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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
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)
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)
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)
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.2012.11.017
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.87.022910
Zelinka, I., Celikovsky, S., Richter, H., Chen, G.: Evolutionary Algorithms and Chaotic Systems, p. 550. Springer, Germany (2010)
Lozi, R.: Emergence Of Randomness From Chaos. International Journal of Bifurcation and Chaos 22(2), 1250021 (2012), doi:10.1142/S0218127412500216
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/s11071-012-0558-0
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)
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/S0217979212502086
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)
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 0898-1221
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: 1860-949X
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)
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)
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.2006.873410
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.2009.511
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.
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)
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.2012.07.017
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this paper
Cite this paper
Zelinka, I., Chadli, M., Davendra, D., Senkerik, R., Pluhacek, M., Lampinen, J. (2013). Hidden Periodicity – Chaos Dependance on Numerical Precision. In: Zelinka, I., Chen, G., Rössler, O., Snasel, V., Abraham, A. (eds) Nostradamus 2013: Prediction, Modeling and Analysis of Complex Systems. Advances in Intelligent Systems and Computing, vol 210. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00542-3_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-00542-3_7
Publisher Name: Springer, Heidelberg
Print ISBN: 978-3-319-00541-6
Online ISBN: 978-3-319-00542-3
eBook Packages: EngineeringEngineering (R0)