Table of Contents Author Guidelines Submit a Manuscript
Complexity
Volume 2017 (2017), Article ID 6020213, 8 pages
https://doi.org/10.1155/2017/6020213
Research Article

Complexity in Linear Systems: A Chaotic Linear Operator on the Space of Odd -Periodic Functions

1Department of Fluid Mechanics, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, Budapest, Hungary
2Institute of Mathematics, Faculty of Natural Sciences, Budapest University of Technology and Economics, Budapest, Hungary

Correspondence should be addressed to Tamás Kalmár-Nagy

Received 28 July 2016; Revised 6 December 2016; Accepted 28 December 2016; Published 22 February 2017

Academic Editor: Sylvain Sené

Copyright © 2017 Tamás Kalmár-Nagy and Márton Kiss. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. Rolewicz, “On orbits of elements,” Studia Mathematica, vol. 32, no. 1, pp. 17–22, 1969. View at Google Scholar · View at MathSciNet
  2. J. H. Shapiro, Notes on the Dynamics of Linear Operators, Lecture Notes, 2001, http://users.math.msu.edu/users/mshapiro/.
  3. N. S. Feldman, “Linear chaos,” 2001, http://home.wlu.edu/~feldmann/Papers/LinearChaos.html.
  4. F. Bayart and É. Matheron, Dynamics of Linear Operators, vol. 179, Cambridge University Press, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  5. K.-G. Grosse-Erdmann and A. P. Manguillot, Linear Chaos, Springer Science & Business Media, 2011.
  6. D. A. Herrero, “Hypercyclic operators and chaos,” Journal of Operator Theory, vol. 28, no. 1, pp. 93–103, 1992. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. K. C. Chan, “The density of hypercyclic operators on a Hilbert space,” Journal of Operator Theory, vol. 47, no. 1, pp. 131–143, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. H. N. Salas, “Hypercyclic weighted shifts,” Transactions of the American Mathematical Society, vol. 347, no. 3, pp. 993–1004, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. P. Bourdon and J. H. Shapiro, Cyclic Phenomena for Composition Operators, vol. 596, American Mathematical Society, 1997.
  10. R. M. Gethner and J. H. Shapiro, “Universal vectors for operators on spaces of holomorphic functions,” Proceedings of the American Mathematical Society, vol. 100, no. 2, pp. 281–288, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. X.-C. Fu and J. Duan, “Infinite-dimensional linear dynamical systems with chaoticity,” Journal of Nonlinear Science, vol. 9, no. 2, pp. 197–211, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. F. Martínez-Giménez and A. Peris, “Chaos for backward shift operators,” International Journal of Bifurcation and Chaos, vol. 12, no. 8, pp. 1703–1715, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. V. Protopopescu, “Linear vs nonlinear and infinite vs finite: an interpretation of chaos,” Tech. Rep. Oak Ridge, Tenn, USA, Oak Ridge National Lab, 1990. View at Google Scholar
  14. V. Protopopescu and Y. Y. Azmy, “Topological chaos for a class of linear models,” Mathematical Models and Methods in Applied Sciences, vol. 2, no. 1, pp. 79–90, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  15. B. Hou, G. Tian, and S. Zhu, “Approximation of chaotic operators,” Journal of Operator Theory, vol. 67, no. 2, pp. 469–493, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. F. Martínez-Giménez, P. Oprocha, and A. Peris, “Distributional chaos for operators with full scrambled sets,” Mathematische Zeitschrift, vol. 274, no. 1-2, pp. 603–612, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. J. Bernardes, A. Bonilla, V. Müller, and A. Peris, “Li-yorke chaos in linear dynamics,” Ergodic Theory and Dynamical Systems, vol. 35, no. 6, pp. 1723–1745, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. I. N. Bronshtein and K. A. Semendyayev, Handbook of Mathematics, Springer-Verlag, Berlin, Germany, 2013. View at MathSciNet
  19. K. T. Davies, M. L. Glasser, V. Protopopescu, and F. Tabakin, “The mathematics of principal value integrals and applications to nuclear physics, transport theory, and condensed matter physics,” Mathematical Models and Methods in Applied Sciences, vol. 6, no. 6, pp. 833–885, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. S. M. Cohen, K. T. R. Davies, R. W. Davies, and M. H. Lee, “Principal-value integrals—revisited,” Canadian Journal of Physics, vol. 83, no. 5, pp. 565–575, 2005. View at Publisher · View at Google Scholar · View at Scopus
  21. E. Belbruno, Fly Me to the Moon: An Insider's Guide to the New Science of Space Travel, Princeton University Press, Princeton, NJ, USA, 2007. View at MathSciNet
  22. C.-Y. Lee, C.-L. Chang, Y.-N. Wang, and L.-M. Fu, “Microfluidic mixing: a review,” International Journal of Molecular Sciences, vol. 12, no. 5, pp. 3263–3287, 2011. View at Publisher · View at Google Scholar · View at Scopus
  23. F. G. Gascon and D. Peralta-Salas, “Escape to infinity in a Newtonian potential,” Journal of Physics A: Mathematical and General, vol. 33, no. 30, pp. 5361–5368, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. V. S. Vladimirov, Methods of the Theory of Generalized Functions, vol. 6 of Analytical Methods and Special Functions, Taylor & Francis, London, UK, 2002. View at MathSciNet
  25. R. P. Kanwal, Generalized Functions: Theory and Applications, Springer, Boston, Mass, USA, 3rd edition, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  26. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, San Diego, Calif, USA, 2014.