Skip to main content
SpringerLink
Log in

On the simplest fractional-order memristor-based chaotic system

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

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.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Cafagna, D.: Fractional calculus: a mathematical tool from the past for present engineers. IEEE Ind. Electron. Mag. 1, 35–40 (2007)

    Article  MathSciNet  Google Scholar 

  2. Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  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. Podlubny, I.: Fractional-order systems and PIλ Dμ-controllers. IEEE Trans. Autom. Control 44, 208–213 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Tseng, Ch.: Design of FIR and IIR fractional order Simpson digital integrators. Signal Process. 87, 1045–1057 (2007)

    Article  MATH  Google Scholar 

  9. Sheu, L.J.: A speech encryption using fractional chaotic systems. Nonlinear Dyn. 65, 103–108 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  11. Wu, X., Lu, Y.: Generalized projective synchronization of the fractional-order Chen hyperchaotic system. Nonlinear Dyn. 57, 25–35 (2009)

    Article  MATH  Google Scholar 

  12. Wu, X., Wang, H.: A new chaotic system with fractional order and its projective synchronization. Nonlinear Dyn. 61, 407–417 (2010)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  16. Pinto, C.M.A., Tenreiro Machado, J.A.: Complex order van der Pol oscillator. Nonlinear Dyn. 65, 247–254 (2011)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, C.G., Chen, G.: Chaos in the fractional-order Chen system and its control. Chaos Solitons Fractals 22, 549–554 (2004)

    Article  MATH  Google Scholar 

  23. Lu, J.G., Chen, G.: A note on the fractional-order Chen system. Chaos Solitons Fractals 27, 685–688 (2006)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, C.G., Chen, G.R.: Chaos and hyperchaos in the fractional-order Rössler equations. Physica A 341, 55–61 (2004)

    Article  MathSciNet  Google Scholar 

  26. Cafagna, D., Grassi, G.: Hyperchaos in the fractional-order Rössler system with lowest-order. Int. J. Bifurc. Chaos 19, 339–347 (2009)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  28. Deng, W., Lu, J.: Design of multidirectional multiscroll chaotic attractors based on fractional differential systems via switching control. Chaos 16, 043120 (2006)

    Article  MathSciNet  Google Scholar 

  29. Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Petras, I.: Fractional-order memristor-based Chua’s circuit. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 57, 975–979 (2010)

    Article  Google Scholar 

  33. Itoh, M., Chua, L.O.: Memristor oscillators. Int. J. Bifurc. Chaos 18, 3183–3206 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Chua, L.O.: Memristor—the missing circuit element. IEEE Trans. Circuit Theory 18, 507–519 (1971)

    Article  Google Scholar 

  35. Strukov, D.B., Snider, G.S., Stewart, G.R., Williams, R.S.: The missing memristor found. Nature 453, 80–83 (2008)

    Article  Google Scholar 

  36. Muthuswamy, B., Chua, L.O.: Simplest chaotic circuit. Int. J. Bifurc. Chaos 20, 1567–1580 (2010)

    Article  Google Scholar 

  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. Cafagna, D., Grassi, G.: An effective method for detecting chaos in fractional-order systems. Int. J. Bifurc. Chaos 20, 669–678 (2010)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Part II. J. Roy. Astr. Soc. 13, 529–539 (1967)

    Article  Google Scholar 

  42. Davison, M., Essex, G.C.: Fractional differential equations and initial value problems. Math. Sci. 23, 108–116 (1998)

    MathSciNet  MATH  Google Scholar 

  43. Chua, L.O., Kang, S.M.: Memristive devices and systems. Proc. IEEE 64, 209–223 (1976)

    Article  MathSciNet  Google Scholar 

  44. Chua, L.O.: Local activity is the origin of complexity. Int. J. Bifurc. Chaos 15, 3435–3456 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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. Maundy, B., Elwakil, A., Gift, S.: On a multivibrator that employs a fractional capacitor. Analog Integr. Circuits Signal Process. 62, 99–103 (2010)

    Article  Google Scholar 

  47. Elwakil, A.S.: Fractional-order circuits and systems: an emerging interdisciplinary research area. IEEE Circuits Syst. Mag. 4, 40–50 (2010)

    Article  Google Scholar 

  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. 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. Khalil, H.K.: Nonlinear Systems. Prentice Hall, New Jersey (2002)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  52. Delavari, H., Baleanu, D., Sadati, J.: Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn. 67, 2433–2439 (2012)

    Article  MATH  Google Scholar 

  53. Tavazoei, M.S., Haeri, M.: Limitations of frequency domain approximation for detecting chaos in fractional order systems. Nonlinear Anal. 69, 1299–1320 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  55. Tavazoei, M.S.: A note on fractional-order derivatives of periodic functions. Automatica 46, 945–948 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  56. Yazdani, M., Salarieh, H.: On the existence of periodic solutions in time-invariant fractional order systems. Automatica 47, 1834–1837 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  60. Galeone, L., Garrappa, R.: Explicit methods for fractional differential equations and their stability properties. J. Comput. Appl. Math. 228, 548–560 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  61. Li, C., Peng, G.: Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22, 443–450 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  62. Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  63. Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31–52 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  65. Wang, X., Chen, G.: A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17, 1264–1272 (2012)

    Article  MathSciNet  Google Scholar 

  66. Wei, Z.: Dynamical behaviors of a chaotic system with no equilibria. Phys. Lett. A 376, 102–108 (2011)

    Article  MathSciNet  Google Scholar 

  67. Gottwald, G.A., Melbourne, I.: A new test for chaos in deterministic systems. Proc. R. Soc. Lond. A 460, 603–611 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento Ingegneria Innovazione, Università del Salento, Lecce, Italy

    Donato Cafagna & Giuseppe Grassi

Authors
  1. Donato Cafagna
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giuseppe Grassi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Donato Cafagna.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cafagna, D., Grassi, G. On the simplest fractional-order memristor-based chaotic system. Nonlinear Dyn 70, 1185–1197 (2012). https://doi.org/10.1007/s11071-012-0522-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-012-0522-z

Keywords

Search

Navigation