Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Attractors and Lyapunov Functions Read chapter Read first chapter

2021 | OriginalPaper | Chapter

5. Dimension and Entropy Estimates for Dynamical Systems

Authors : Nikolay Kuznetsov, Volker Reitmann

Published in: Attractor Dimension Estimates for Dynamical Systems: Theory and Computation

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In the present chapter various approaches to estimate the fractal dimension and the Hausdorff dimension, which involve Lyapunov functions, are developed. One of the main results of this chapter is a theorem called by us the limit theorem for the Hausdorff measure of a compact set under differentiable maps. One of the sections of Chap. 5 is devoted to applications of this theorem to the theory of ordinary differential equations. The use of Lyapunov functions in the estimates of fractal dimension and of topological entropy is also considered. The representation is illustrated by examples of concrete systems.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Dimensional Aspects of Almost Periodic Dynamics
next chapter Lyapunov Dimension for Dynamical Systems in Euclidean Spaces
Literature
1.
Almeida, J., Barreira, L.: Hausdorff dimension in convex bornological spaces. J. Math. Anal. Appl. 266, 590–601 (2002)MathSciNetCrossRef
2.
Anikushin, M.M., Reitmann, V.: Development of the topological entropy conception for dynamical systems with multiple time. Electr. J. Diff. Equ. Contr. Process. 4 (2016). (Russian); English transl. J. Diff. Equ. 52(13), 1655–1670 (2016)
3.
Arnédo, A., Coullet, P.H., Spiegel, E.A.: Chaos in a finite macroscopic system. Phys. Lett. 92A, 369–373 (1982)
4.
Benedicks, M. Carleson, L.: The dynamics of the Hénon map. Ann. Math. II. Ser. 133(1), 73–169 (1991)
5.
Boichenko, V.A.: Frequency theorems on estimates of the dimension of attractors and global stability of nonlinear systems. Leningrad, Dep. at VINITI 04.04.90, No. 1832–B90 (1990) (Russian)
6.
Boichenko, V.A.: The Lyapunov function in estimates of the Hausdorff dimension of attractors of an equation of third order. Diff. Urav. 30, 913–915 (1994). (Russian)MathSciNet
7.
Boichenko, V.A., Leonov, G.A.: On estimates of attractors dimension and global stability of generalized Lorenz equations. Vestn. Lening. Gos. Univ. Ser. 1, 2, 7–13 (1990) (Russian); English transl. Vestn. Lening. Univ. Math. 23(2), 6–12 (1990)
8.
Boichenko, V.A., Leonov, G.A.: On Lyapunov functions in estimates of the Hausdorff dimension of attractors. Leningrad, Dep. at VINITI 28.10.91, No. 4 123–B 91 (1991) (Russian)
9.
Boichenko, V.A., Leonov, G.A.: Lyapunov functions in the estimation of the topological entropy. Preprint, University of Technology Dresden, Dresden (1994)
10.
Boichenko, V.A., Leonov, G.A.: Lyapunov’s direct method in estimates of topological entropy. Zap. Nauchn. Sem. POMI 231, 62–75 (1995) (Russian); English transl. J. Math. Sci. 91(6), 3370–3379 (1998)
11.
Boichenko, V.A., Leonov, G.A.: Lyapunov functions, Lozinskii norms, and the Hausdorff measure in the qualitative theory of differential equations. Am. Math. Soc. Transl. 2(193), 1–26 (1999)MathSciNetMATH
12.
Boichenko, V.A., Leonov, G.A.: On dimension estimates for the attractors of the Hénon map. Vestn. S. Peterburg Gos. Univ. Ser. 1, Matematika, (3), 8–13 (2000) (Russian); English transl. Vestn. St. Petersburg Univ. Math. 33 (3), 5–9 (2000)
13.
Boichenko, V.A., Leonov, G.A., Franz, A., Reitmann, V.: Hausdorff and fractal dimension estimates of invariant sets of non-injective maps. Zeitschrift für Analysis und ihre Anwendungen (ZAA) 17(1), 207–223 (1998)MathSciNetCrossRef
14.
Chen, Z.-M.: A note on Kaplan-Yorke-type estimates on the fractal dimension of chaotic attractors. Chaos, Solitons & Fractals 3(5), 575–582 (1993)
15.
Coullet, P., Tresser, C., Arnédo, A.: Transition to stochasticity for a class of forced oscillators. Phys. Lett. 72 A, 268–270 (1979)
16.
Courant, R.: Dirichlet’s Principle, Conformed Mappings, and Minimal Surfaces. Springer, New York (1977)CrossRef
17.
Denisov, G.G.: On the rotation of a solid body in a resisting medium. Izv. Akad. Nauk SSSR Mekh. Tverd. Tela 4, 37–43 (1989) (Russian)
18.
Douady, A., Oesterlé, J.: Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris, Ser. A 290, 1135–1138 (1980)
19.
Eden, A., Foias, C., Temam, R.: Local and global Lyapunov exponents. J. Dyn. Diff. Equ. 3, 133–177 (1991) [Preprint No. 8804, The Institute for Applied Mathematics and Scientific Computing, Indiana University, 1988]
20.
21.
Falconer, K.J.: The Geometry of Fractal sets. Cambridge Tracts in Mathematics 85, Cambridge University Press (1985)
22.
Fathi, A.: Some compact invariant subsets for hyperbolic linear automorphisms of Torii. Ergod. Theory Dyn. Syst. 8, 191–202 (1988)CrossRef
23.
Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel (1984)CrossRef
24.
Glukhovsky, A.B.: Nonlinear systems in the form of superposition of gyrostats. Dokl. Akad. Nauk SSSR 266 (7) (1982) (Russian)
25.
Glukhovsky, A.B., Dolzhanskii, F.V: Three-component geostrophic model of convection in rotating fluid. Izv. Akad. Nauk SSSR, Fiz. Atmos. i Okeana 16, 451–462 (1980) (Russian)
26.
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)CrossRef
27.
28.
Ito, S.: An estimate from above for the entropy and the topological entropy of a $C^1$-diffeomorphism. Proc. Jpn. Acad. 46, 226–230 (1970)MathSciNetCrossRef
29.
Leonov, G.A.: On a method for investigating global stability of nonlinear systems. Vestn. Leningr. Gos. Univ. Mat. Mekh. Astron. 4, 11–14 (1991) (Russian); English transl. Vestn. Leningr. Univ. Math. 24(4), 9–11 (1991)
30.
Leonov, G.A.: On estimates of the Hausdorff dimension of attractors. Vestn. Leningr. Gos. Univ. Ser. 1 15, 41–44 (1991) (Russian); English transl. Vestn. Leningr. Univ. Math. 24(3), 38–41 (1991)
31.
Leonov, G.A., Boichenko, V.A.: Lyapunov’s direct method in the estimation of the Hausdorff dimension of attractors. Acta Appl. Math. 26, 1–60 (1992)MathSciNetCrossRef
32.
Leonov, G.A., Lyashko, S.A.: Surprising property of Lyapunov dimension for invariant sets of dynamical systems. DFG Priority Research Program “Dynamics: Analysis, Efficient Simulation, and Ergodic Theory”. Preprint 04 (2000)
33.
Ott, E., Yorke, E.D., Yorke, J.A.: A scaling law: how an attractor’s volume depends on noise level. Physica D 16(1), 62–78 (1985)MathSciNetCrossRef
34.
Pikovsky, A.S., Rabinovich, M.I., Trakhtengerts, VYu.: Appearance of stochasticity on decay confinement of parametric instability. JTEF 74, 1366–1374 (1978). (Russian)
35.
Pochatkin, M.A., Reitmann, V.: The Douady-Oesterlé theorem for dynamical systems in convex bornological spaces. Preprint, St. Petersburg State University, St. Petersburg (2017). (Russian)
36.
37.
Reitmann, V.: Dynamical Systems, Attractors and their Dimension Estimates. St. Petersburg State University Press, St. Petersburg (2013). (Russian)
38.
Rössler, O.E.: Different types of chaos in two simple differential equations. Z. Naturforsch. 31 A, 1664–1670 (1976)
39.
40.
Smith, R.A.: Some applications of Hausdorff dimension inequalities for ordinary differential equations. Proc. R. Soc. Edinburgh 104A, 235–259 (1986)MathSciNetCrossRef
41.
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York, Berlin (1988)CrossRef
42.
Thi, D.T., Fomenko, A.T.: Minimal Surfaces, Stratified Multivarifolds, and the Plateau Problem, Nauka, Moscow (1987) (Russian)
Metadata
Title
Dimension and Entropy Estimates for Dynamical Systems
Authors
Nikolay Kuznetsov
Volker Reitmann
Copyright Year
2021
Book
Attractor Dimension Estimates for Dynamical Systems: Theory and Computation

Print ISBN: 978-3-030-50986-6

Electronic ISBN: 978-3-030-50987-3

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-50987-3_5