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Nonlinear Dynamics

, Volume 70, Issue 2, pp 1185–1197 | Cite as

On the simplest fractional-order memristor-based chaotic system

  • Donato CafagnaEmail author
  • Giuseppe Grassi
Original Paper

Abstract

In 1695, G. Leibniz laid the foundations of fractional calculus, but mathematicians revived it only 300 years later. In 1971, L.O. Chua postulated the existence of a fourth circuit element, called memristor, but Williams’s group of HP Labs realized it only 37 years later. By looking at these interdisciplinary and promising research areas, in this paper, a novel fractional-order system including a memristor is introduced. In particular, chaotic behaviors in the simplest fractional-order memristor-based system are shown. Numerical integrations (via a predictor–corrector method) and stability analysis of the system equilibria are carried out, with the aim to show that chaos can be found when the order of the derivative is 0.965. Finally, the presence of chaos is confirmed by the application of the recently introduced 0-1 test.

Keywords

Fractional chaotic systems Chaotic attractors Noninteger-order dynamics Memristor 

References

  1. 1.
    Cafagna, D.: Fractional calculus: a mathematical tool from the past for present engineers. IEEE Ind. Electron. Mag. 1, 35–40 (2007) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) zbMATHGoogle Scholar
  3. 3.
    Sun, H., Abdelwahed, A., Onaral, B.: Linear approximation for transfer function with a pole of fractional order. IEEE Trans. Autom. Control 29, 441–444 (1984) zbMATHCrossRefGoogle Scholar
  4. 4.
    Diethelm, K., Ford, N.J., Freed, A.D.: A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999) zbMATHGoogle Scholar
  6. 6.
    Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Nonlinear Non-integer Order Circuits and Systems—An Introduction. World Scientific, Singapore (2000) Google Scholar
  7. 7.
    Podlubny, I.: Fractional-order systems and PIλ Dμ-controllers. IEEE Trans. Autom. Control 44, 208–213 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Tseng, Ch.: Design of FIR and IIR fractional order Simpson digital integrators. Signal Process. 87, 1045–1057 (2007) zbMATHCrossRefGoogle Scholar
  9. 9.
    Sheu, L.J.: A speech encryption using fractional chaotic systems. Nonlinear Dyn. 65, 103–108 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974) zbMATHGoogle Scholar
  11. 11.
    Wu, X., Lu, Y.: Generalized projective synchronization of the fractional-order Chen hyperchaotic system. Nonlinear Dyn. 57, 25–35 (2009) zbMATHCrossRefGoogle Scholar
  12. 12.
    Wu, X., Wang, H.: A new chaotic system with fractional order and its projective synchronization. Nonlinear Dyn. 61, 407–417 (2010) zbMATHCrossRefGoogle Scholar
  13. 13.
    Chang, C.M., Chen, H.K.: Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen–Lee systems. Nonlinear Dyn. 62, 851–858 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Dadras, S., Momeni, H.R., Qi, G., Wang, Z.L.: Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form. Nonlinear Dyn. 67, 1161–1173 (2012) zbMATHCrossRefGoogle Scholar
  15. 15.
    Zeng, C., Yang, Q., Wang, J.: Chaos and mixed synchronization of a new fractional-order system with one saddle and two stable node-foci. Nonlinear Dyn. 65, 457–466 (2011) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pinto, C.M.A., Tenreiro Machado, J.A.: Complex order van der Pol oscillator. Nonlinear Dyn. 65, 247–254 (2011) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Chua, L.O., Komuro, M., Matsumoto, T.: The double scroll family. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 33, 1073–1118 (1986) Google Scholar
  18. 18.
    Hartley, T., Lorenzo, C., Qammer, H.: Chaos in a fractional order Chua’s system. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 42, 485–490 (1995) CrossRefGoogle Scholar
  19. 19.
    Li, C.P., Deng, W.H., Xu, D.: Chaos synchronization of the Chua system with a fractional order. Physica A 360, 171–185 (2006) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Cafagna, D., Grassi, G.: Fractional-order Chua’s circuit: time-domain analysis, bifurcation, chaotic behaviour and test for chaos. Int. J. Bifurc. Chaos 18, 615–639 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Li, C.G., Chen, G.: Chaos in the fractional-order Chen system and its control. Chaos Solitons Fractals 22, 549–554 (2004) zbMATHCrossRefGoogle Scholar
  23. 23.
    Lu, J.G., Chen, G.: A note on the fractional-order Chen system. Chaos Solitons Fractals 27, 685–688 (2006) zbMATHCrossRefGoogle Scholar
  24. 24.
    Cafagna, D., Grassi, G.: Bifurcation and chaos in the fractional-order Chen system via a time-domain approach. Int. J. Bifurc. Chaos 18, 1845–1863 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Li, C.G., Chen, G.R.: Chaos and hyperchaos in the fractional-order Rössler equations. Physica A 341, 55–61 (2004) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Cafagna, D., Grassi, G.: Hyperchaos in the fractional-order Rössler system with lowest-order. Int. J. Bifurc. Chaos 19, 339–347 (2009) zbMATHCrossRefGoogle Scholar
  27. 27.
    Cafagna, D., Grassi, G.: Fractional-order chaos: a novel four-wing attractor in coupled Lorenz systems. Int. J. Bifurc. Chaos 19, 3329–3338 (2009) zbMATHCrossRefGoogle Scholar
  28. 28.
    Deng, W., Lu, J.: Design of multidirectional multiscroll chaotic attractors based on fractional differential systems via switching control. Chaos 16, 043120 (2006) MathSciNetCrossRefGoogle Scholar
  29. 29.
    Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Grassi, G., Severance, F.L., Mashev, E.D., Bazuin, B.J., Miller, D.A.: Generation of a four-wing chaotic attractor by two weakly-coupled Lorenz systems. Int. J. Bifurc. Chaos 18, 2089–2094 (2008) zbMATHCrossRefGoogle Scholar
  31. 31.
    Cafagna, D., Grassi, G.: Hyperchaotic coupled Chua circuits: an approach for generating new nxm-scroll attractors. Int. J. Bifurc. Chaos 13, 2537–2550 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Petras, I.: Fractional-order memristor-based Chua’s circuit. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 57, 975–979 (2010) CrossRefGoogle Scholar
  33. 33.
    Itoh, M., Chua, L.O.: Memristor oscillators. Int. J. Bifurc. Chaos 18, 3183–3206 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Chua, L.O.: Memristor—the missing circuit element. IEEE Trans. Circuit Theory 18, 507–519 (1971) CrossRefGoogle Scholar
  35. 35.
    Strukov, D.B., Snider, G.S., Stewart, G.R., Williams, R.S.: The missing memristor found. Nature 453, 80–83 (2008) CrossRefGoogle Scholar
  36. 36.
    Muthuswamy, B., Chua, L.O.: Simplest chaotic circuit. Int. J. Bifurc. Chaos 20, 1567–1580 (2010) CrossRefGoogle Scholar
  37. 37.
    Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Proceedings of IMACS, IEEE-SMC, Lille, France, pp. 963–968 (1996) Google Scholar
  38. 38.
    Cafagna, D., Grassi, G.: An effective method for detecting chaos in fractional-order systems. Int. J. Bifurc. Chaos 20, 669–678 (2010) zbMATHCrossRefGoogle Scholar
  39. 39.
    Sun, K.H., Liu, X., Zhu, C.X.: The 0-1 test algorithm for chaos and its applications. Chin. Phys. B 19, 110510 (2010) CrossRefGoogle Scholar
  40. 40.
    Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Fractal and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Wien (1997) Google Scholar
  41. 41.
    Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Part II. J. Roy. Astr. Soc. 13, 529–539 (1967) CrossRefGoogle Scholar
  42. 42.
    Davison, M., Essex, G.C.: Fractional differential equations and initial value problems. Math. Sci. 23, 108–116 (1998) MathSciNetzbMATHGoogle Scholar
  43. 43.
    Chua, L.O., Kang, S.M.: Memristive devices and systems. Proc. IEEE 64, 209–223 (1976) MathSciNetCrossRefGoogle Scholar
  44. 44.
    Chua, L.O.: Local activity is the origin of complexity. Int. J. Bifurc. Chaos 15, 3435–3456 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Martin, R., Quintana, J.: Modeling of electrochemical double layer capacitors by means of fractional impedance. J. Comput. Nonlinear Dyn. 3, 1303–1309 (2008) Google Scholar
  46. 46.
    Maundy, B., Elwakil, A., Gift, S.: On a multivibrator that employs a fractional capacitor. Analog Integr. Circuits Signal Process. 62, 99–103 (2010) CrossRefGoogle Scholar
  47. 47.
    Elwakil, A.S.: Fractional-order circuits and systems: an emerging interdisciplinary research area. IEEE Circuits Syst. Mag. 4, 40–50 (2010) CrossRefGoogle Scholar
  48. 48.
    Petras, I., Chen, Y.Q., Coopmans, C.: Fractional-order memristive systems. In: Proc. of IEEE Conf. on Emerging Technologies & Factory Automation (ETFA), Mallorca, Spain (2009) Google Scholar
  49. 49.
    Coopmans, C., Petras, I., Chen, Y.Q.: Analogue fractional-order generalized memristive devices. In: Proc. of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, San Diego, CA, USA (2009) Google Scholar
  50. 50.
    Khalil, H.K.: Nonlinear Systems. Prentice Hall, New Jersey (2002) zbMATHGoogle Scholar
  51. 51.
    Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A.: Equilibrium points, stability and numerical solutions of fractional–order predator-prey and rabies models. J. Math. Anal. Appl. 325, 542–553 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Delavari, H., Baleanu, D., Sadati, J.: Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn. 67, 2433–2439 (2012) zbMATHCrossRefGoogle Scholar
  53. 53.
    Tavazoei, M.S., Haeri, M.: Limitations of frequency domain approximation for detecting chaos in fractional order systems. Nonlinear Anal. 69, 1299–1320 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Tavazoei, M.S., Haeri, M.: A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica 45, 1886–1890 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Tavazoei, M.S.: A note on fractional-order derivatives of periodic functions. Automatica 46, 945–948 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Yazdani, M., Salarieh, H.: On the existence of periodic solutions in time-invariant fractional order systems. Automatica 47, 1834–1837 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Tavazoei, M.S., Haeri, M., Nazari, N.: Analysis of undamped oscillations generated by marginally stable fractional order systems. Signal Process. 88, 2971–2978 (2008) zbMATHCrossRefGoogle Scholar
  58. 58.
    Wang, Y., Li, C.: Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle? Phys. Lett. A 363, 414–419 (2007) CrossRefGoogle Scholar
  59. 59.
    Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Berlin (2007) zbMATHGoogle Scholar
  60. 60.
    Galeone, L., Garrappa, R.: Explicit methods for fractional differential equations and their stability properties. J. Comput. Appl. Math. 228, 548–560 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Li, C., Peng, G.: Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22, 443–450 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31–52 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Yang, Q., Wei, Z.C., Chen, G.: An unusual 3D autonomous quadratic chaotic system with two stable node-foci. Int. J. Bifurc. Chaos 20, 1061–1083 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Wang, X., Chen, G.: A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17, 1264–1272 (2012) MathSciNetCrossRefGoogle Scholar
  66. 66.
    Wei, Z.: Dynamical behaviors of a chaotic system with no equilibria. Phys. Lett. A 376, 102–108 (2011) MathSciNetCrossRefGoogle Scholar
  67. 67.
    Gottwald, G.A., Melbourne, I.: A new test for chaos in deterministic systems. Proc. R. Soc. Lond. A 460, 603–611 (2004) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Dipartimento Ingegneria InnovazioneUniversità del SalentoLecceItaly

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