Dynamics, Synchronization and SPICE Implementation of a Memristive System with Hidden Hyperchaotic Attractor

  • Viet-Thanh Pham
  • Sundarapandian Vaidyanathan
  • Christos K. Volos
  • Thang Manh Hoang
  • Vu Van Yem
Chapter

Abstract

The realization of memristor in nanoscale size has received considerate attention recently because memristor can be applied in different potential areas such as spiking neural network, high-speed computing, synapses of biological systems, flexible circuits, nonvolatile memory, artificial intelligence, modeling of complex systems or low power devices and sensing. Interestingly, memristor has been used as a nonlinear element to generate chaos in memristive system. In this chapter, a new memristive system is proposed. The fundamental dynamics properties of such memristive system are discovered through equilibria, Lyapunov exponents, and Kaplan–York dimension. Especially, hidden attractor and hyperchaos can be observed in this new system. Moreover, synchronization for such system is studied and simulation results are presented showing the accuracy of the introduced synchronization scheme. An electronic circuit modelling such hyperchaotic memristive system is also reported to verify its feasibility.

Keywords

Chaos Hyperchaos Lyapunov exponents Hidden attractor No-equilibrium Memristor Synchronization Circuit SPICE 

References

  1. 1.
    Lorenz EN (1963) Deterministic non-periodic flow. J Atmos Sci 20:130–141CrossRefGoogle Scholar
  2. 2.
    Azar AT, Vaidyanathan S (2015) Chaos modeling and control systems design. Springer, GermanyGoogle Scholar
  3. 3.
    Azar AT, Vaidyanathan S (2015) Computational intelligence applications in modeling and control. Springer, GermanyGoogle Scholar
  4. 4.
    Azar AT, Vaidyanathan S (2015) Handbook of research on advanced intelligent control engineering and automation. IGI Global, USAGoogle Scholar
  5. 5.
    Chen G, Yu X (2003) Chaos control: theory and applications. Springer, BerlinCrossRefMATHGoogle Scholar
  6. 6.
    Chen GR (1999) Controlling chaos and bifurcations in engineering systems. CRC Press, Boca RatonGoogle Scholar
  7. 7.
    Sprott JC (2003) Chaos and times-series analysis. Oxford University Press, OxfordMATHGoogle Scholar
  8. 8.
    Strogatz SH (1994) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Perseus Books, MassachusettsGoogle Scholar
  9. 9.
    Yalcin ME, Suykens JAK, Vandewalle J (2005) Cellular neural networks, multi-scroll chaos and synchronization. World Scientific, SingaporeMATHGoogle Scholar
  10. 10.
    Rössler OE (1976) An equation for continuous chaos. Phys Lett A 57:397–398CrossRefGoogle Scholar
  11. 11.
    Arneodo A, Coullet P, Tresser C (1981) Possible new strange attractors with spiral structure. Comm Math Phys 79:573–579MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lü J, Chen G (2002) A new chaotic attractor coined. Int J Bif Chaos 12:659–661MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Vaidyanathan S (2013) A new six-term 3-D chaotic system with an exponential nonlineariry. Far East J Math Sci 79:135–143MATHGoogle Scholar
  14. 14.
    Barnerjee T, Biswas D, Sarkar BC (2012) Design and analysis of a first order time-delayed chaotic system. Nonlinear Dyn 70:721–734MathSciNetCrossRefGoogle Scholar
  15. 15.
    Pham V-T, Volos C, Vaidyanathan S (2015b) Multi-scroll chaotic oscillator based on a first-order delay differential equation. In: Azar AT, Vaidyanathan S (eds) Chaos modelling and control systems design, vol 581., Studies in computational intelligenceSpringer, Germany, pp 59–72Google Scholar
  16. 16.
    Vaidyanathan S, Azar AT (2015b) Analysis, control and synchronization of a nine-term 3-D novel chaotic system. In: Azar AT, Vaidyanathan S (eds) Chaos modelling and control systems design, vol 581., Studies in computational intelligenceSpringer, Germany, pp 19–38Google Scholar
  17. 17.
    Yalcin ME, Suykens JAK, Vandewalle J (2004) True random bit generation from a double-scroll attractor. IEEE Trans Circuits Syst I Regul Papers 51:1395–1404MathSciNetCrossRefGoogle Scholar
  18. 18.
    Volos CK, Kyprianidis IM, Stouboulos IN (2012) A chaotic path planning generator for autonomous mobile robots. Robot Auto Syst 60:651–656CrossRefGoogle Scholar
  19. 19.
    Hoang TM, Nakagawa M (2008) A secure communication system using projective-lag and/or projective-anticipating synchronizations of coupled multidelay feedback systems. Chaos Solitions Fractals 38:1423–1438CrossRefGoogle Scholar
  20. 20.
    Rössler OE (1979) An equation for hyperchaos. Phys Lett A 71:155–157MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Vaidyanathan S, Azar AT (2015a) Analysis and control of a 4-D novel hyperchaotic system. In: Azar AT, Vaidyanathan S (eds) Chaos modeling and control systems design, vol 581., Studies in computational intelligenceSpringer, Germany, pp 19–38Google Scholar
  22. 22.
    Sadoudi S, Tanougast C, Azzaz MS, Dandache A (2013) Design and FPGA implementation of a wireless hyperchaotic communication system for secure realtime image transmission. EURASIP J Image Video Process 943:1–18Google Scholar
  23. 23.
    Udaltsov VS, Goedgebuer JP, Larger L, Cuenot JB, Levy P, Rhodes WT (2003) Communicating with hyperchaos: the dynamics of a DNLF emitter and recovery of transmitted information. Optics Spectrosc 95:114–118CrossRefGoogle Scholar
  24. 24.
    Grassi G, Mascolo S (1999) A system theory approach for designing cryptosystems based on hyperchaos. IEEE Trans Cir Sys I: Fund Theory Appl 46:1135–1138CrossRefMATHGoogle Scholar
  25. 25.
    Huang Y, Yang X (2006) Hyperchaos and bifurcation in a new class of four-dimensional hopfield neural networks. Neurocomputing 69:1787–1795CrossRefGoogle Scholar
  26. 26.
    Vicente R, Dauden J, Colet P, Toral R (2005) Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop. IEEE J Quantum Electr 41:541–548CrossRefGoogle Scholar
  27. 27.
    Buscarino A, Fortuna L, Frasca M, Gambuzza LV (2012) A chaotic circuit based on Hewlett-Packard memristor. Chaos 22:023136Google Scholar
  28. 28.
    Fitch AL, Yu D, Iu HHC, Sreeram V (2012) Hyperchaos in an memristor-based modified canonical chua’s circuit. Int J Bif Chaos 22:1250133–1250138CrossRefMATHGoogle Scholar
  29. 29.
    Buscarino A, Fortuna L, Frasca M, Gambuzza LV (2012) A gallery of chaotic oscillators based on hp memristor. Int J Bif Chaos 22:1330014–1330015MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Li Q, Hu S, Tang S, Zeng G (2014) Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation. Int J Cir Theory Appl 42:1172–1188CrossRefGoogle Scholar
  31. 31.
    Li Q, Zeng H, Li J (2015) Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria. Nonlinear Dyn 79:2295–2308MathSciNetCrossRefGoogle Scholar
  32. 32.
    Pham VT, Volos CK, Vaidyanathan S, Le TP, Vu VY (2015c) A memristor-based hyperchaotic system with hidden attractors: dynamics, sychronization and circuital emulating. J Eng Sci Tech Rev 8:205–214Google Scholar
  33. 33.
    Leonov GA, Kuznetsov NV (2013) Hidden attractors in dynamical systems: from hidden oscillation in Hilbert-Kolmogorov, Aizerman and Kalman problems to hidden chaotic attractor in Chua circuits. Int J Bifurc Chaos 23:1330002MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Leonov GA, Kuznetsov NV, Kuznetsova OA, Seldedzhi SM, Vagaitsev VI (2011) Hidden oscillations in dynamical systems. Trans Syst Contr 6:54–67Google Scholar
  35. 35.
    Jafari S, Sprott JC (2013) Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals 57:79–84MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kuznetsov NV, Leonov GA, Seledzhi SM (2011) Hidden oscillations in nonlinear control systems. IFAC Proc 18:2506–2510Google Scholar
  37. 37.
    Pham V-T, Jafari S, Volos C, Wang X, Golpayegani SMRH (2014a) Is that really hidden? The presence of complex fixed-points in chaotic flows with no equilibria. Int J Bifur Chaos 24:1450146CrossRefMATHGoogle Scholar
  38. 38.
    Pham V-T, Volos CK, Jafari S, Wei Z, Wang X (2014b) Constructing a novel no-equilibrium chaotic system. Int J Bifurc Chaos 24:1450073MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Sharma PR, Shrimali MD, Prasad A, Kuznetsov NV, Leonov GA (2015) Control of multistability in hidden attractors. Eur Phys J Special Topics 224:1485–1491CrossRefGoogle Scholar
  40. 40.
    Leonov GA, Kuznetsov NV, Vagaitsev VI (2012) Hidden attractor in smooth Chua system. Phys D 241:1482–1486Google Scholar
  41. 41.
    Leonov GA, Kuznetsov NV, Kiseleva MA, Solovyeva EP, Zaretskiy AM (2014) Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor. Nonlinear Dyn 77:277–288CrossRefGoogle Scholar
  42. 42.
    Leonov GA, Kuznetsov NV (2011) Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems. IFAC Proc 18:2494–2505Google Scholar
  43. 43.
    Brezetskyi S, Dudkowski D, Kapitaniak T (2015) Rare and hidden attractors in van der pol-duffing oscillators. Eur Phys J Special Topics 224:1459–1467CrossRefGoogle Scholar
  44. 44.
    Jafari S, Sprott JC, Nazarimehr F (2015) Recent new examples of hidden attractors. Eur Phys J Special Topics 224:1469–1476CrossRefGoogle Scholar
  45. 45.
    Shahzad M, Pham VT, Ahmad MA, Jafari S, Hadaeghi F (2015) Synchronization and circuit design of a chaotic system with coexisting hidden attractors. Eur Phys J Special Topics 224:1637–1652CrossRefGoogle Scholar
  46. 46.
    Sprott JC (2015) Strange attractors with various equilibrium types. Eur Phys J Special Topics 224:1409–1419CrossRefGoogle Scholar
  47. 47.
    Vaidyanathan S, Volos CK, Pham VT (2015c) Analysis, control, synchronization and spice implementation of a novel 4-d hyperchaotic rikitake dynamo system without equilibrium. J Eng Sci Tech Rev 8:232–244Google Scholar
  48. 48.
    Vaidyanathan S, Pham VT, Volos CK (2015b) A 5-d hyperchaotic rikitake dynamo system with hidden attractors. Eur Phys J Special Topics 224:1575–1592CrossRefGoogle Scholar
  49. 49.
    Pham VT, Vaidyanathan S, Volos CK, Jafari S (2015a) Hidden attractors in a chaotic system with an exponential nonlinear term. Eur Phys J Special Topics 224:1507–1517CrossRefGoogle Scholar
  50. 50.
    Leonov GA, Kuznetsov NV (2011) Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems. Dokl Math 84:475–481Google Scholar
  51. 51.
    Leonov GA, Kuznetsov NV, Vagaitsev VI (2011) Localization of hidden Chua’s attractors. Phys Lett A 375:2230–2233Google Scholar
  52. 52.
    Bao B, Liu Z, Xu B (2010) Dynamical analysis of memristor chaotic oscillator. Acta Physica Sinica 59:3785–3793Google Scholar
  53. 53.
    Muthuswamy B (2010) Implementing memristor based chaotic circuits. Int J Bif Chaos 20:1335–1350CrossRefMATHGoogle Scholar
  54. 54.
    Sprott JC (2010) Elegant chaos: algebraically simple chaotic flows. World Scientific, SingaporeCrossRefMATHGoogle Scholar
  55. 55.
    Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Phys D 16:285–317Google Scholar
  56. 56.
    Frederickson P, Kaplan JL, Yorke ED, York J (1983) The lyapunov dimension of strange attractors. J Differ Equ 49:185–207CrossRefMATHGoogle Scholar
  57. 57.
    Boccaletti S, Kurths J, Osipov G, Valladares DL, Zhou CS (2002) The synchronization of chaotic systems. Phys Rep 366:1–101MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Fortuna L, Frasca M (2007) Experimental synchronization of single-transistor-based chaotic circuits. Chaos 17:043118-1–043118-5MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Kapitaniak T (1994) Synchronization of chaos using continuous control. Phys Rev E 50:1642–1644Google Scholar
  60. 60.
    Pecora LM, Carroll TL (1990) Synchronization in chaotic signals. Phys Rev A 64:821–824MathSciNetMATHGoogle Scholar
  61. 61.
    Buscarino A, Fortuna L, Frasca M (2009) Experimental robust synchronization of hyperchaotic circuits. Phys D 238:1917–1922CrossRefMATHGoogle Scholar
  62. 62.
    Gamez-Guzman L, Cruz-Hernandez C, Lopez-Gutierrez R, Garcia-Guerrero EE (2009) Synchronization of Chua’s circuits with multi-scroll attractors: application to communication. Commun Nonlinear Sci Numer Simul 14:2765–2775CrossRefGoogle Scholar
  63. 63.
    Karthikeyan R, Vaidyanathan S (2014) Hybrid chaos synchronization of four-scroll systems via active control. J Electr Eng 65:97–103Google Scholar
  64. 64.
    Srinivasan K, Senthilkumar DV, Murali K, Lakshmanan M, Kurths J (2011) Synchronization transitions in coupled time-delay electronic circuits with a threshold nonlinearity. Chaos 21:023119CrossRefMATHGoogle Scholar
  65. 65.
    Vaidyanathan S (2014) Analysis and adaptive synchronization of eight-term novel 3-D chaotic system with three quadratic nonlinearities. Eur Phys J Special Topics 223:1519–1529CrossRefGoogle Scholar
  66. 66.
    Vaidyanathan S, Azar AT (2015c) Anti-synchronization of identical chaotic systems using sliding mode control and an application to Vaidhyanathan-Madhavan chaotic systems. Stud Comput Intell 576:527–547CrossRefGoogle Scholar
  67. 67.
    Vaidyanathan S, Azar AT (2015d) Hybrid synchronization of identical chaotic systems using sliding mode control and an application to Vaidhyanathan chaotic systems. Stud Comput Intell 576:549–569CrossRefGoogle Scholar
  68. 68.
    Vaidyanathan S, Idowu BA, Azar AT (2015a) Backstepping controller design for the global chaos synchronization of Sprott’s jerk systems. Stud Comput Intell 581:39–58CrossRefGoogle Scholar
  69. 69.
    Zhu Q, Azar AT (2015) Complex system modelling and control through intelligent soft computations. Springer, GermanyCrossRefMATHGoogle Scholar
  70. 70.
    Woafo P, Kadji HGE (2004) Synchronized states in a ring of mutually coupled self-sustained electrical oscillators. Phys Rev E 69:046206CrossRefGoogle Scholar
  71. 71.
    Stefanski A, Perlikowski P, Kapitaniak T (2007) Ragged synchronizability of coupled oscillators. Phys Rev E 75:016210MathSciNetCrossRefGoogle Scholar
  72. 72.
    Volos CK, Kyprianidis IM, Stouboulos IN (2011) Various synchronization phenomena in bidirectionally coupled double scroll circuits. Commun Nonlinear Sci Numer Simul 71:3356–3366MathSciNetCrossRefGoogle Scholar
  73. 73.
    Aguilar-Lopez R, Martinez-Guerra R, Perez-Pinacho C (2014) Nonlinear observer for synchronization of chaotic systems with application to secure data transmission. Eur Phys J Special Topics 223:1541–1548CrossRefGoogle Scholar
  74. 74.
    Rosenblum MG, Pikovsky AS, Kurths J (1997) From phase to lag synchronization in coupled chaotic oscillators. Phys Rev Lett 78:4193–4196CrossRefMATHGoogle Scholar
  75. 75.
    Akopov A, Astakhov V, Vadiasova T, Shabunin A, Kapitaniak T (2005) Frequency synchronization in clusters in coupled extended systems. Phys Lett A 334:169–172CrossRefMATHGoogle Scholar
  76. 76.
    Hoang TM, Nakagawa M (2007) Anticipating and projective–anticipating synchronization of coupled multidelay feedback systems. Phys Lett A 365:407–411Google Scholar
  77. 77.
    Vaidyanathan S (2012) Anti-synchronization of four-wing chaotic systems via sliding mode control. Int J Auto Comput 9:274–279CrossRefGoogle Scholar
  78. 78.
    Vaidyanathan S, Volos C, Pham VT, Madhavan K, Idowo BA (2014) Adaptive backstepping control, synchronization and circuit simualtion of a 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities. Arch Cont Sci 33:257–285MATHGoogle Scholar
  79. 79.
    Khalil H (2002) Nonlinear systems. Prentice Hall, New JerseyMATHGoogle Scholar
  80. 80.
    Sastry S (1999) Nonlinear systems: analysis, stability, and control. Springer, USACrossRefMATHGoogle Scholar
  81. 81.
    Barakat M, Mansingka A, Radwan AG, Salama KN (2013) Generalized hardware post processing technique for chaos-based pseudorandom number generators. ETRI J 35:448–458CrossRefGoogle Scholar
  82. 82.
    Volos CK, Kyprianidis IM, Stouboulos IN (2013) Image encryption process based on chaotic synchronization phenomena. Signal Process 93:1328–1340CrossRefGoogle Scholar
  83. 83.
    Sundarapandian V, Pehlivan I (2012) Analysis, control, synchronization, and circuit design of a novel chaotic system. Math Comp Model 55:1904–1915MathSciNetCrossRefMATHGoogle Scholar
  84. 84.
    Bouali S, Buscarino A, Fortuna L, Frasca M, Gambuzza LV (2012) Emulating complex business cycles by using an electronic analogue. Nonlinear Anal Real World Appl 13:2459–2465Google Scholar
  85. 85.
    Fortuna L, Frasca M, Xibilia MG (2009) Chua’s circuit implementation: yesterday, today and tomorrow. World Scientific, SingaporeGoogle Scholar
  86. 86.
    Tetzlaff R (2014) Memristors and memristive systems. Springer, New YorkCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Viet-Thanh Pham
    • 1
  • Sundarapandian Vaidyanathan
    • 2
  • Christos K. Volos
    • 3
  • Thang Manh Hoang
    • 1
  • Vu Van Yem
    • 1
  1. 1.School of Electronics and TelecommunicationsHanoi University of Science and TechnologyHanoiVietnam
  2. 2.Research and Development CentreVel Tech UniversityTamil NaduIndia
  3. 3.Physics DepartmentAristotle University of ThessalonikiThessalonikiGreece

Personalised recommendations