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Circuits, Systems, and Signal Processing

, Volume 37, Issue 9, pp 3702–3724 | Cite as

Hyperchaotic Memcapacitor Oscillator with Infinite Equilibria and Coexisting Attractors

  • Karthikeyan Rajagopal
  • Sajad Jafari
  • Anitha Karthikeyan
  • Ashokkumar Srinivasan
  • Biniyam Ayele
Article

Abstract

A newly introduced charge-controlled memcapacitor-based hyperchaotic oscillator with coexisting chaotic attractors is investigated. Dynamic analysis of the oscillator shows that it has infinite number of equilibrium points and shows multistability. Its multistability analysis in the parameter space shows the existence of chaotic and hyperchaotic attractors. Fractional-order analysis of the hyperchaotic oscillator shows that the hyperchaos remains in the fractional order too. Field programmable gate arrays are used to realize the proposed oscillator.

Keywords

Memcapacitor Hyperchaos Multistability Fractional order FPGA 

References

  1. 1.
    G. Adomian, A review of the decomposition method and some recent results for nonlinear equations. Math. Comput. Model. 13(7), 17–43 (1990)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    M.P. Aghababa, Robust finite-time stabilization of fractional-order chaotic systems based on fractional Lyapunov stability theory. J. Comput. Nonlinear Dyn. 7(2), 021010 (2012)CrossRefGoogle Scholar
  3. 3.
    D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods, vol. 5 (World Scientific, Singapore, 2016)MATHGoogle Scholar
  4. 4.
    K. Barati, S. Jafari, J.C. Sprott, V.-T. Pham, Simple chaotic flows with a curve of equilibria. Int. J. Bifurc. Chaos 26(12), 1630034 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    R. Barboza, L.O. Chua, The four-element Chua’s circuit. Int. J. Bifurc. Chaos 18(04), 943–955 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    G. Bianchi, N. Kuznetsov, G. Leonov, M. Yuldashev, R. Yuldashev: Limitations of PLL simulation: hidden oscillations in MatLab and SPICE, in Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 2015 7th International Congress on 2015 (IEEE), pp. 79–84Google Scholar
  7. 7.
    B. Blażejczyk-Okolewska, T. Kapitaniak, Co-existing attractors of impact oscillator. Chaos Solitons Fractals 9(8), 1439–1443 (1998)CrossRefMATHGoogle Scholar
  8. 8.
    B. Bo-Cheng, S. GuoDong, X. JianPing, L. Zhong, P. SaiHu, Dynamics analysis of chaotic circuit with two memristors. Sci. China Technol. Sci. 54(8), 2180–2187 (2011).  https://doi.org/10.1007/s11431-011-4400-6 CrossRefMATHGoogle Scholar
  9. 9.
    E.A. Boroujeni, H.R. Momeni, Non-fragile nonlinear fractional order observer design for a class of nonlinear fractional order systems. Sig. Process. 92(10), 2365–2370 (2012)CrossRefGoogle Scholar
  10. 10.
    A. Buscarino, L. Fortuna, M. Frasca, L.V. Gambuzza, A gallery of chaotic oscillators based on HP memristor. Int. J. Bifurc. Chaos 23(05), 1330015 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    A. Buscarino, L. Fortuna, M. Frasca, L. Valentina Gambuzza, A chaotic circuit based on Hewlett–Packard memristor. Chaos Interdiscip. J. Nonlinear Sci. 22(2), 023136 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    D. Cafagna, G. Grassi, Fractional-order systems without equilibria: the first example of hyperchaos and its application to synchronization. Chin. Phys. B 24(8), 080502 (2015)CrossRefGoogle Scholar
  13. 13.
    R. Caponetto, S. Fazzino, An application of Adomian decomposition for analysis of fractional-order chaotic systems. Int. J. Bifurc. Chaos 23(03), 1350050 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    A. Charef, H. Sun, Y. Tsao, B. Onaral, Fractal system as represented by singularity function. IEEE Trans. Autom. Control 37(9), 1465–1470 (1992)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    L. Chua, Memristor-the missing circuit element. IEEE Trans. Circuit Theory 18(5), 507–519 (1971)CrossRefGoogle Scholar
  16. 16.
    L.O. Chua, S.M. Kang, Memristive devices and systems. Proc. IEEE 64(2), 209–223 (1976)MathSciNetCrossRefGoogle Scholar
  17. 17.
    A. Chudzik, P. Perlikowski, A. Stefanski, T. Kapitaniak, Multistability and rare attractors in van der Pol–Duffing oscillator. Int. J. Bifurc. Chaos 21(07), 1907–1912 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    F. Corinto, V. Krulikovskyi, S.D. Haliuk: Memristor-based chaotic circuit for pseudo-random sequence generators, in 2016 18th Mediterranean 2016 Electrotechnical Conference (MELECON) (IEEE), pp. 1–3Google Scholar
  19. 19.
    M.-F. Danca, N. Kuznetsov, Hidden chaotic sets in a Hopfield neural system. Chaos Solitons Fractals 103, 144–150 (2017)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    M.-F. Danca, N. Kuznetsov, G. Chen, Unusual dynamics and hidden attractors of the Rabinovich–Fabrikant system. Nonlinear Dyn. 88(1), 791–805 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    M.-F. Danca, W.K. Tang, G. Chen, Suppressing chaos in a simplest autonomous memristor-based circuit of fractional order by periodic impulses. Chaos Solitons Fractals 84, 31–40 (2016)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type (Springer, Berlin, 2010)CrossRefMATHGoogle Scholar
  23. 23.
    E. Dong, Z. Liang, S. Du, Z. Chen, Topological horseshoe analysis on a four-wing chaotic attractor and its FPGA implement. Nonlinear Dyn. 83(1–2), 623–630 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    D. Dudkowski, S. Jafari, T. Kapitaniak, N.V. Kuznetsov, G.A. Leonov, A. Prasad, Hidden attractors in dynamical systems. Phys. Rep. 637, 1–50 (2016).  https://doi.org/10.1016/j.physrep.2016.05.002 MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    S. Ellner, A.R. Gallant, D. McCaffrey, D. Nychka, Convergence rates and data requirements for Jacobian-based estimates of Lyapunov exponents from data. Phys. Lett. A 153(6–7), 357–363 (1991)MathSciNetCrossRefGoogle Scholar
  26. 26.
    A.L. Fitch, H.H. Iu, D. Yu: Chaos in a memcapacitor based circuit, in 2014 IEEE International Symposium on 2014 Circuits and Systems (ISCAS) (IEEE), pp. 482–485Google Scholar
  27. 27.
    S. He, K. Sun, H. Wang, Complexity analysis and DSP implementation of the fractional-order lorenz hyperchaotic system. Entropy 17(12), 8299–8311 (2015)CrossRefGoogle Scholar
  28. 28.
    S. Jafari, V.-T. Pham, T. Kapitaniak, Multiscroll chaotic sea obtained from a simple 3D system without equilibrium. Int. J. Bifurc. Chaos 26(02), 1650031 (2016)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    S. Jafari, J.C. Sprott, Erratum to:“Simple chaotic flows with a line equilibrium” [Chaos, Solitons and Fractals 57 (2013) 79–84]. Chaos Solitons Fractals 77, 341–342 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    S. Jafari, J.C. Sprott, Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals 57, 79–84 (2013)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    P. Jaros, L. Borkowski, B. Witkowski, K. Czolczynski, T. Kapitaniak, Multi-headed chimera states in coupled pendula. Eur. Phys. J. Spec. Top. 224(8), 1605–1617 (2015)CrossRefGoogle Scholar
  32. 32.
    H. Kim, M.P. Sah, C. Yang, S. Cho, L.O. Chua, Memristor emulator for memristor circuit applications. IEEE Trans. Circuits Syst. I Regul. Pap. 59(10), 2422–2431 (2012)MathSciNetCrossRefGoogle Scholar
  33. 33.
    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)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    N. Kuznetsov, The Lyapunov dimension and its estimation via the Leonov method. Phys. Lett. A 380(25), 2142–2149 (2016)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    N. Kuznetsov, T. Alexeeva, G. Leonov: Invariance of Lyapunov characteristic exponents, Lyapunov exponents, and Lyapunov dimension for regular and non-regular linearizations. arXiv preprint arXiv:1410.2016 (2014)
  36. 36.
    N. Kuznetsov, G. Leonov, M. Yuldashev, R. Yuldashev, Hidden attractors in dynamical models of phase-locked loop circuits: limitations of simulation in MATLAB and SPICE. Commun. Nonlinear Sci. Numer. Simul. 51, 39–49 (2017)CrossRefGoogle Scholar
  37. 37.
    N. Kuznetsov, T. Mokaev, P. Vasilyev, Numerical justification of Leonov conjecture on Lyapunov dimension of Rossler attractor. Commun. Nonlinear Sci. Numer. Simul. 19(4), 1027–1034 (2014)MathSciNetCrossRefGoogle Scholar
  38. 38.
    V. Lakshmikantham, A. Vatsala, Basic theory of fractional differential equations. Nonlinear Anal. Theory Methods Appl. 69(8), 2677–2682 (2008)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    D.M. Leenaerts, Higher-order spectral analysis to detect power-frequency mechanisms in a driven Chua’s circuit. Int. J. Bifurc. Chaos 7(06), 1431–1440 (1997)CrossRefMATHGoogle Scholar
  40. 40.
    G.A. Leonov, N.V. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurc. Chaos 23(01), 1330002 (2013)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    C. Li, Z. Gong, D. Qian, Y. Chen, On the bound of the Lyapunov exponents for the fractional differential systems. Chaos Interdisc. J. Nonlinear Sci. 20(1), 013127 (2010)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    C. Li, W. Hu, J.C. Sprott, X. Wang, Multistability in symmetric chaotic systems. Eur. Phys. J. Spec. Top. 224(8), 1493–1506 (2015)CrossRefGoogle Scholar
  43. 43.
    R. Li, W. Chen, Fractional order systems without equilibria. Chin. Phys. B 22, 040503 (2013)CrossRefGoogle Scholar
  44. 44.
    Y. Maistrenko, T. Kapitaniak, P. Szuminski, Locally and globally riddled basins in two coupled piecewise-linear maps. Phys. Rev. E 56(6), 6393 (1997)MathSciNetCrossRefGoogle Scholar
  45. 45.
    A. Maus, J. Sprott, Evaluating Lyapunov exponent spectra with neural networks. Chaos Solitons Fractals 51, 13–21 (2013)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    B. Muthuswamy, Implementing memristor based chaotic circuits. Int. J. Bifurc. Chaos 20(05), 1335–1350 (2010)CrossRefMATHGoogle Scholar
  47. 47.
    B. Muthuswamy, P.P. Kokate, Memristor-based chaotic circuits. IETE Tech. Rev. 26(6), 417–429 (2009)CrossRefGoogle Scholar
  48. 48.
    Y.V. Pershin, M. Di Ventra, Emulation of floating memcapacitors and meminductors using current conveyors. Electron. Lett. 47(4), 243–244 (2011)CrossRefGoogle Scholar
  49. 49.
    I. Petráš, Method for simulation of the fractional order chaotic systems. Acta Montan. Slovaca 11(4), 273–277 (2006)Google Scholar
  50. 50.
    C. Pezeshki, S. Elgar, R. Krishna, Bispectral analysis of possessing chaotic motion. J. Sound Vib. 137(3), 357–368 (1990)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    V.-T. Pham, S. Jafari, T. Kapitaniak, Constructing a chaotic system with an infinite number of equilibrium points. Int. J. Bifurc. Chaos 26(13), 1650225 (2016)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    V.T. Pham, S. Jafari, C. Volos, T. Kapitaniak, A gallery of chaotic systems with an infinite number of equilibrium points. Chaos Solitons Fractals 93, 58–63 (2016)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    C. Pradhan, S.K. Jena, S.R. Nadar, N. Pradhan, Higher-order spectrum in understanding nonlinearity in EEG rhythms. Comput. Math. Methods Med. 2012, 206857 (2012).  https://doi.org/10.1155/2012/206857 MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    H. Qing-Hui, L. Zhi-Jun, Z. Jin-Fang, Z. Yi-Cheng, Design and simulation of a memristor chaotic circuit based on current feedback op amp. Acta. Phys. Sin. (Chinese Edition) 63(18) (2014).  https://doi.org/10.7498/aps.63.180502
  55. 55.
    K. Rajagopal, L. Guessas, A. Karthikeyan, A. Srinivasan, G. Adam, Fractional order memristor no equilibrium chaotic system with its adaptive sliding mode synchronization and genetically optimized fractional order PID synchronization. Complexity 2017, 1892618 (2017).  https://doi.org/10.1155/2017/1892618 MathSciNetMATHGoogle Scholar
  56. 56.
    K. Rajagopal, L. Guessas, S. Vaidyanathan, A. Karthikeyan, A. Srinivasan, Dynamical analysis and FPGA implementation of a novel hyperchaotic system and its synchronization using Adaptive Sliding mode control and genetically optimized PID control. Math. Prob. Eng. 2017, 7307452 (2017).  https://doi.org/10.1155/2017/7307452 MathSciNetCrossRefGoogle Scholar
  57. 57.
    K. Rajagopal, A. Karthikeyan, P. Duraisamy, Hyperchaotic Chameleon: fractional order FPGA mentation. Complexity 2017, 8979408 (2017).  https://doi.org/10.1155/2017/8979408 Google Scholar
  58. 58.
    K. Rajagopal, A. Karthikeyan, A.K. Srinivasan, FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization. Nonlinear Dyn. 87, 1–24 (2017)CrossRefGoogle Scholar
  59. 59.
    V. Rashtchi, M. Nourazar, FPGA Implementation of a real-time weak signal detector using a duffing oscillator. Circuits Syst. Signal Process. 34(10), 3101–3119 (2015)CrossRefGoogle Scholar
  60. 60.
    H. Shao-Bo, S. Ke-Hui, W. Hui-Hai, Solution of the fractional-order chaotic system based on Adomian decomposition algorithm and its complexity analysis. Acta Phys. Sin. (Chinese Edition) 63(3) (2014). https://doi.org/10.7498/aps.63.030502Google Scholar
  61. 61.
    P. Sharma, M. Shrimali, A. Prasad, N. Kuznetsov, G. Leonov, Control of multistability in hidden attractors. Eur. Phys. J. Spec. Top. 224(8), 1485–1491 (2015)CrossRefGoogle Scholar
  62. 62.
    A. Silchenko, T. Kapitaniak, V. Anishchenko, Noise-enhanced phase locking in a stochastic bistable system driven by a chaotic signal. Phys. Rev. E 59(2), 1593 (1999)CrossRefGoogle Scholar
  63. 63.
    J.C. Sprott, A proposed standard for the publication of new chaotic systems. Int. J. Bifurc. Chaos 21(09), 2391–2394 (2011)CrossRefGoogle Scholar
  64. 64.
    J.C. Sprott, C. Li, Asymmetric bistability in the Rössler system. Acta Phys. Pol. B 32, 97–107 (2017)CrossRefGoogle Scholar
  65. 65.
    J.C. Sprott, X. Wang, G. Chen, Coexistence of point, periodic and strange attractors. Int. J. Bifurc. Chaos 23(05), 1350093 (2013)MathSciNetCrossRefGoogle Scholar
  66. 66.
    A. Stefanski, A. Dabrowski, T. Kapitaniak, Evaluation of the largest Lyapunov exponent in dynamical systems with time delay. Chaos Solitons Fractals 23(5), 1651–1659 (2005)MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    D.B. Strukov, G.S. Snider, D.R. Stewart, R.S. Williams, The missing memristor found. Nature 453(7191), 80–83 (2008)CrossRefGoogle Scholar
  68. 68.
    H. Sun, A. Abdelwahab, B. Onaral, Linear approximation of transfer function with a pole of fractional power. IEEE Trans. Autom. Control 29(5), 441–444 (1984)CrossRefMATHGoogle Scholar
  69. 69.
    F.R. Tahir, S. Jafari, V.-T. Pham, C. Volos, X. Wang, A novel no-equilibrium chaotic system with multiwing butterfly attractors. Int. J. Bifurc. Chaos 25(04), 1550056 (2015)MathSciNetCrossRefGoogle Scholar
  70. 70.
    M. Tavazoei, M. Haeri, Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems. IET Signal Proc. 1(4), 171–181 (2007)CrossRefGoogle Scholar
  71. 71.
    E. Tlelo-Cuautle, V. Carbajal-Gomez, P. Obeso-Rodelo, J. Rangel-Magdaleno, J.C. Nuñez-Perez, FPGA realization of a chaotic communication system applied to image processing. Nonlinear Dyn. 82(4), 1879–1892 (2015)MathSciNetCrossRefGoogle Scholar
  72. 72.
    E. Tlelo-Cuautle, A. Pano-Azucena, J. Rangel-Magdaleno, V. Carbajal-Gomez, G. Rodriguez-Gomez, Generating a 50-scroll chaotic attractor at 66 MHz by using FPGAs. Nonlinear Dyn. 85(4), 2143–2157 (2016)CrossRefGoogle Scholar
  73. 73.
    E. Tlelo-Cuautle, J. Rangel-Magdaleno, A. Pano-Azucena, P. Obeso-Rodelo, J.C. Nuñez-Perez, FPGA realization of multi-scroll chaotic oscillators. Commun. Nonlinear Sci. Numer. Simul. 27(1), 66–80 (2015)MathSciNetCrossRefGoogle Scholar
  74. 74.
    Z. Trzaska: Matlab solutions of chaotic fractional order circuits. Chapter 19, in All Assi. Engineering Education and Research Using MATLAB. Intech, Rijeka (2011)Google Scholar
  75. 75.
    G.-Y. Wang, P.-P. Jin, X.-W. Wang, Y.-R. Shen, F. Yuan, X.-Y. Wang, A flux-controlled model of meminductor and its application in chaotic oscillator. Chin. Phys. B 25(9), 090502 (2016)CrossRefGoogle Scholar
  76. 76.
    G. Wang, M. Cui, B. Cai, X. Wang, T. Hu, A chaotic oscillator based on HP memristor model. Math. Probl. Eng. 2015, 561901 (2015).  https://doi.org/10.1155/2015/561901 MathSciNetGoogle Scholar
  77. 77.
    G. Wang, S. Jiang, X. Wang, Y. Shen, F. Yuan, A novel memcapacitor model and its application for generating chaos. Math. Probl. Eng. 2016, 3173696 (2016).  https://doi.org/10.1155/2016/3173696 MathSciNetGoogle Scholar
  78. 78.
    G. Wang, C. Shi, X. Wang, F. Yuan, Coexisting oscillation and extreme multistability for a memcapacitor-based circuit. Math. Prob. Eng. 2017, 6504969 (2017).  https://doi.org/10.1155/2017/6504969 MathSciNetGoogle Scholar
  79. 79.
    Q. Wang, S. Yu, C. Li, J. Lü, X. Fang, C. Guyeux, J.M. Bahi, Theoretical design and FPGA-based implementation of higher-dimensional digital chaotic systems. IEEE Trans. Circuits Syst. I Regul. Pap. 63(3), 401–412 (2016)MathSciNetCrossRefGoogle Scholar
  80. 80.
    A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series. Physica D 16(3), 285–317 (1985)MathSciNetCrossRefMATHGoogle Scholar
  81. 81.
    X. Ya-Ming, W. Li-Dan, D. Shu-KaiL, A memristor-based chaotic system and its field programmable gate array implementation. ACTA Phys. Sin. 65(12), 120503 (2016).  https://doi.org/10.7498/aps.65.120503 Google Scholar
  82. 82.
    C. Yang, Q. Hu, Y. Yu, R. Zhang, Y. Yao, J. Cai: Memristor-based chaotic circuit for text/image encryption and decryption, in 2015 8th International Symposium on 2015 Computational Intelligence and Design (ISCID) (IEEE), pp. 447–450Google Scholar
  83. 83.
    D. Yu, Y. Liang, H. Chen, H.H. Iu, Design of a practical memcapacitor emulator without grounded restriction. IEEE Trans. Circuits Syst. II Express Briefs 60(4), 207–211 (2013)CrossRefGoogle Scholar
  84. 84.
    R. Zhang, J. Gong, Synchronization of the fractional-order chaotic system via adaptive observer. Syst. Sci. Control Eng. Open Access J. 2(1), 751–754 (2014)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Electrical and Communication Engineering, Center for Nonlinear DynamicsThe PNG University of TechnologyLaePapua New Guinea
  2. 2.Center for Nonlinear DynamicsDefence UniversityBishoftuEthiopia
  3. 3.Biomedical Engineering DepartmentAmirkabir University of TechnologyTehranIran

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