Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

About History Editorial Page Addresses Scope Editorial Staff Submission Review For Authors Publication Ethics Issues Русская версия

Development of the Topological Entropy Conception for Dynamical Systems with Multiple Time

Author(s):

Mikhail M. Anikushin

demolishka@gmail.com

Volker Reitmann

70 corp.3, Botanicheskaya st,
Peterhof, Saint-Petersburg,
198516, Russia
Saint-Petersburg State University
professor of the Department of Applied Cybernetics
Prof. Dr.

vreitmann@aol.com

Abstract:

We study the topological entropy for dynamical systems with discrete or continuous multiple time. Due to the generalization of a well-known one time-dimensional result we show that the definition of topological entropy, using the approach for subshifts, leads to zero entropy for many systems different from subshift. We define a new type of relative topological entropy to avoid this phenomenon. The generalization of Bowen's power rule allows us to define topological and relative topological entropies for systems with continuous multiple time. As an application we find the relation between relative topological entropy and controllability of linear system with continuous multiple time.

Keywords

References:

  1. Boichenko, V. A., Leonov, G. A. Lyapunov’s direct method in estimates of topological entropy, Zap. nauchn. sem. POMI, 1995, vol. 231, pp. 62-75. (In Russ. )
  2. Gaishun, I. V., Kirillova, F. M. Vpolne razreshimye mnogomernye differencial'nye uravneniya[Completely Integrable Differential Equations], Nauka i Tekhnika Publ., 1983
  3. Kolmogorov, A. N. New Metric Invariant of Transitive Dynamical Systems and Endomorphisms of Lebesgue Spaces, DAN SSSR, 1958, vol. 119, no. 5, pp. 861-864. (In Russ. )
  4. Lakshtanov, E. L., Langvagen, E. S. Criteria for infinity value of topological entropy of multidimensional cellular automata, Problemy peredachi informacii, 2004, vol. 40, no. 2, pp. 70-72. (In Russ. )
  5. Lakshtanov, E. L., Langvagen, E. S. Entropy of multi-dimensional cellular automata, Problemy peredachi informacii, 2006, vol. 42, no. 1, pp. 43-51. (In Russ. )
  6. Sinaj, YA. G. On the Notion of Entropy of a Dynamical System, DAN SSSR, 1959, vol. 124, no. 4, pp. 768-771. (In Russ. )
  7. Sinaj, YA. G. Sovremennye problemy ehrgodicheskoj teorii[Modern problems of the ergodic theory], Moscow, Fizmatlit Publ., 1995
  8. Adler, R. L., Konheim, A. G., McAndrew, M. H. Topological entropy, Trans. Amer. Math. Soc. , 1965, vol. 114, no. 2, pp. 309-319
  9. Anosov, D. V. Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. , 1967, vol. 90, pp. 3-210
  10. Boichenko, V. A., Leonov, G. A., Reitmann, V. Dimension Theory for Ordinary Differential Equations, Teubner Wiesbaden, 2005
  11. Bowen, R. Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. , 1971, vol. 153, no. 2, pp. 401-414
  12. Ito, S. An estimate from above for the entropy and the topological entropy of a C1-diffeomorphism, Proc. Japan Acad. , 1970, vol. 46, no. 3, pp. 226-230
  13. Kuznetsov, N. V. The Lyapunov dimension and its estimation via the Leonov method, Physics LettersA, 2016, vol. 380, no. 25, pp. 2142-2149
  14. Kuznetsov, N. V., Alexeeva, T. A., Leonov, G. A. Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations, Nonlinear Dynamics, 2016, pp. 1-7 (http://dx.doi.org/10.1007/s11071-016-2678-4)
  15. Leonov, G. A. Formulas for the Lyapunov dimension of attractors of the generalized Lorenz system, Doklady Mathematics, 2013, vol. 87, no. 3, pp. 264-268
  16. Leonov, G. A., Kuznetsov, N. V., Korzhemanova, N. A., Kusakin, D. V. Lyapunov dimension formula for the global attractor of the Lorenz system, Communications in Nonlinear Science and NumericalSimulation, 2016, http://dx.doi.org/10.1016/j.cnsns.2016.04.032
  17. Leonov, G. A., Alexeeva, T. A., Kuznetsov, N. V. Analytic exact upper bound for the Lyapunov dimension of the Shimizu-Morioka system, Entropy, 2015, vol. 17, no. 7, pp. 5101-5116
  18. Lind, D., Schmidt, K. Symbolic and algebraic dynamical systems, in book: Handbook of DynamicalSystems, 2002, vol. 1, pp. 765-812
  19. Millionshchikov, V. M. A formula for the entropy of a smooth dynamical system Differentsial'nyeUravneniya, 1976, vol. 12, pp. 2188-2192
  20. Newhouse, S. E. Entropy and volume, Ergodic Theory and Dynamical Systems, 1988, vol. 8, no. 8, pp. 283-299
  21. Oseledets, V. I. A multiplicative ergodic theorem: Lyapunov characteristic exponents for dynamical systems, Trudy Moskovskogo Matematicheskogo Obshchestva, 1968, vol. 19, pp. 179-210
  22. Pesin, Y. B. Dimension theory in dynamical systems: contemporary views and applications, University of Chicago Press, 2008
  23. Schmidt, K. Multi-dimensional symbolic dynamical systems, in book: Codes, Systems, and GraphicalModels, Springer, 2001, pp. 67-82
  24. Sun, H. Topological entropy of linear systems and its application to optimal control, Master's Thesis, Hong Kong University of Science and Technology, 2008
  25. Udriste, C. Multitime controllability, observability and bang-bang principle, J. Optim. Theory Appl. , 2008, vol. 139, no. 1, pp. 141-157

Full text (pdf)