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Hidden Periodicity – Chaos Dependance on Numerical Precision

  • Ivan ZelinkaEmail author
  • Mohammed Chadli
  • Donald Davendra
  • Roman Senkerik
  • Michal Pluhacek
  • Jouni Lampinen
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 210)

Abstract

Deterministic chaos has been observed in many systems and seems to be random-like for external observer. Chaos, especially of discrete systems, has been used on numerous occasions in place of random number generators in so called evolutionary algorithms. When compared to random generators, chaotic systems generate values via so called map function that is deterministic and thus, the next value can be calculated, i.e. between elements of random series is no deterministic relation, while in the case of chaotic system it is. Despite this fact, the very often use of chaotic generators improves the performance of evolutionary algorithms. In this paper, we discuss the behavior of two selected chaotic system (logistic map and Lozi system) with dependance on numerical precision and show that numerical precision causes the appearance of many periodic orbits and explain reason why it is happens.

Keywords

Periodic Orbit Evolutionary Algorithm Chaotic System Deterministic Chaos Numerical Precision 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Persohn, K.J., Povinelli, R.J.: Analyzing logistic map pseudorandom number generators for periodicity induced by finite precision floating-point representation. Chaos, Solitons and Fractals 45, 238–245 (2012)CrossRefGoogle Scholar
  2. 2.
    Drutarovsky, M., Galajda, P.: A robust chaos-based true random number generator embedded in reconfigurable switched-capacitor hardware. In: 17th International Conference Radioelektronika, Brno, Czech Republic, April 24-25, vol. 1, 2, pp. 29–34 (2007)Google Scholar
  3. 3.
    Bucolo, M., Caponetto, R., Fortuna, L., Frasca, M., Rizzo, A.: Does chaos work better than noise? IEEE Circuits and Systems Magazine 2(3), 4–19 (2002)CrossRefGoogle Scholar
  4. 4.
    Caponetto, R., Fortuna, L., Fazzino, S., Xibilia, M.: Chaotic sequences to improve the performance of evolutionary algorithms. IEEE Trans. Evol. Comput. 7(3), 289–304 (2003)CrossRefGoogle Scholar
  5. 5.
    Hu, H., Liu, L., Ding, N.: Pseudorandom sequence generator based on the Chen chaotic system. Computer Physics Communications 184(3), 765–768 (2013), doi:10.1016/j.cpc.2012.11.017CrossRefGoogle Scholar
  6. 6.
    Pluchino, A., Rapisarda, A., Tsallis, C.: Noise, synchrony, and correlations at the edge of chaos. Physical Review E 87(2) (2013), doi:10.1103/PhysRevE.87.022910Google Scholar
  7. 7.
    Zelinka, I., Celikovsky, S., Richter, H., Chen, G.: Evolutionary Algorithms and Chaotic Systems, p. 550. Springer, Germany (2010)zbMATHCrossRefGoogle Scholar
  8. 8.
    Lozi, R.: Emergence Of Randomness From Chaos. International Journal of Bifurcation and Chaos 22(2), 1250021 (2012), doi:10.1142/S0218127412500216MathSciNetCrossRefGoogle Scholar
  9. 9.
    Wang, X.-Y., Qin, X.: A new pseudo-random number generator based on CML and chaotic iteration. International Journal of Nonlinear Dynamics and Chaos in Engineering Systems 70(2), 1589–1592 (2012), doi:10.1007/s11071-012-0558-0MathSciNetCrossRefGoogle Scholar
  10. 10.
    Pareek, N.K., Patidar, V., Sud, K.K.: A Random Bit Generator Using Chaotic Maps. International Journal of Network Security 10(1), 32–38 (2010)Google Scholar
  11. 11.
    Wang, X.-Y., Yang, L.: Design Of Pseudo-Random Bit Generator Based On Chaotic Maps. International Journal of Modern Physics B 26(32), 1250208 (9 pages) (2012), doi:10.1142/S0217979212502086Google Scholar
  12. 12.
    Davendra, D., Zelinka, I., Senkerik, R.: Chaos driven evolutionary algorithms for the task of PID control. Computers and Mathematics with Applications 60(4), 1088–1104 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Pluhacek, M., Senkerik, R., Davendra, D., Kominkova Oplatkova, Z.: On the Behaviour and Performance of Chaos Driven PSO Algorithm with Inertia Weight. Computers and Mathematics with Applications (in print), ISSN 0898-1221Google Scholar
  14. 14.
    Pluhacek, M., Budikova, V., Senkerik, R., Kominkova Oplatkova, Z., Zelinka, I.: On the Performance of Enhanced PSO Algorithm with Lozi Chaotic Map. In: Application of Modern Methods of Prediction, Modeling and Analysis of Nonlinear Systems. SCI, vol. 1, p. 18. Springer, Heidelberg (November 2012) (accepted for publication) ISSN: 1860-949XGoogle Scholar
  15. 15.
    Senkerik, R., Davendra, D., Zelinka, I., Oplatkova, Z., Pluhacek, M.: Optimization of the batch reactor by means of chaos driven differential evolution. In: Snasel, V., Abraham, A., Corchado, E.S. (eds.) SOCO Models in Industrial & Environmental Appl. AISC, vol. 188, pp. 93–102. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  16. 16.
    Yang, M., Guan, J., Cai, Z., Wang, L.: Self-adapting differential evolution algorithm with chaos random for global numerical optimization. In: Cai, Z., Hu, C., Kang, Z., Liu, Y. (eds.) ISICA 2010. LNCS, vol. 6382, pp. 112–122. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Coelho, L., Mariani, V.: Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with valve-point effect. IEEE Transactions on Power Systems 21(2), 989–996 (2006), doi:10.1109/TPWRS.2006.873410CrossRefGoogle Scholar
  18. 18.
    Hu, G.-W.: Chaos-differential evolution for multiple sequence alignment. In: 3rd International Symposium on Intelligent Information Technology Application, Nanchang, Peoples R China, vol. 2, pp. 556–558., doi:10.1109/IITA.2009.511Google Scholar
  19. 19.
    Zhao, Q., Ren, J., Zhang, Z., Duan, F.: Immune co-evolution algorithm based on chaotic optimization. In: Workshop on Intelligent Information Technology Application (IITA 2007), Zhang Jiajie, Peoples R China, pp. 149–152.Google Scholar
  20. 20.
    Liua, B., Wanga, L., Jina, Y.-H., Tangb, F., Huanga, D.-X.: Improved particle swarm optimization combined with chaos. Chaos, Solitons & Fractals 25(5), 1261–1271 (2005)CrossRefGoogle Scholar
  21. 21.
    Gandomi, A., Yun, G., Yang, X., Talatahari, S.: Chaos-enhanced accelerated particle swarm optimization. Communications In Nonlinear Science and Numerical Simulation 18(2), 327–340 (2013), doi:10.1016/j.cnsns.2012.07.017MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Ivan Zelinka
    • 1
    Email author
  • Mohammed Chadli
    • 3
  • Donald Davendra
    • 1
  • Roman Senkerik
    • 2
  • Michal Pluhacek
    • 2
  • Jouni Lampinen
    • 4
  1. 1.VSB-Technical University of OstravaOstrava-PorubaCzech Republic
  2. 2.Faculty of Applied InformaticsTomas Bata University in ZlinZlinCzech Republic
  3. 3.Laboratory of Modeling, Information & Systems (MIS)University of Picardie Jules Verne (UPJV)Amiens Cedex 1France
  4. 4.Department of Computer ScienceUniversity of VaasaVaasaFinland

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