Skip to main content
SpringerLink
Log in

FPGA Implementation of Integer/Fractional Chaotic Systems

  • Chapter
  • First Online:
Multimedia Security Using Chaotic Maps: Principles and Methodologies

Part of the book series: Studies in Computational Intelligence ((SCI,volume 884 ))

Abstract

Chaotic systems have remarkable importance in capturing some complex features of the physical process. Recently, fractional calculus becomes a vigorous tool in characterizing the dynamics of complex systems. The fractional-order chaotic systems increase the chaotic behavior in new dimensions and add extra degrees of freedom, which increase system controllability. In this chapter, FPGA implementation of different integer and fractional-order chaotic systems is presented. The investigated integer-order systems include Chua double scroll chaotic system and the modified Chua N-scroll chaotic system. The investigated fractional-order systems include Chua, Yalcin et al., Ozuogos et al., and Tang et al., chaotic systems. These systems are implemented and simulated based on the Grunwald–Letnikov (GL) definition with different window sizes. The parameters effect, along with different GL window sizes is investigated where some interesting chaotic behaviors are obtained. The proposed FPGA implementation utilizes fewer resources and has high throughput. Experimental results are provided on a digital oscilloscope.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Murphy RP (2010) Chaos theory. Ludwig von Mises Institute

    Google Scholar 

  2. Lai Q, Zhao XW, Rajagopal K, Xu G, Akgul A, Guleryuz E (2018) Dynamic analyses, FPGA implementation and engineering applications of multi-butterfly chaotic attractors generated from generalised Sprott C system. Pramana J Phys 90(1):6

    Article  Google Scholar 

  3. Garnett PW (1997) Chaos theory tamed. Joseph Henry Press, Washington, DC

    MATH  Google Scholar 

  4. Ruelle D (2012) From the theory of chaos to nonequilibrium statistical mechanics. Procedia IUTAM 5:27–33

    Article  Google Scholar 

  5. Parker TS, Chua LO (1987) Chaos: a tutorial for engineers. Proc IEEE 75(8):982–1008

    Article  Google Scholar 

  6. Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Elsevier

    Google Scholar 

  7. Baumann G, Stenger F (2011) Fractional calculus and Sinc methods. Fract Calc Appl Anal 14(4):568–622

    Article  MathSciNet  MATH  Google Scholar 

  8. Petráš I (2011) Fractional-order nonlinear systems: modeling, analysis and simulation. Springer Science and Business Media, London

    Book  MATH  Google Scholar 

  9. Chi C-M, Gao F (2013) Simulating fractional derivatives using Matlab. JSW 8(3):572–578

    Article  Google Scholar 

  10. Kimeu JM (2009) Fractional calculus: definitions and applications

    Google Scholar 

  11. Tolba MF et al (2017) FPGA implementation of two fractional order chaotic systems. AEU Int J Electron Commun 78:162–172

    Article  Google Scholar 

  12. El-Maksoud AJA et al (2018) FPGA implementation of fractional-order Chua’s chaotic system. In: 2018 7th international conference on modern circuits and systems technologies (MOCAST), pp 1–4

    Google Scholar 

  13. Jiang CX, Carletta JE, Hartley TT, Veillette RJ (2013) A systematic approach for implementing fractional-order operators and systems. IEEE J Emerg Sel Top Circuits Syst 3(3):301–312

    Article  Google Scholar 

  14. Jiang CX, Carletta JE, Hartley T (2007) Implementation of fractional-order operators on field programmable gate arrays. In: Advances in fractional calculus. Springer Netherlands, Dordrecht, pp 333–346

    Chapter  MATH  Google Scholar 

  15. Tolba MF et al (2017) FPGA realization of Caputo and Grünwald-Letnikov operators. In: 2017 6th international conference on modern circuits and systems technologies, MOCAST 2017

    Google Scholar 

  16. Tolba MF, Said LA, Madian AH, Radwan AG (2018) FPGA implementation of fractional-order integrator and differentiator based on Grünwald–Letnikov’s definition. In: Proceedings of the international conference on microelectronics, ICM, vol 2017

    Google Scholar 

  17. Tolba MF, AboAlNaga BM, Said LA, Madian AH, Radwan AG (2019) Fractional order integrator/differentiator: FPGA implementation and FOPID controller application. AEU-Int J Electron Commun 98:220–229

    Article  Google Scholar 

  18. Tolba MF, Said LA, Madian AH, Radwan AG (2018) FPGA implementation of the fractional order integrator/differentiator: two approaches and applications. IEEE Trans Circuits Syst I Regul Pap 66(4):1484–1495

    Article  MathSciNet  Google Scholar 

  19. Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20(2):130–141

    Article  MathSciNet  MATH  Google Scholar 

  20. Ren G, Duan YS (2017) Extended theory of harmonic maps connects general relativity to chaos and quantum mechanism. Chaos, Solitons Fractals 103:567–570

    Article  MathSciNet  Google Scholar 

  21. Shah DK, Chaurasiya RB, Vyawahare VA, Pichhode K, Patil MD (2017) FPGA implementation of fractional-order chaotic systems. AEU-Int J Electron Commun 78:245–257

    Article  Google Scholar 

  22. Sadoudi S, Tanougast C, Azzaz MS, Dandache A, Bouridane A (2009) Real-time FPGA Implementation of Lu’ s Chaotic Generator. In: Signals, circuits and systems, pp 4–7

    Google Scholar 

  23. Ghys É (2013) The Lorenz attractor, a paradigm for chaos. Prog Math Phys 66:1–54

    Article  MathSciNet  MATH  Google Scholar 

  24. Chua L, Komuro M, Matsumoto T (1986) The double scroll family. IEEE Trans Circuits Syst 33(11):1072–1118

    Article  MATH  Google Scholar 

  25. Hénon M (1976) A two-dimensional mapping with a strange attractor. Commun Math Phys 50(1):69–77

    Article  MathSciNet  MATH  Google Scholar 

  26. Han FHF, Yu XYX, Han SHS (2006) Improved baker map for image encryption. In: 2006 1st international symposium of systems and control in aerospace and astronautics, pp 1273–1276

    Google Scholar 

  27. Radwan AG (2013) On some generalized discrete logistic maps. J Adv Res 4(2):163–171

    Article  Google Scholar 

  28. May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261(5560):459–467

    Article  MATH  Google Scholar 

  29. Ismail SM et al (2017) Biomedical image encryption based on double-humped and fractional logistic maps, pp 6–9

    Google Scholar 

  30. Ismail SM et al (2017) Generalized fractional logistic map encryption system based on FPGA. AEU-Int J Electron Commun 80:114–126

    Article  Google Scholar 

  31. Ismail SM, Said LA, Radwan AG, Madian AH, Abu-Elyazeed MF (2018) Dynamics of fractional and double-humped logistic maps versus the conventional one. In: Proceedings of international conference on microelectronics (ICM), vol 2017, pp 1–4

    Google Scholar 

  32. Lux T (1998) The socio-economic dynamics of speculative markets: interacting agents, chaos, and the fat tails of return distributions. J Econ Behav Organ 33(2):143–165

    Article  Google Scholar 

  33. Habib Z, Khan JS, Ahmad J, Khan MA, Khan FA (2017) Secure speech communication algorithm via DCT and TD-ERCS chaotic map. In: 2017 4th international conference on electrical and electronic engineering (ICEEE), pp 246–250

    Google Scholar 

  34. Xin B, Liu L, Hou G, Ma Y (2017) Chaos synchronization of nonlinear fractional discrete dynamical systems via linear control. Entropy 19(7):351

    Article  Google Scholar 

  35. Aihara K (2002) Chaos engineering and its application to parallel distributed processing with chaotic neural networks. Proc IEEE 90(5):919–930

    Article  Google Scholar 

  36. Zhen W, Xia H, Ning L, Xiao-Na S (2012) Image encryption based on a delayed fractional-order chaotic logistic system. Chin Phys B 21(5):50506

    Article  Google Scholar 

  37. Safi A (2017) Improving the security of internet of things using encryption algorithms. Int J Comput Electr Autom Control Inf Eng 11:5

    Google Scholar 

  38. Chen S, Yu S, Lü J, Chen G, He J (2017) Design and FPGA-based realization of a chaotic secure video communication system. IEEE Trans Circuits Syst Video Technol 28(9):2359–2371

    Article  Google Scholar 

  39. Azzaz MS, Tanougast C, Sadoudi S, Dandache A (2011) New hardware cryptosystem based chaos for the secure real-time of embedded applications. In: 2011 IEEE workshop on signal processing systems (siPS), pp 251–254

    Google Scholar 

  40. Sobhy MI, Shehata A-E (2001) Chaotic algorithms for data encryption. In: 2001 IEEE international conference on acoustics, speech, and signal processing. Proceedings (Cat. No. 01CH37221), vol 2, pp 997–1000

    Google Scholar 

  41. Sheu LJ (2011) A speech encryption using fractional chaotic systems. Nonlinear Dyn 65(1–2):103–108

    Article  MathSciNet  MATH  Google Scholar 

  42. Yousri DA, AbdelAty AM, Said LA, AboBakr A, Radwan AG (2017) Biological inspired optimization algorithms for cole-impedance parameters identification. AEU-Int J Electron Commun 78:79–89

    Article  Google Scholar 

  43. AboBakr A, Said LA, Madian AH, Elwakil AS, Radwan AG (2017) Experimental comparison of integer/fractional-order electrical models of plant. AEU-Int J Electron Commun 80:1–9

    Article  Google Scholar 

  44. Said LA, Radwan AG, Madian AH, Soliman AM (2016) Fractional-order inverting and non-inverting filters based on CFOA. In: 2016 39th international conference on telecommunications and signal processing (TSP), pp 599–602

    Google Scholar 

  45. Said LA, Radwan AG, Madian AH, Soliman AM (2018) Survey on two-port network-based fractional-order oscillators. In: Fractional order systems, Elsevier, pp 305–327

    Chapter  Google Scholar 

  46. Said LA, Radwan AG, Madian AH, Soliman AM (2017) Three fractional-order-capacitors-based oscillators with controllable phase and frequency. J. Circuits Syst Comput 26(10):1750160

    Article  Google Scholar 

  47. Said LA, Radwan AG, Madian AH, Soliman AM (2015) Fractional order oscillator design based on two-port network. Circuits Syst Signal Process 35(9):3086–3112

    Article  MathSciNet  Google Scholar 

  48. Tolba MF, Elsafty AH, Armanyos M, Said LA, Madian AH, Radwan AG (2019) Synchronization and FPGA realization of fractional-order Izhikevich neuron model. Microelectron J 89:56–69

    Article  Google Scholar 

  49. Tolba MF, Sayed WS, Radwan AG, Abd-El-Hafiz SK (2018) Chaos-based hardware speech encryption scheme using modified tent map and bit permutation. In: 2018 7th international conference on modern circuits and systems technologies (MOCAST), pp 1–4

    Google Scholar 

  50. Cai X, Liu F (2007) Numerical simulation of the fractional-order control system. J Appl Math Comput 23(1–2):229–241

    Article  MathSciNet  MATH  Google Scholar 

  51. Mainardi F, Gorenflo R (2000) On Mittag-Leffler-type functions in fractional evolution processes. J Comput Appl Math 118(1–2):283–299

    Article  MathSciNet  MATH  Google Scholar 

  52. Petráš I (2011) An effective numerical method and its utilization to solution of fractional models used in bioengineering applications. Adv Differ Eqn

    Google Scholar 

  53. Osborne GA (1906) Differential and integral calculus with example and applications, p 30

    Google Scholar 

  54. Mansingka AS, Barakat ML, Radwan AG, Salama KN (2014) Hardware stream cipher with controllable chaos generator for colour image encryption. IET Image Process 8(1):33–43

    Article  Google Scholar 

  55. Chung SK (2007) Understanding basic calculus. p 284

    Google Scholar 

  56. Ames WF, Brezinski C (1993) Numerical recipes in Fortran. Art Sci Comput 35(5)

    Google Scholar 

  57. Yalçin ME, Suykens JAK, Vandewalle J, Özoğuz S (2008) Families of scroll grid attractors. Int J Bifurc Chaos 12(1):23–41

    Article  MathSciNet  MATH  Google Scholar 

  58. Zidan MA, Radwan AG, Salama KN (2012) Controllable V-shape multiscroll butterfly attractor: system and circuit implementation. Int J Bifurc Chaos 22(6):1250143

    Article  MATH  Google Scholar 

  59. Radwan AG, Soliman AM, Elwakil AS (2008) The fractional domain 17(1):55–66

    Google Scholar 

  60. Elwakil AS, Kennedy MP (2000) Improved implementation of Chua’s chaotic oscillator using current feedback op amp. IEEE Trans Circuits Syst I Fundam Theory Appl 47(1):76–79

    Article  Google Scholar 

  61. Özoǧuz S, Elwakil AS, Salama KN (2002) N-scroll chaos generator using nonlinear transconductor. Electron Lett 38(14):685–686

    Article  Google Scholar 

  62. Tang WKS, Zhong GQ, Chen G, Man KF (2001) Generation of N-scroll attractors via sine function. IEEE Trans Circuits Syst I Fundam Theory Appl 48(11):1369–1372

    Article  Google Scholar 

  63. Pano-Azucena AD, Tlelo-Cuautle E, Muñoz-Pacheco JM, de la Fraga LG (2019) FPGA-based implementation of different families of fractional-order chaotic oscillators applying Grünwald–Letnikov method. Commun Nonlinear Sci Numer Simul 72:516–527

    Article  MathSciNet  Google Scholar 

  64. Pham V-T, Kingni ST, Volos C, Jafari S, Kapitaniak T (2017) A simple three-dimensional fractional-order chaotic system without equilibrium: dynamics, circuitry implementation, chaos control and synchronization. AEU-Int J Electron Commun 78:220–227

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Electronics and Communication Engineering Department, Cairo University, Giza, Egypt

    Ahmed J. Abd El-Maksoud, Ayman A. Abd El-Kader, Bahy G. Hassan, Nader G. Rihan & Mohamed F. Abu-Elyazeed

  2. NISC Research Center, Nile University, Cairo, Egypt

    Mohamed F. Tolba & Lobna A. Said

  3. Department of Engineering Mathematics and Physics, Cairo University, Giza, Egypt

    Ahmed G. Radwan

  4. School of Engineering and Applied Sciences, Nile University, Giza, Egypt

    Ahmed G. Radwan

Authors
  1. Ahmed J. Abd El-Maksoud
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ayman A. Abd El-Kader
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bahy G. Hassan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nader G. Rihan
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Mohamed F. Tolba
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Lobna A. Said
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Ahmed G. Radwan
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Mohamed F. Abu-Elyazeed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lobna A. Said .

Editor information

Editors and Affiliations

  1. Department of Information Technology, Faculty of Computers and Informatics, Zagazig University, Zagazig, Egypt

    Khalid M. Hosny

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Abd El-Maksoud, A.J. et al. (2020). FPGA Implementation of Integer/Fractional Chaotic Systems. In: Hosny, K. (eds) Multimedia Security Using Chaotic Maps: Principles and Methodologies. Studies in Computational Intelligence, vol 884 . Springer, Cham. https://doi.org/10.1007/978-3-030-38700-6_9

Download citation

Publish with us

Policies and ethics

Search

Navigation