Skip Navigation Links
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland

Email: pawel.olejnik@p.lodz.pl

C. Steve Suh (editor)

Texas A&M University, USA

Email: ssuh@tamu.edu

Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China

Email: tuoxianguo@suse.edu.cn


Strange Non-Chaotic Attractors with Unpredictable Trajectories

Journal of Vibration Testing and System Dynamics 6(3) (2022) 317--327 | DOI:10.5890/JVTSD.2022.09.004

Marat Akhmet$^1$, Mehmet Onur Fen$^{2}$, Astrit Tola$^1$

$^1$ Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey

$^2$ Department of Mathematics, TED University, 06420 Ankara, Turkey

Download Full Text PDF

 

Abstract

Continuous and discrete time systems possessing strange non-chaotic attractors are under investigation. It is demonstrated that unpredictable trajectories exist in the dynamics. A recent numerical technique, the sequential test, is utilized to show the presence of unpredictability.

References

  1. [1]  Grebogi, C., Ott, E., Pelikan, S., and Yorke, J.A. (1984), Strange attractors that are not chaotic, Physica D: Nonlinear Phenomena, 13, 261-268.
  2. [2]  Kapitaniak, T. (1993), Generating strange nonchaotic trajectories, Physical Review E, 47, 1408-1410.
  3. [3]  Prasad, A., Nandi, A., and Ramaswamy, R. (2007), Aperiodic nonchaotic attractors, strange and otherwise, International Journal of Bifurcation and Chaos, 17, 3397-3407.
  4. [4]  Ruelle, D. and Takens, F. (1971), On the nature of turbulence, Communications in Mathematical Physics, 20, 167-192.
  5. [5]  Badard, R. (2008), A lot of strange attractors: Chaotic or not?, Chaos, 18, 023127.
  6. [6]  Brown, R. and Chua, L.O. (1998), Clarifying chaos II: Bernoulli chaos, zero Lyapunov exponents and strange attractors, International Journal of Bifurcation and Chaos, 8, 1-32.
  7. [7]  Li, G., Yue, Y., Xie, J., and Grebogi, C. (2019), Strange nonchaotic attractors in a nonsmooth dynamical system, Communications in Nonlinear Science and Numerical Simulation, 78, 1-10.
  8. [8]  de Moura, A.P.S. (2007), Strange nonchaotic repellers, Physical Review E, 76, 1-4.
  9. [9]  Prasad, A., Mehra, V., and Ramakrishna, R. (1998), Strange nonchaotic attractors in the quasiperiodically forced logistic map, Physical Review E, 57, 1576-1584.
  10. [10]  Glendinning, P., J\"{a}ger, T.H., and Keller, G. (2006), How chaotic are strange non-chaotic attractors?, Nonlinearity, 19, 2005-2022.
  11. [11]  Pikovsky, A. and Feudel, U. (1995), Characterizing strange nonchaotic attractors, Chaos, 5, 253.
  12. [12]  Akhmet, M. and Fen, M.O. (2016), Unpredictable points and chaos, Communications in Nonlinear Science and Numerical Simulation, 40, 1-5.
  13. [13]  Devaney, R. (1987), An Introduction to Chaotic Dynamical Systems, Addison-Wesley: United States of America.
  14. [14]  Li, T.Y. and Yorke, J.A. (1975), Period three implies chaos, The American Mathematical Monthly, 82, 985-992.
  15. [15]  Akhmet, M. and Fen, M.O. (2017), Poincar{e} chaos and unpredictable functions, Communications in Nonlinear Science and Numerical Simulation, 48, 85-94.
  16. [16]  Akhmet, M. and Fen, M.O. (2017), Existence of unpredictable solutions and chaos, Turkish Journal of Mathematics, 41, 254-266.
  17. [17]  Akhmet, M. and Fen, M.O. (2018), Non-autonomous equations with unpredictable solutions, Communications in Nonlinear Science and Numerical Simulation, 59, 657-670.
  18. [18]  Fen, M.O. and Tokmak Fen, F (2021), Unpredictable oscillations of SICNNs with delay, Neurocomputing, 464, 119-129.
  19. [19]  Akhmet, M., Fen, M.O., and Tola, A., Existence of Poincar{e} chaos by a numerical approach, Discontinuity, Nonlinearity, and Complexity, in press.
  20. [20]  Blanchard, F. (2009), Topological chaos: what may this mean?, Journal of Difference Equations and Applications, 15, 23-46.
  21. [21]  Glasner, E. and Weiss, B. (1993), Sensitive dependence on initial conditions, Nonlinearity, 6, 1067-1075.
  22. [22]  Sell, G.R. (1971), Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold Company: London.
  23. [23]  Roman-Flor{e}s, H. (2003), A note on transitivity in set-valued discrete systems, Chaos, Solitons \& Fractals, 17, 99-104.
  24. [24]  Ko\c{c}ak, H. and Palmer, K.J. (2010), Lyapunov exponents and sensitive dependence, Journal of Dynamics and Differential Equations, 22, 381-398.
  25. [25]  Vellekoop, M. and Berglund, R. (1994), On intervals, transitivity = chaos, The American Mathematical Monthly, 101, 353-355.
  26. [26]  Alligood, K.T., Sauer T.D., and Yorke, J.A. (1996), Chaos: An Introduction to Dynamical Systems, Springer: United States of America.
  27. [27]  Gottwald, G.A. and Melbourne, I. (2004), A new test for chaos in deterministic systems, Proceedings of the Royal Society A, 460, 603-611.
  28. [28]  Gottwald, G.A. and Melbourne, I. (2005), Testing for chaos in deterministic systems with noise, Physica D: Nonlinear Phenomena, 212, 100-110.
  29. [29]  Gottwald, G.A. and Melbourne, I. (2009), On the implementation of the 0-1 test for chaos, SIAM Journal on Applied Dynamical Systems, 8, 129-145.
  30. [30]  Gopal, R., Venkatesan, A., and Lakshmanan, M. (2013), Applicability of 0-1 test for strange nonchaotic attractors, Chaos, 23, 023123.
  31. [31]  Akhmet, M. and Topla, A. (2020), Unpredictable strings, Kazakh Mathematical Journal, 20, 16-22.