respect to Diffeomorphisms . . . . . . . . . 2796.5 Analytical Formulas of Exact Lyapunov Dimension

for Well-Known Dynamical Systems . . . . . . . . . . . . . . . . . . . . 2866.5.1 Hénon Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2866.5.2 Lorenz System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2876.5.3 Glukhovsky-Dolzhansky System . . . . . . . . . . . . . . . . . 2916.5.4 Yang and Tigan Systems . . . . . . . . . . . . . . . . . . . . . . 2926.5.5 Shimizu-Morioka System . . . . . . . . . . . . . . . . . . . . . . 293

6.6 Attractors of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . 2946.6.1 Computation of Attractors and Lyapunov

Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2946.7 Computation of the Finite-Time Lyapunov Exponents

and Dimension in MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . 298References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

xvi Contents

Part III Dimension Estimates on Riemannian Manifolds

7 Basic Concepts for Dimension Estimation on Manifolds . . . . . . . . . 3097.1 Exterior Calculus in Linear Spaces, Singular Values

of an Operator and Covering Lemmas . . . . . . . . . . . . . . . . . . . 3097.1.1 Multiplicative and Additive Compounds

of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3097.1.2 Singular Values of an Operator Acting Between

Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3187.1.3 Lemmas on Covering of Ellipsoids in an Euclidean

Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3217.1.4 Singular Value Inequalities for Operators . . . . . . . . . . . 323

7.2 Orbital Stability for Flows on Manifolds . . . . . . . . . . . . . . . . . 3257.2.1 The Andronov-Vitt Theorem . . . . . . . . . . . . . . . . . . . . 3257.2.2 Various Types of Variational Equations . . . . . . . . . . . . 3267.2.3 Asymptotic Orbital Stability Conditions . . . . . . . . . . . . 3297.2.4 Characteristic Exponents . . . . . . . . . . . . . . . . . . . . . . . 3437.2.5 Orbital Stability Conditions in Terms of Exponents . . . 3477.2.6 Estimating the Singular Values and Orbital Stability . . . 3487.2.7 Frequency-Domain Conditions for Orbital Stability

in Feedback Control Equations on the Cylinder . . . . . . 3547.2.8 Dynamical Systems with a Local Contraction

Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

8 Dimension Estimates on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 3658.1 Hausdorff Dimension Estimates for Invariant Sets

of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3658.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3658.1.2 Hausdorff Dimension Bounds for Invariant

Sets of Maps on Manifolds . . . . . . . . . . . . . . . . . . . . . 3668.1.3 Time-Dependent Vector Fields on Manifolds . . . . . . . . 3718.1.4 Convergence for Autonomous Vector Fields . . . . . . . . 376

8.2 The Lyapunov Dimension as Upper Bound of the FractalDimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3798.2.1 Statement of the Results . . . . . . . . . . . . . . . . . . . . . . . 3798.2.2 Proof of Theorem 8.5 . . . . . . . . . . . . . . . . . . . . . . . . . 3808.2.3 Global Lyapunov Exponents and Upper Lyapunov

Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3878.2.4 Application to the Lorenz System . . . . . . . . . . . . . . . . 388

8.3 Hausdorff Dimension Estimates by Use of a TubularCarathéodory Structure and their Application to StabilityTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

Contents xvii

8.3.1 The System in Normal Variation . . . . . . . . . . . . . . . . . 3908.3.2 Tubular Carathéodory Structure . . . . . . . . . . . . . . . . . . 3948.3.3 Dimension Estimates for Sets Which are Negatively

Invariant for a Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 3968.3.4 Flow Invariant Sets with an Equivariant Tangent

Bundle Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4038.3.5 Generalizations of the Theorems of Hartman-Olech

and Borg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

9 Dimension and Entropy Estimates for Global Attractorsof Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4119.1 Basic Facts from Cocycle Theory in Non-fibered Spaces . . . . . . 411

9.1.1 Definition of a Cocycle . . . . . . . . . . . . . . . . . . . . . . . . 4119.1.2 Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4139.1.3 Global B-Attractors of Cocycles . . . . . . . . . . . . . . . . . 4139.1.4 Extension System Over the Bebutov Flow

on a Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4169.2 Local Cocycles Over the Base Flow in Non-fibered Spaces . . . . 418

9.2.1 Definition of a Local Cocycle . . . . . . . . . . . . . . . . . . . 4189.2.2 Upper Bounds of the Hausdorff Dimension

for Local Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 4199.2.3 Upper Estimates for the Hausdorff Dimension

of Local Cocycles Generated by DifferentialEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

9.2.4 Upper Estimates for the Hausdorff Dimensionof a Negatively Invariant Set of the Non-autonomousRössler System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

9.3 Dimension Estimates for Cocycles on Manifolds (Non-fiberedCase) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4299.3.1 The Douady-Oesterlé Theorem for Cocycles

on a Finite Dimensional Riemannian Manifold . . . . . . . 4299.3.2 Upper Bounds for the Haussdorff Dimension of

Negatively Invariant Sets of Discrete-TimeCocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

9.3.3 Frequency Conditions for Dimension Estimatesfor Discrete Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . 438

9.3.4 Upper Bound for Hausdorff Dimension of InvariantSets and B-attractors of Cocycles Generated byOrdinary Differential Equations on Manifolds . . . . . . . 440

9.3.5 Upper Bounds for the Hausdorff Dimension ofAttractors of Cocycles Generated by DifferentialEquations on the Cylinder . . . . . . . . . . . . . . . . . . . . . . 444

xviii Contents

9.4 Discrete-Time Cocycles on Fibered Spaces . . . . . . . . . . . . . . . . 4509.4.1 Definition of Cocycles on Fibered Spaces . . . . . . . . . . 4509.4.2 Global Pullback Attractors . . . . . . . . . . . . . . . . . . . . . 4519.4.3 The Topological Entropy of Fibered Cocycles . . . . . . . 451

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

10 Dimension Estimates for Dynamical Systems with SomeDegree of Non-injectivity and Nonsmoothness . . . . . . . . . . . . . . . . . 45710.1 Dimension Estimates for Non-injective Smooth Maps . . . . . . . . 457

10.1.1 Hausdorff Dimension Estimates . . . . . . . . . . . . . . . . . . 45710.1.2 Fractal Dimension Estimates . . . . . . . . . . . . . . . . . . . . 465

10.2 Dimension Estimates for Piecewise C1-Maps . . . . . . . . . . . . . . 47110.2.1 Decomposition of Invariant Sets of Piecewise

Smooth Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47110.2.2 A Class of Piecewise C1-Maps . . . . . . . . . . . . . . . . . . 47210.2.3 Douady-Oesterlé-Type Estimates . . . . . . . . . . . . . . . . . 47610.2.4 Consideration of the Degree of Non-injectivity . . . . . . 47910.2.5 Introduction of Long Time Behavior Information . . . . . 48110.2.6 Estimation of the Hausdorff Dimension for Invariant

Sets of Piecewise Smooth Vector Fields . . . . . . . . . . . 48410.3 Dimension Estimates for Maps with Special Singularity Sets . . . 491

10.3.1 Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . 49210.3.2 Proof of the Main Result . . . . . . . . . . . . . . . . . . . . . . 49310.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

10.4 Lower Dimension Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 49910.4.1 Frequency-Domain Conditions for Lower

Topological Dimension Bounds of GlobalB-Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

10.4.2 Lower Estimates of the Hausdorff Dimensionof Global B-Attractors . . . . . . . . . . . . . . . . . . . . . . . . 503

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

Appendix A: Basic Facts from Manifold Theory . . . . . . . . . . . . . . . . . . . 509

Appendix B: Miscellaneous Facts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

Contents xix