A Chaotic Map with Amplitude Control

  • Chuanfu Wang
  • Qun DingEmail author
Conference paper
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 157)


A general approach based on the control factor for controlling the amplitude of the Logistic map is discussed in this paper. We consider that the approach is illustrated using the Logistic map as a typical example. It is proved that the amplitude of the Logistic map can be controlled completely. Since the approach is derived from the general quadratic map, it is suitable for all quadratic chaotic maps.


Amplitude control Logistic map Quadratic map 


  1. 1.
    Chen, G., Mao, Y., Chui, C.: A symmetric image encryption scheme based on 3D chaotic cat maps. Chaos Solitons Fractals 21, 749–761 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Chen, C.-M., Linlin, X., Tsu-Yang, W., Li, C.-R.: On the security of a chaotic maps-based three-party authenticated key agreement protocol. J. Netw. Intell. 1(2), 61–66 (2016)Google Scholar
  3. 3.
    Chen, C.-M., Wang, K.-H., Wu, T.-Y., Wang, E.K.: On the security of a three-party authenticated key agreement protocol based on chaotic maps. Data Sci. Pattern Recogn. 1(2), 1–10 (2017)Google Scholar
  4. 4.
    Fan, C., Ding, Q.: ARM-embedded implementation of H.264 selective encryption based on chaotic stream cipher. J. Netw. Intell. 3(1), 9–15 (2018)Google Scholar
  5. 5.
    Wu, T.-Y., Fan, X., Wang, K.-H., Pan, J.-S., Chen, C.-M.: Security analysis and improvement on an image encryption algorithm using Chebyshev generator. J. Internet Technol. 20(1), 13–23 (2019)Google Scholar
  6. 6.
    Wu, T.-Y., Fan, X., Wang, K.-H., Pan, J.-S., Chen, C.-M., Wu, J.M.-T.: Security analysis and improvement of an image encryption scheme based on chaotic tent map. J. Inf. Hiding Multimed. Signal Process. 9(4), 1050–1057 (2018)Google Scholar
  7. 7.
    Chen, C.-M., Linlin, X., Wang, K.-H., Liu, S., Wu, T.-Y.: Cryptanalysis and improvements on three-party-authenticated key agreement protocols based on chaotic maps. J. Internet Technol. 19(3), 679–687 (2018)Google Scholar
  8. 8.
    Chen, C.-M., Fang, W., Liu, S., Tsu-Yang, W., Pan, J.-S., Wang, K.-H.: Improvement on a chaotic map-based mutual anonymous authentication protocol. J. Inf. Sci. Eng. 34, 371–390 (2018)MathSciNetGoogle Scholar
  9. 9.
    Wu, T.-Y., Wang, K.-H., Chen, C.-M., Wu, J.M.-T., Pan, J.-S.: A simple image encryption algorithm based on logistic map. Adv. Intell. Syst. Comput. 891, 241–247 (2018)Google Scholar
  10. 10.
    Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Rössler, O.E: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976)zbMATHCrossRefGoogle Scholar
  12. 12.
    Chua, L.O., Lin, G.N.: Canonical realization of Chua’s circuit family. IEEE Trans. Circuits Syst. 37, 885–902 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Chen, G., Ueta, T: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 3, 659–661 (2000)zbMATHGoogle Scholar
  15. 15.
    Qi, G., Chen, G., Du, S., Chen, Z., Yuan, Z: Analysis of a new chaotic system. Phys. A Stat. Mech. Appl. 352, 295–308 (2005)CrossRefGoogle Scholar
  16. 16.
    May, R.M: Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976)zbMATHCrossRefGoogle Scholar
  17. 17.
    Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50, 69–77 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Chen, G., Lai, D.: Feedback control of Lyapunov exponents for discrete-time dynamical systems. Int. J. Bifurc. Chaos 06, 1341–1349 (1996)zbMATHCrossRefGoogle Scholar
  19. 19.
    Lin, Z., Yu, S., Lü, J., Cai, S., Chen, G.: Design and ARM-embedded implementation of a chaotic map-based real-time secure video communication system. IEEE. Trans. Circ. Syst. Video 25, 1203–1216 (2015)CrossRefGoogle Scholar
  20. 20.
    Wang, C.F., Fan, C.L., Ding, Q.: Constructing discrete chaotic systems with positive Lyapunov exponents. Int. J. Bifurcat. Chaos 28, 1850084 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hilbert, D.: Mathematical problems. Bull. Amer. Math. Soc. 8, 437C479 (1902)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Gubar, N.A.: Investigation of a piecewise linear dynamical system with three parameters. J. Appl. Math. Mech. 25, 1011C1023 (1961)Google Scholar
  23. 23.
    Markus, L., Yamabe, H.: Global stability criteria for differential systems. Osaka Math. J. 12, 305C317 (1960)Google Scholar
  24. 24.
    Leonov, G.A.: Algorithms for finding hidden oscillations in nonlinear systems. The Aiz-erman and Kalman conjectures and Chuas circuits. J. Comput. Syst. Sci. Int. 50, 511C543 (2011)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Electronic Engineering CollegeHeilongjiang UniversityHarbinChina

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