Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A Novel Simple 4-D Hyperchaotic System with a Saddle-Point Index-2 Equilibrium Point and Multistability: Design and FPGA-Based Applications

  • 44 Accesses

Abstract

This paper presents a novel simple 4-D dissipative autonomous hyperchaotic system with a saddle-point index-2 equilibrium point. The dynamics of the new system consists of nine terms with two nonlinear terms. Thus, the proposed system is one of the simplest 4-D hyperchaotic systems. The new system also shows multistability behavior for some range of its parameters. Bifurcation diagrams, Lyapunov spectrum, Kaplan–Yorke dimension, and phase plots are used to analyze the complex dynamical behavior of the system. The discrete-time equivalent of the presented hyperchaotic system is made by using the Euler algorithm for improving the embedded chaos-based engineering applications on FPGA. Then, the design of a novel TRNG having chaotic entropy seed has been performed utilizing FPGA-based 4-D hyperchaotic system structure. The Euler-based model in the creation phase on FPGA has been described using VHDL with 32-bit IEEE 754-1985 floating-point format. The designed hyperchaotic system and TRNG unit have been synthesized using Xilinx ISE Design Tools, and chip statistics obtained from place-and-route process have been presented by testing in the Virtex-6 FPGA chip of Xilinx family. The numbers generated in high data rates of 185.447 Mbit/s have been obtained with 370.894 MHz clock frequency for the implementation of the novel TRNG unit. In the last stage, randomness tests of the generated random numbers have been carried out and the generated random numbers have passed successfully all randomness tests.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

References

  1. 1.

    A. Akhshani, A. Akhavan, A. Mobaraki, S.C. Lim, Z. Hassan, Pseudo random number generator based on quantum chaotic map. Commun. Nonlinear Sci. Numer. Simul. 19(1), 101–111 (2014)

  2. 2.

    A. Al-Khedhairi, A. Elsonbaty, A.H. Abdel Kader, A.A. Elsadany, Dynamic analysis and circuit implementation of a new 4D Lorenz-type hyperchaotic system. Math. Probl. Eng. 2019, 1–17 (2019)

  3. 3.

    M. Alcin, I. Koyuncu, M. Tuna, M. Varan, I. Pehlivan, A novel high speed artificial neural network-based chaotic true random number generator on field programmable gate array. Int. J. Circuit Theory Appl. 47(3), 365–378 (2019)

  4. 4.

    A. Anzo-Hernández, E. Campos-Cantón, M. Nicol, Itinerary synchronization between PWL systems coupled with unidirectional links. Commun. Nonlinear Sci. Numer. Simul. 70, 102–124 (2019)

  5. 5.

    P. Arena, S. Baglio, L. Fortuna, G. Manganaro, Hyperchaos from cellular neural networks. Electron. Lett. 31(4), 250–251 (1995)

  6. 6.

    E. Avaroğlu, Pseudorandom number generator based on Arnold cat map and statistical analysis. Turk. J. Electr. Eng. Comput. Scı. 25(1), 633–643 (2017)

  7. 7.

    E. Avaroğlu, İ. Koyuncu, A.B. Özer, M. Türk, Hybrid pseudo-random number generator for cryptographic systems. Nonlinear Dyn. 82(1–2), 239–248 (2015)

  8. 8.

    E. Avaroğlu, A.B. Özer, M. Türk, The study of the reasons for not giving the results of NIST tests of random walk, random walk variable and Lempel Ziv. Turk. J. Sci. Technol. 10(1), 1–8 (2015)

  9. 9.

    E. Avaroğlu, T. Tuncer, A.B. Özer, M. Türk, A new method for hybrid pseudo random number generator. J. Microelectron. Electron. Compon. Mater. 44(4), 303–311 (2014)

  10. 10.

    V.V. Bonde, A.D. Kale, Design and implementation of a random number generator on FPGA. Int. J. Sci. Res. 4(5), 203–208 (2015)

  11. 11.

    T. Bonny, R. Al Debsi, S. Majzoub, A.S. Elwakil, Hardware optimized FPGA implementations of high-speed true random bit generators based on switching-type chaotic oscillators. Circuits Syst. Signal Process. 38(3), 1342–1359 (2019)

  12. 12.

    Ü. Çavuşoğlu, A. Akgül, S. Kaçar, İ. Pehli̇van, A. Zengi̇n, A novel chaos-based encryption algorithm over TCP data packet for secure communication. Secur. Commun. Netw. 9(11), 1285–1296 (2016)

  13. 13.

    G. Chen, T. Ueta, Yet another chaotic attractor. Int. J. Bifurc. Chaos 09(07), 1465–1466 (1999)

  14. 14.

    L. Chen, S. Tang, Q. Li, S. Zhong, A new 4D hyperchaotic system with high complexity. Math. Comput. Simul. 146, 44–56 (2018)

  15. 15.

    Y. Chen, Q. Yang, A new Lorenz-type hyperchaotic system with a curve of equilibria. Math. Comput. Simul. 112, 40–55 (2015)

  16. 16.

    Z. Chen, Y. Yang, G. Qi, Z. Yuan, A novel hyperchaos system only with one equilibrium. Phys. Lett. A 360(6), 696–701 (2007)

  17. 17.

    I. Cicek, A.E. Pusane, G. Dundar, A new dual entropy core true random number generator. Analog Integr. Circuits Signal Process. 81(1), 61–70 (2014)

  18. 18.

    A.M.A. El-Sayed, H.M. Nour, A. Elsaid, A.E. Matouk, A. Elsonbaty, Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional order hyperchaotic system. Appl. Math. Model. 40(5–6), 3516–3534 (2016)

  19. 19.

    R.A. Elmanfaloty, E. Abou-Bakr, Random property enhancement of a 1D chaotic PRNG with finite precision implementation. Chaos, Solitons Fractals 118, 134–144 (2019)

  20. 20.

    S. Ergün, U. Güler, K. Asada, IC truly random number generators based on regular and chaotic sampling of chaotic waveforms. Nonlinear Theory Appl. 2(2), 246–261 (2011)

  21. 21.

    S. Ergün, S. Özoguz, Truly random number generators based on a non-autonomous chaotic oscillator. AEU Int. J. Electron. Commun. 61(4), 235–242 (2007)

  22. 22.

    S. Ergün, S. Özog̃uz, Truly random number generators based on non-autonomous continuous-time chaos. Int. J. Circuit Theory Appl. 38(1), 1–24 (2010)

  23. 23.

    E. Fatemi-Behbahani, K. Ansari-Asl, E. Farshidi, A new approach to analysis and design of chaos-based random number generators using algorithmic converter. Circuits Syst. Signal Process. 35(11), 3830–3846 (2016)

  24. 24.

    A.M. Garipcan, E. Erdem, Implementation and performance analysis of true random number generator on FPGA environment by using non-periodic chaotic signals obtained from chaotic maps. Arab. J. Sci. Eng. 44(11), 9427–9441 (2019)

  25. 25.

    A.M. Garipcan, E. Erdem, Implementation of a Digital TRNG Using Jitter Based Multiple Entropy Source on FPGA. Inf. MIDEM J. Microelectron. Electron Compon. Mater. 49(2), 79–90 (2019)

  26. 26.

    T. Gotthans, J. Petržela, New class of chaotic systems with circular equilibrium. Nonlinear Dyn. 81(3), 1143–1149 (2015)

  27. 27.

    G.A. Gottwald, I. Melbourne, On the implementation of the 0–1 test for chaos. SIAM J. Appl. Dyn. Syst. 8(1), 129–145 (2009)

  28. 28.

    G.A. Gottwald, I. Melbourne, A new test for chaos in deterministic systems. Proc. R. Soc. A Math. Phys. Eng. Sci. 460(2042), 603–611 (2004)

  29. 29.

    X. Huang, L. Liu, X. Li, M. Yu, Z. Wu, A new two-dimensional mutual coupled logistic map and its application for pseudorandom number generator. Math. Probl. Eng. 2019(1), 1–10 (2019)

  30. 30.

    S. Jafari, J.C. Sprott, Simple chaotic flows with a line equilibrium. Chaos, Solitons Fractals 57, 79–84 (2013)

  31. 31.

    S. Jafari, J.C. Sprott, F. Nazarimehr, Recent new examples of hidden attractors. Eur. Phys. J. Spec. Top. 224(8), 1469–1476 (2015)

  32. 32.

    H. Jahanshahi, A. Yousefpour, J.M. Munoz-Pacheco, I. Moroz, Z. Wei, O. Castillo, A new multi-stable fractional-order four-dimensional system with self-excited and hidden chaotic attractors: dynamic analysis and adaptive synchronization using a novel fuzzy adaptive sliding mode control method. Appl. Soft Comput. 87, 105943 (2020)

  33. 33.

    T. Kaya, A true random number generator based on a Chua and RO-PUF: design, implementation and statistical analysis. Analog Integr. Circuits Signal Process. 102, 415–426 (2020)

  34. 34.

    S.T. Kingni, V.T. Pham, S. Jafari, G.R. Kol, P. Woafo, Three-dimensional chaotic autonomous system with a circular equilibrium: analysis, circuit implementation and its fractional-order form. Circuits Syst. Signal Process. 35(6), 1933–1948 (2016)

  35. 35.

    I. Koyuncu, M. Alcin, M. Tuna, I. Pehlivan, M. Varan, S. Vaidyanathan, Real-time high-speed 5-D hyperchaotic Lorenz system on FPGA. Int. J. Comput. Appl. Technol. 61(3), 152–165 (2019)

  36. 36.

    İ. Koyuncu, M. Tuna, İ. Pehlivan, C.B. Fidan, M. Alçın, Design, FPGA implementation and statistical analysis of chaos-ring based dual entropy core true random number generator. Analog Integr. Circuits Signal Process. 102, 445–456 (2020)

  37. 37.

    İ. Koyuncu, A. Turan Özcerit, The design and realization of a new high speed FPGA-based chaotic true random number generator. Comput. Electr. Eng. 58, 203–214 (2017)

  38. 38.

    G.A. Leonov, N.V. Kuznetsov, T.N. Mokaev, Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur. Phys. J. Spec. Top. 224(8), 1421–1458 (2015)

  39. 39.

    C. Li, J.C. Sprott, W. Thio, Bistability in a hyperchaotic system with a line equilibrium. J. Exp. Theor. Phys. 118(3), 494–500 (2014)

  40. 40.

    Q. Li, S. Hu, S. Tang, G. Zeng, Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation. Int. J. Circuit Theory Appl. 42(11), 1172–1188 (2014)

  41. 41.

    X. Li, X. Fan, J. Yin, Y. Zhang, X. Lv, Adaptive control of a four-dimensional hyperchaotic system. Asian Res. J. Math. 13(1), 1–17 (2019)

  42. 42.

    J.A. López-Leyva, A. Arvizu-Mondragón, Generador dual simultáneo de números verdaderamente aleatorios. DYNA 83(195), 93–98 (2016)

  43. 43.

    D. López-Mancilla, G. López-Cahuich, C. Posadas-Castillo, C.E. Castañeda, J.H. García-López et al., Synchronization of complex networks of identical and nonidentical chaotic systems via model-matching control. PLoS ONE 14(5), e0216349 (2019)

  44. 44.

    E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)

  45. 45.

    J. Ma, Z. Chen, Z. Wang, Q. Zhang, A four-wing hyper-chaotic attractor generated from a 4-D memristive system with a line equilibrium. Nonlinear Dyn. 81(3), 1275–1288 (2015)

  46. 46.

    H. Martin, P. Martin-Holgado, P. Peris-Lopez, Y. Morilla, L. Entrena, On the entropy of oscillator-based true random number generators under ionizing radiation. Entropy 20(513), 1–11 (2018)

  47. 47.

    A.E. Matouk, Dynamics and control in a novel hyperchaotic system. Int. J. Dyn. Control 7(1), 241–255 (2019)

  48. 48.

    H. Moqadasi, M.B. Ghaznavi-Ghoushchi, A new Chua’s circuit with monolithic Chua’s diode and its use for efficient true random number generation in CMOS 180 nm. Analog Integr. Circuits Signal Process. 82(3), 719–731 (2015)

  49. 49.

    M. Murillo-Escobar, C. Cruz-Hernández, L. Cardoza-Avendaño, R. Méndez-Ramírez, L. Cardoza-Avendaño, A novel pseudorandom number generator based on pseudorandomly enhanced logistic map. Nonlinear Dyn. 87(1), 407–425 (2017)

  50. 50.

    F. Nazarimehr, K. Rajagopal, J. Kengne, S. Jafari, V.T. Pham, A new four-dimensional system containing chaotic or hyper-chaotic attractors with no equilibrium, a line of equilibria and unstable equilibria. Chaos, Solitons Fractals 111, 108–118 (2018)

  51. 51.

    H. Nejati, A. Beirami, W.H. Ali, Discrete-time chaotic-map truly random number generators: design, implementation, and variability analysis of the zigzag map. Analog Integr. Circuits Signal Process. 73(1), 363–374 (2012)

  52. 52.

    L. Palacios-Luengas, J. Pichardo-Méndez, J. Díaz-Méndez, F. Rodríguez-Santos, R. Vázquez-Medina, PRNG based on skew tent map. Arab. J. Sci. Eng. 44, 3817–3830 (2019)

  53. 53.

    A.D. Pano-Azucena, E. Tlelo-Cuautle, G. Rodriguez-Gomez, L.G. De La Fraga, FPGA-based implementation of chaotic oscillators by applying the numerical method based on trigonometric polynomials. AIP Adv. 8(7), 075217 (2018)

  54. 54.

    V.T. Pham, S. Jafari, X. Wang, J. Ma, A chaotic system with different shapes of equilibria. Int. J. Bifurc. Chaos 26(04), 1650069 (2016)

  55. 55.

    K. Rajagopal, S. Jafari, A. Karthikeyan, A. Srinivasan, B. Ayele, B. Karthikeyan Rajagopal, Hyperchaotic memcapacitor oscillator with infinite equilibria and coexisting attractors. Circuits Syst. Signal Process. 37, 3702–3724 (2018)

  56. 56.

    K. Rajagopal, M. Tuna, A. Karthikeyan, İ. Koyuncu, P. Duraisamy, A. Akgul, Dynamical analysis, sliding mode synchronization of a fractional-order memristor Hopfield neural network with parameter uncertainties and its non-fractional-order FPGA implementation. Eur. Phys. J. Spec. Top. 228(10), 2065–2080 (2019)

  57. 57.

    A.A. Rezk, A.H. Madian, A.G. Radwan, A.M. Soliman, Reconfigurable chaotic pseudo random number generator based on FPGA. AEU Int. J. Electron. Commun. 98, 174–180 (2019)

  58. 58.

    W.S. Sayed, A.G. Radwan, M. Elnawawy, H. Orabi, A. Sagahyroon et al., Two-dimensional rotation of chaotic attractors: demonstrative examples and FPGA realization. Circuits Syst. Signal Process. 38(10), 1–14 (2019)

  59. 59.

    J.P. Singh, J. Koley, A. Akgul, B. Gurevin, B.K. Roy, A new chaotic oscillator containing generalised memristor, single op-amp and RLC with chaos suppression and an application for the random number generation. Eur. Phys. J. Spec. Top. 228(10), 2233–2245 (2019)

  60. 60.

    J.P. Singh, B.K. Roy, Crisis and inverse crisis route to chaos in a new 3-D chaotic system with saddle, saddle foci and stable node foci nature of equilibria. Optik (Stuttg) 127(24), 11982–12002 (2016)

  61. 61.

    J.P. Singh, B.K. Roy, The nature of Lyapunov exponents is (+, +, −, −). Is it a hyperchaotic system? Chaos, Solitons Fractals 92, 73–85 (2016)

  62. 62.

    J.P. Singh, B.K. Roy, Analysis of an one equilibrium novel hyperchaotic system and its circuit validation. Int. J. Control Theory Appl. 8(3), 1015–1023 (2015)

  63. 63.

    J.P. Singh, B.K. Roy, A novel asymmetric hyperchaotic system and its circuit validation. Int. J. Control Theory Appl. 8(3), 1005–1013 (2015)

  64. 64.

    P.P. Singh, J.P. Singh, B.K. Roy, Synchronization and anti-synchronization of Lu and Bhalekar-Gejji chaotic systems using nonlinear active control. Chaos, Solitons Fractals 69, 31–39 (2014)

  65. 65.

    J.C. Sprott, Strange attractors with various equilibrium types. Eur. Phys. J. Spec. Top. 224(8), 1409–1419 (2015)

  66. 66.

    E. Tlelo-Cuautle, J.J. Rangel-Magdaleno, A.D. Pano-Azucena, P.J. Obeso-Rodelo, J.C. Nunez-Perez, FPGA realization of multi-scroll chaotic oscillators. Commun. Nonlinear Sci. Numer. Simul. 27(1–3), 66–80 (2015)

  67. 67.

    M. Tuna, M. Alçın, İ. Koyuncu, C.B. Fidan, İ. Pehlivan, High speed FPGA-based chaotic oscillator design. Microprocess. Microsyst. 66, 72–80 (2019)

  68. 68.

    M. Tuna, C.B. Fidan, A Study on the importance of chaotic oscillators based on FPGA for true random number generating (TRNG) and chaotic systems. J. Fac. Eng. Archit. Gazi Univ. 33(2), 469–486 (2018)

  69. 69.

    M. Tuna, A. Karthikeyan, K. Rajagopal, M. Alçın, İ. Koyuncu, Hyperjerk multiscroll oscillators with megastability: analysis, FPGA implementation and a novel ANN-ring-based true random number generator. AEU Int. J. Electron. Commun. 112, 152941 (2019)

  70. 70.

    T. Tuncer, E. Avaroglu, M. Türk, A.B. Ozer, Implementation of non-periodic sampling true random number generator on FPGA. Inf. MIDEM. 44(4), 296–302 (2015)

  71. 71.

    S. Vaidyanathan, A.T. Azar, A. Boulkroune, A novel 4-D hyperchaotic system with two quadratic nonlinearities and its adaptive synchronisation. Int. J. Autom. Control 12(1), 5–26 (2018)

  72. 72.

    S. Vaidyanathan, I. Pehlivan, L.G. Dolvis, K. Jacques, M. Alcin et al., A novel ANN-based four-dimensional two-disk hyperchaotic dynamical system, bifurcation analysis, circuit realisation and FPGA-based TRNG implementation. Int. J. Comput. Appl. Technol. 62(1), 20–35 (2020)

  73. 73.

    C. Volos, J.O. Maaita, S. Vaidyanathan, V.T. Pham, I. Stouboulos, I. Kyprianidis, A novel four-dimensional hyperchaotic four-wing system with a saddle–focus equilibrium. IEEE Trans. Circuits Syst. II Express Briefs 64(3), 339–343 (2017)

  74. 74.

    H. Wang, G. Dong, New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system. Appl. Math. Comput. 346, 272–286 (2019)

  75. 75.

    X. Wang, G. Chen, Constructing a chaotic system with any number of equilibria. Nonlinear Dyn. 71(3), 429–436 (2013)

  76. 76.

    Y. Wang, C. Hui, C. Liu, C. Xu, Theory and implementation of a very high throughput true random number generator in field programmable gate array. Rev. Sci. Instrum. 87(4), 044704 (2016)

  77. 77.

    Y. Wang, Z. Liu, J. Ma, H. He, A pseudorandom number generator based on piecewise logistic map. Nonlinear Dyn. 83(4), 2373–2391 (2016)

  78. 78.

    C. Wannaboon, M. Tachibana, W. San-Um, A 0.18-μm CMOS high-data-rate true random bit generator through ΔΣ modulation of chaotic jerk circuit signals. Chaos 28(6), 063126 (2018)

  79. 79.

    Z. Wei, W. Zhang, Hidden hyperchaotic attractors in a modified Lorenz–Stenflo system with only one stable equilibrium. Int. J. Bifurc. Chaos 24(10), 1450127 (2014)

  80. 80.

    A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series. Phys. D Nonlinear Phenom. 16(3), 285–317 (1985)

  81. 81.

    C. Zhang, Theoretical design approach of four-dimensional piecewise-linear multi-wing hyperchaotic differential dynamic system. Optik (Stuttg) 127(11), 4575–4580 (2016)

  82. 82.

    P. Zhou, F. Yang, Hyperchaos, chaos, and horseshoe in a 4D nonlinear system with an infinite number of equilibrium points. Nonlinear Dyn. 76(1), 473–480 (2014)

Download references

Author information

Correspondence to Murat Tuna.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Prakash, P., Rajagopal, K., Koyuncu, I. et al. A Novel Simple 4-D Hyperchaotic System with a Saddle-Point Index-2 Equilibrium Point and Multistability: Design and FPGA-Based Applications. Circuits Syst Signal Process (2020). https://doi.org/10.1007/s00034-020-01367-0

Download citation

Keywords

  • Hyperchaotic system
  • Saddle-point index-2 equilibrium point
  • Multistability
  • FPGA-based application