Skip to main content
SpringerLink
Log in

Discrete one-dimensional piecewise chaotic systems without fixed points

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper presents a new method of generating chaotic maps that do not have fixed points. The method uses an appropriate transformation of functions defined on the unit square in such a way that the newly created projection has no fixed points. In this way, countless new chaotic mappings can be created, both piecewise linear and nonlinear, that belong to the family of systems with hidden attractors. The new family of maps is presented in three examples showing phase diagrams, bifurcation diagrams, and space parameters of the Lyapunov exponent. The discussed examples of mappings were created by appropriate logistic and tent mapping transformations, and their combination. In addition, this paper analyzes the applications of the new families of chaotic mappings in chaotic cryptography. Furthermore, an algorithm for generating pseudorandom bit values (PRBG) is also presented based on the examples of proposed mappings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Data availability

Not applicable.

References

  1. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press, Boca Raton (2018)

    Book  Google Scholar 

  2. Berezowski, M., Dubaj, D.: Chaotic oscillations of coupled chemical reactors. Chaos, Solitons Fractals 78, 22–25 (2015). https://doi.org/10.1016/j.chaos.2015.07.001

    Article  MathSciNet  Google Scholar 

  3. Foley, D.: In: Lines, M. (ed.) Complex and Chaotic Dynamics in Economics, pp. 27–66. Springer, Vienna (2005). https://doi.org/10.1007/3-211-38043-4_2

  4. Skinner, J.E., Molnar, M., Vybiral, T., Mitra, M.: Application of chaos theory to biology and medicine. Integr. Physiol. Behav. Sci. 27(1), 39–53 (1992)

    Article  Google Scholar 

  5. Lawnik, M., Moysis, L., Volos, C.: Chaos-based cryptography: Text encryption using image algorithms. Electronics (2022). https://doi.org/10.3390/electronics11193156

    Article  Google Scholar 

  6. Zhang, B., Liu, L.: Chaos-based image encryption: Review, application, and challenges. Mathematics (2023). https://doi.org/10.3390/math11112585

    Article  Google Scholar 

  7. Wu, R., Gao, S., Wang, X., Liu, S., Li, Q., Erkan, U., Tang, X.: Aea-ncs: An audio encryption algorithm based on a nested chaotic system. Chaos, Solitons Fractals 165, 112770 (2022). https://doi.org/10.1016/j.chaos.2022.112770

    Article  Google Scholar 

  8. Su, Z., Lian, S., Zhang, G., Jiang, J.: In: Kocarev, L., Lian, S. (eds.) Chaos-Based Video Encryption Algorithms, pp. 205–226. Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-20542-2_6

  9. Alawida, M., Teh, J.S., Oyinloye, D.P., Alshoura, W.H., Ahmad, M., Alkhawaldeh, R.S.: A new hash function based on chaotic maps and deterministic finite state automata. IEEE Access 8, 113163–113174 (2020). https://doi.org/10.1109/ACCESS.2020.3002763

    Article  Google Scholar 

  10. Bin Faheem, Z., Ali, A., Khan, M.A., Ul-Haq, M.E., Ahmad, W.: Highly dispersive substitution box (s-box) design using chaos. ETRI J. 42(4), 619–632 (2020). https://doi.org/10.4218/etrij.2019-0138

    Article  Google Scholar 

  11. Ye, G., Jiao, K., Wu, H., Pan, C., Huang, X.: An asymmetric image encryption algorithm based on a fractional-order chaotic system and the rsa public-key cryptosystem. Int. J. Bifurc. Chaos 30(15), 2050233 (2020). https://doi.org/10.1142/S0218127420502338

    Article  MathSciNet  Google Scholar 

  12. Nasr, S., Mekki, H., Bouallegue, K.: A multi-scroll chaotic system for a higher coverage path planning of a mobile robot using flatness controller. Chaos, Solitons Fractals 118, 366–375 (2019). https://doi.org/10.1016/j.chaos.2018.12.002

    Article  MathSciNet  Google Scholar 

  13. Feng, J., Zhang, J., Zhu, X., Lian, W.: A novel chaos optimization algorithm. Multimedia Tools Appl. 76(16), 17405–17436 (2017). https://doi.org/10.1007/s11042-016-3907-z

    Article  Google Scholar 

  14. Wang, R., Du, P., Zhong, W., Han, H., Sun, H.: Analyses and encryption implementation of a new chaotic system based on semitensor product. Complexity (2020). https://doi.org/10.1155/2020/1230804

    Article  Google Scholar 

  15. Liang, B., Hu, C., Tian, Z., Wang, Q., Jian, C.: A 3d chaotic system with multi-transient behavior and its application in image encryption. Phys. A 616, 128624 (2023). https://doi.org/10.1016/j.physa.2023.128624

    Article  Google Scholar 

  16. Guo, Y., Zhang, J., Xie, Q., Hou, J.: Multi-vortex hyperchaotic systems based on memristors and their application to image encryption. Optik 287, 171119 (2023). https://doi.org/10.1016/j.ijleo.2023.171119

    Article  Google Scholar 

  17. Xu, S., Wang, X., Ye, X.: A new fractional-order chaos system of hopfield neural network and its application in image encryption. Chaos, Solitons Fractals 157, 111889 (2022). https://doi.org/10.1016/j.chaos.2022.111889

    Article  MathSciNet  Google Scholar 

  18. Hosny, K.M., Kamal, S.T., Darwish, M.M.: Novel encryption for color images using fractional-order hyperchaotic system. J. Ambient. Intell. Hum. Comput. 13(2), 973–988 (2022). https://doi.org/10.1007/s12652-021-03675-y

    Article  Google Scholar 

  19. Khan, N.A., Qureshi, M.A., Akbar, S., Ara, A.: From chaos to encryption using fractional order Lorenz–Stenflo model with flux-controlled feedback memristor. Phys. Scr. 98(1), 014002 (2022). https://doi.org/10.1088/1402-4896/aca1e8

    Article  Google Scholar 

  20. Lin, L., Zhuang, Y., Xu, Z., Yang, D., Wu, D.: Encryption algorithm based on fractional order chaotic system combined with adaptive predefined time synchronization. Front. Phys. (2023). https://doi.org/10.3389/fphy.2023.1202871

    Article  Google Scholar 

  21. Khairullah, M.K., Alkahtani, A.A., Bin Baharuddin, M.Z., Al-Jubari, A.M.: Designing 1d chaotic maps for fast chaotic image encryption. Electronics (2021). https://doi.org/10.3390/electronics10172116

    Article  Google Scholar 

  22. Dua, M., Makhija, D., Manasa, P.Y.L., Mishra, P.: 3d chaotic map-cosine transformation based approach to video encryption and decryption. Open Comput. Sci. 12(1), 37–56 (2022). https://doi.org/10.1515/comp-2020-0225

    Article  Google Scholar 

  23. Liang, Q., Zhu, C.: A new one-dimensional chaotic map for image encryption scheme based on random dna coding. Opt. Laser Technol. 160, 109033 (2023). https://doi.org/10.1016/j.optlastec.2022.109033

    Article  Google Scholar 

  24. Azar, A.T., Volos, C., Gerodimos, N.A., Tombras, G.S., Pham, V.-T., Radwan, A.G., Vaidyanathan, S., Ouannas, A., Munoz-Pacheco, J.M.: A novel chaotic system without equilibrium: Dynamics, synchronization, and circuit realization. Complexity 2017, 7871467 (2017). https://doi.org/10.1155/2017/7871467

    Article  MathSciNet  Google Scholar 

  25. Wang, Z., Akgul, A., Pham, V.-T., Jafari, S.: Chaos-based application of a novel no-equilibrium chaotic system with coexisting attractors. Nonlinear Dyn. 89(3), 1877–1887 (2017). https://doi.org/10.1007/s11071-017-3558-2

    Article  Google Scholar 

  26. Tamba, V.K., Pham, V.-T., Hoang, D.V., Jafari, S., Alsaadi, F.E., Alsaadi, F.E.: Dynamic system with no equilibrium and its chaos anti-synchronization. Automatika 59(1), 35–42 (2018). https://doi.org/10.1080/00051144.2018.1491934

    Article  Google Scholar 

  27. Lai, Q., Wan, Z., Kamdem Kuate, P.D.: Modelling and circuit realisation of a new no-equilibrium chaotic system with hidden attractor and coexisting attractors. Electron. Lett. 56(20), 1044–1046 (2020). https://doi.org/10.1049/el.2020.1630

    Article  Google Scholar 

  28. Zhang, S., Wang, X., Zeng, Z.: A simple no-equilibrium chaotic system with only one signum function for generating multidirectional variable hidden attractors and its hardware implementation. Chaos Interdiscipl. J. Nonlinear Sci. 30(5), 053129 (2020). https://doi.org/10.1063/5.0008875

    Article  MathSciNet  Google Scholar 

  29. Wang, X., Chen, G.: In: Wang, X., Kuznetsov, N.V., Chen, G. (eds.) Chaotic Systems Without Equilibria, pp. 55–75. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75821-9_4

  30. Wang, C., Ding, Q.: A new two-dimensional map with hidden attractors. Entropy (2018). https://doi.org/10.3390/e20050322

    Article  MathSciNet  Google Scholar 

  31. Almatroud, O.A., Pham, V.-T.: Building fixed point-free maps with memristor. Mathematics (2023). https://doi.org/10.3390/math11061319

    Article  Google Scholar 

  32. García-Grimaldo, C., Campos-Cantón, E.: Comparative analysis of chaotic features of maps without fixed points. In: Huerta Cuéllar, G., Campos Cantón, E., Tlelo-Cuautle, E. (eds.) Complex Syst. Their Appl., pp. 151–176. Springer, Cham (2022)

    Chapter  Google Scholar 

  33. García-Grimaldo, C., Bermudez-Marquez, C.F., Tlelo-Cuautle, E., Campos-Cantón, E.: Fpga implementation of a chaotic map with no fixed point. Electronics (2023). https://doi.org/10.3390/electronics12020444

    Article  Google Scholar 

  34. García-Grimaldo, C., Campos, E.: Chaotic features of a class of discrete maps without fixed points. Int. J. Bifurc. Chaos 31(13), 2150200 (2021). https://doi.org/10.1142/S021812742150200X

    Article  MathSciNet  Google Scholar 

  35. Jafari, S., Pham, V.-T., Golpayegani, S.M.R.H., Moghtadaei, M., Kingni, S.T.: The relationship between chaotic maps and some chaotic systems with hidden attractors. Int. J. Bifurc. Chaos 26(13), 1650211 (2016)

    Article  MathSciNet  Google Scholar 

  36. García-Grimaldo, C., Campos-Cantón, E.: One-dimensional map without fixed points and with amplitude control. In: 15th Chaotic Modeling and Simulation International Conference, pp. 87–97 (2022). Springer

  37. García-Grimaldo, C., Campos-Cantón, E.: Exploring a family of bernoulli-like shift chaotic maps and its amplitude control. Chaos, Solitons Fractals 175, 113951 (2023). https://doi.org/10.1016/j.chaos.2023.113951

    Article  MathSciNet  Google Scholar 

  38. Berezowski, M., Lawnik, M.: Hidden attractors in discrete dynamical systems. Entropy (2021). https://doi.org/10.3390/e23050616

  39. Lawnik, M., Moysis, L., Volos, C.: A family of 1d chaotic maps without equilibria. Symmetry (2023). https://doi.org/10.3390/sym15071311

    Article  Google Scholar 

  40. Baptista, M.S., Grebogi, C., Barreto, E.: Topology of windows in the high-dimensional parameter space of chaotic maps. Int. J. Bifurc. Chaos 13(09), 2681–2688 (2003). https://doi.org/10.1142/S0218127403008181

    Article  MathSciNet  Google Scholar 

  41. de Sousa, F.F.G., Rubinger, R.M., Sartorelli, J.C., Albuquerque, H.A., Baptista, M.S.: Parameter space of experimental chaotic circuits with high-precision control parameters. Chaos Interdiscipl. J. Nonlinear Sci. 26(8), 083107 (2016). https://doi.org/10.1063/1.4960582

    Article  MathSciNet  Google Scholar 

  42. Maranhão, D.M., Baptista, M.S., Sartorelli, J.C., Caldas, I.L.: Experimental observation of a complex periodic window. Phys. Rev. E 77, 037202 (2008). https://doi.org/10.1103/PhysRevE.77.037202

    Article  Google Scholar 

  43. Fowler, A., McGuinness, M.: Homoclinic Bifurcations, pp. 99–142. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-32538-1_4

  44. Rukhin, A., Soto, J., Nechvatal, J., Smid, M., Barker, E.: A statistical test suite for random and pseudorandom number generators for cryptographic applications. Technical report, Booz-Allen and Hamilton Inc Mclean Va (2001)

Download references

Acknowledgements

The authors are thankful to the anonymous reviewers for their constructive feedback.

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics Applications and Methods for Artificial Intelligence, Faculty of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100, Gliwice, Poland

    Marcin Lawnik

  2. Laboratory of Nonlinear Systems - Circuits & Complexity, Physics Department, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Lazaros Moysis & Christos Volos

  3. Department of Mechanical Engineering, University of Western Macedonia, 50100, Kozani, Greece

    Lazaros Moysis

  4. Department of Physics, Institute for Complex Systems and Mathematical Biology, SUPA, University of Aberdeen, AB24 3UX, UK

    Murilo S. Baptista

Authors
  1. Marcin Lawnik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lazaros Moysis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Murilo S. Baptista
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Christos Volos
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

ML and LM contributed to conceptualization, software, writing-original draft preparation; ML contributed to methodology and visualization; ML, LM, MSB and CV contributed to writing-review and editing; and MSB and CV supervised the study.

Corresponding authors

Correspondence to Marcin Lawnik or Lazaros Moysis.

Ethics declarations

Competing interests

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lawnik, M., Moysis, L., Baptista, M.S. et al. Discrete one-dimensional piecewise chaotic systems without fixed points. Nonlinear Dyn 112, 6679–6693 (2024). https://doi.org/10.1007/s11071-024-09349-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-024-09349-6

Keywords

Search

Navigation