Two-Dimensional Rotation of Chaotic Attractors: Demonstrative Examples and FPGA Realization

  • W. S. Sayed
  • A. G. RadwanEmail author
  • M. Elnawawy
  • H. Orabi
  • A. Sagahyroon
  • F. Aloul
  • A. S. Elwakil
  • H. A. Fahmy
  • A. El-Sedeek
Short Paper


In this work, we demonstrate the possibility of performing two-dimensional rotation on a chaotic system. This enables the rotation of its attractor in space without changing its chaotic dynamics. In particular, the rotated system preserves the same eigenvalues at all equilibrium points and its largest Lyapunov exponent remains unchanged. Two chaotic systems, one of which is the classical Lorenz system, are used to illustrate and validate the rotation operation using numerical simulations and further experimentally using a digital FPGA platform.


Chaotic oscillators Digital chaos generation FPGA Two-dimensional rotation 


Compliance with Ethical Standards

Conflict of Interest

The authors declare that they have no conflict of interest.


  1. 1.
    K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer, Berlin, 1996)zbMATHGoogle Scholar
  2. 2.
    M.L. Barakat, A.S. Mansingka, A.G. Radwan, K.N. Salama, Generalized hardware post-processing technique for chaos-based pseudorandom number generators. ETRI J. 35(3), 448–458 (2013)CrossRefGoogle Scholar
  3. 3.
    T. Bonny, A.S. Elwakil, FPGA realizations of high speed switching-type chaotic oscillators using compact VHDL codes. Nonlinear Dyn. 93(2), 819–833 (2018)CrossRefGoogle Scholar
  4. 4.
    V.H. Carbajal-Gomez, E. Tlelo-Cuautle, J.M. Muñoz-Pacheco, L.G. de la Fraga, C. Sanchez-Lopez, F.V. Fernandez-Fernandez, Optimization and CMOS design of chaotic oscillators robust to PVT variations: INVITED. Integration (2018).
  5. 5.
    V. Carbajal-Gomez, E. Tlelo-Cuautle, C. Sanchez-Lopez, F. Fernandez-Fernandez, PVT-robust CMOS programmable chaotic oscillator: synchronization of two 7-scroll attractors. Electronics 7(10), 252 (2018a)CrossRefGoogle Scholar
  6. 6.
    A.S. Elwakil, M.P. Kennedy, Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices. IEEE Trans. Circuits Syst.-I 48(3), 289–307 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A.S. Elwakil, S. Özoguz, A system and circuit for generating “multi-butterflies”. Int. J. Bifurc. Chaos 18(03), 841–844 (2008)CrossRefGoogle Scholar
  8. 8.
    A.S. Elwakil, S. Ozoguz, M.P. Kennedy, Creation of a complex butterfly attractor using a novel Lorenz-type system. IEEE Trans. Circuits Syst.-I: Fundam. Theory Appl. 49(4), 527–530 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D.A. Hsieh, Chaos and nonlinear dynamics: application to financial markets. J. Finance 46(5), 1839–1877 (1991)CrossRefGoogle Scholar
  10. 10. (Release date: September) (2018)
  11. 11.
    S.M. Ismail, L.A. Said, A.A. Rezk, A.G. Radwan, A.H. Madian, M.F. Abu-Elyazeed, A.M. Soliman, Generalized fractional logistic map encryption system based on FPGA. Int. J. Electron. Commun. 80, 114–126 (2017)CrossRefGoogle Scholar
  12. 12.
    G. Kaddoum, Wireless chaos-based communication systems: a comprehensive survey. IEEE Access 4, 2621–2648 (2016)CrossRefGoogle Scholar
  13. 13.
    L. Kocarev, S. Lian, Chaos-Based Cryptography: Theory, Algorithms and Applications, vol. 354 (Springer, Berlin, 2011)CrossRefzbMATHGoogle Scholar
  14. 14.
    F. Lau, C.K. Tse, Chaos-Based Digital Communication Systems (Springer, Berlin, 2003)CrossRefzbMATHGoogle Scholar
  15. 15.
    E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    I. Pan, S. Das, Evolving chaos: identifying new attractors of the generalised lorenz family. Appl. Math. Model. 57, 391–405 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    A. Pano-Azucena, E. Tlelo-Cuautle, G. Rodriguez-Gomez, L. de la Fraga, FPGA-based implementation of chaotic oscillators by applying the numerical method based on trigonometric polynomials. AIP Adv. 8(7), 075217 (2018)CrossRefGoogle Scholar
  18. 18.
    A. Radwan, A. Soliman, A. El-Sedeek, MOS realization of the modified lorenz chaotic system. Chaos Solitons Fract. 21(3), 553–561 (2004)CrossRefzbMATHGoogle Scholar
  19. 19.
    A.G. Radwan, S.H. AbdElHaleem, S.K. Abd-El-Hafiz, Symmetric encryption algorithms using chaotic and non-chaotic generators: a review. J. Adv. Res. 7(2), 193–208 (2016)CrossRefGoogle Scholar
  20. 20.
    E. Schöll, Nonlinear Spatio-temporal Dynamics and Chaos in Semiconductors, vol. 10 (Cambridge University Press, Cambridge, 2001)CrossRefGoogle Scholar
  21. 21.
    D. Shah, R. Charasiys, V. Vyawahare, K. Pichhode, M. Patil, FPGA implementation of fractional-order chaotic systems. Int. J. Electron. Commun. 78, 245–257 (2017)CrossRefGoogle Scholar
  22. 22.
    S.H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering (Westview Press, Boulder, 2014)zbMATHGoogle Scholar
  23. 23.
    M.F. Tolba, A.M. AbdelAty, N.S. Soliman, L.A. Said, A.H. Madian, A.T. Azar, A.G. Radwan, FPGA implementation of two fractional order chaotic systems. Int. J. Electron. Commun. 78, 162–172 (2017)CrossRefGoogle Scholar
  24. 24.
    H. Wang, H.F. Liang, Z.H. Miao, A new color image encryption scheme based on chaos synchronization of time-delay Lorenz system. Adv. Manuf. 4(4), 348–354 (2016)CrossRefGoogle Scholar
  25. 25.
    G.C. Wu, D. Baleanu, Z.X. Lin, Image encryption technique based on fractional chaotic time series. J. Vib. Control 22(8), 2092–2099 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    S. Yu, J. Lü, W.K. Tang, G. Chen, A general multiscroll Lorenz system family and its realization via digital signal processors. Chaos: an interdisciplinary. J. Nonlinear Sci. 16(3), 033126 (2006)zbMATHGoogle Scholar
  27. 27.
    M.A. Zidan, A.G. Radwan, K.N. Salama, Controllable V-shape multiscroll butterfly attractor: system and circuit implementation. Int. J. Bifurc. Chaos 22(06), 1250143 (2012)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Engineering Mathematics and Physics Department, Faculty of EngineeringCairo UniversityGizaEgypt
  2. 2.Nanoelectronics Integrated Systems CenterNile UniversityCairoEgypt
  3. 3.Department of Computer Science and EngineeringAmerican University of SharjahSharjahUnited Arab Emirates
  4. 4.Department of Electrical and Computer EngineeringUniversity of SharjahSharjahUnited Arab Emirates
  5. 5.Department of Electrical and Computer EngineeringUniversity of CalgaryAlbertaCanada
  6. 6.Electronics and Communications Engineering Department, Faculty of EngineeringCairo UniversityGizaEgypt

Personalised recommendations