Skip to main content
SpringerLink
Log in

A fractional-order multistable locally active memristor and its chaotic system with transient transition, state jump

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

Abstract

Fractional calculus is closer to reality and has the same memory characteristics as memristor. Therefore, a fractional-order multistable locally active memristor is proposed for the first time in this paper, which has infinitely many coexisting pinched hysteresis loops under different initial states and wide locally active regions. Through the theoretical and numerical analysis, it is found that the fractional-order memristor has stronger locally active and memory characteristics and wider nonvolatile ranges than the integer-order memristor. Furthermore, this fractional-order memristor is applied in a chaotic system. It is found that oscillations occur only within the locally active regions. This chaotic system not only has complex and rich nonlinear dynamics such as infinitely many discrete equilibrium points, multistability and anti-monotonicity but also produces two new phenomena that have not been found in other chaotic systems. The first one is transient transition: the behavior of local chaos and local period transition alternately occurring. The second is state jump: the behavior of local period-4 oscillation or local chaotic oscillation jumping to local period-2 oscillation. Finally, the circuit simulation of the fractional-order multistable locally active memristive chaotic system using PSIM is carried out to verify the validity of the numerical simulation results.

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

Access this article

Log in via an institution

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
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24

Similar content being viewed by others

References

  1. Petráš, I.: A note on the fractional-order Chua’s system. Chaos, Solitons Fractals 38(1), 140–147 (2008)

  2. Ma, J., Tang, J.: A review for dynamics in neuron and neuronal network. Nonlinear Dyn. 89(3), 1569–1578 (2017)

    Article  MathSciNet  Google Scholar 

  3. Bao, H., Liu, W., Ma, J., Wu, H.: Memristor initial-Offset boosting in memristive HR neuron model with hidden firing patterns. Int. J. Bifurc. Chaos 30(10), 2030029 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Zhou, P., Yao, Z., Ma, J., Zhu, Z.: A piezoelectric sensing neuron and resonance synchronization between auditory neurons under stimulus. Chaos, Solitons Fractals 145, 110751 (2021)

    Article  MathSciNet  Google Scholar 

  5. Li, Z., Zhou, H., Wang, M., et al.: Coexisting firing patterns and phase synchronization in locally active memristor coupled neurons with HR and FN models. Nonlinear Dyn. (2021). https://doi.org/10.1007/s11071-021-06315-4

  6. Yao, W., Wang, C., Sun, Y., Zhou, C., Lin, H.: Exponential multistability of memristive Cohen–Grossberg neural networks with stochastic parameter perturbations. Appl. Math. Comput. 386, 125483 (2020)

    MathSciNet  MATH  Google Scholar 

  7. Lin, H., Wang, C., Yao, W., Tan, Y.: Chaotic dynamics in a neural network with different types of external stimuli. Commun. Nonlinear Sci. Numer. Simul. 90, 105390 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lin, H., Wang, C., Tan, Y.: Hidden extreme multistability with hyperchaos and transient chaos in a Hopfield neural network affected by electromagnetic radiation. Nonlinear Dyn. 90(3), 2369–2386 (2020)

    Article  Google Scholar 

  9. Deng, Q., Wang, C., Yang, L.: Four-wing hidden attractors with one stable equilibrium point. Int. J. Bifurc. Chaos 30(06), 2050086 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Yu, F., Qian, S., Chen, X., et al.: Chaos-based engineering applications with a 6D memristive multistable hyperchaotic system and a 2D SF-SIMM hyperchaotic map. Complexity 2021, 6683284 (2021)

    Article  Google Scholar 

  11. Wang, C., Xia, H., Zhou, L.: A memristive hyperchaotic multiscroll jerk system with controllable scroll numbers. Int. J. Bifurc. Chaos 27(06), 1750091 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Wu, R., Wang, C.: A new simple chaotic circuit based on memristor. Int. J. Bifurc. Chaos 26(09), 1650145 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cheng, G., Wang, C., Xu, C.: A novel hyper-chaotic image encryption scheme based on quantum genetic algorithm and compressive sensing. Multimed. Tools Appl. 79(39), 29243–29263 (2020)

    Article  Google Scholar 

  14. Deng, J., Zhou, M., Wang, C., Wang, S., Xu, C.: Image segmentation encryption algorithm with chaotic sequence generation participated by cipher and multi-feedback loops. Multimed. Tools Appl. (2021). https://doi.org/10.1007/s11042-020-10429-z

  15. Zeng, J., Wang, C.: A novel hyperchaotic image encryption system based on particle swarm optimization algorithm and cellular automata. Secur. Commun. Netw. 2021, 6675565 (2021)

    Article  Google Scholar 

  16. Chen, X., Qian, S., Yu, F., et al.: Pseudorandom number generator based on three kinds of four-wing memristive hyperchaotic system and its application in image encryption. Complexity 2020, 8274685 (2020)

    Article  Google Scholar 

  17. Xu, C., Sun, J., Wang, C.: An image encryption algorithm based on random walk and hyperchaotic systems. Int. J. Bifurc. Chaos 30(4), 2050060 (2020)

    Article  MathSciNet  Google Scholar 

  18. Hong, Q., Yan, R., Wang, C., Sun, J.: Memristive circuit implementation of biological nonassociative learning mechanism and its applications. IEEE Trans. Biomed. Circuits Syst. 14(5), 1036–1050 (2020)

    Article  Google Scholar 

  19. Hong, Q., Shi, Z., Sun, J., Du, S.: Memristive self-learning logic circuit with application to encoder and decoder. Neural Comput. Appl. (2020). https://doi.org/10.1007/s00521-020-05281-z

  20. Coopmans, C., Petráš, I., Chen, Y.: Analogue fractional-order generalized memristive devices. In: Proceedings of the ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, San Diego, USA (2009)

  21. Petráš, I.: Fractional-order memristor-based Chua’s circuit. IEEE Trans. Circuits. Syst. II Exp. Briefs 57(12), 975–979 (2010)

  22. Cafagna, D., Grassi, G.: On the simplest fractional-order memristor-based chaotic system. Nonlinear Dyn. 70(2), 1185–1197 (2012)

    Article  MathSciNet  Google Scholar 

  23. Teng, L., Iu, H.H., Wang, X., Wang, X.: Chaotic behavior in fractional-order memristor-based simplest chaotic circuit using fourth degree polynomial. Nonlinear Dyn. 77(1), 231–241 (2014)

    Article  Google Scholar 

  24. Si, G., Diao, L., Zhu, J.: Fractional-order charge-controlled memristor: theoretical analysis and simulation. Nonlinear Dyn. 87(4), 2625–2634 (2017)

    Article  Google Scholar 

  25. Yang, N., Xu, C., Wu, C., et al.: Fractional-order cubic nonlinear flux-controlled memristor: theoretical analysis, numerical calculation and circuit simulation. Nonlinear Dyn. 97(1), 33–44 (2019)

    Article  MATH  Google Scholar 

  26. Yu, Y., Wang, Z.: A fractional-order memristor model and the fingerprint of the simple series circuits including a fractional-order memristor. Acta Physica Sinica 64(23), 238401 (2015)

    Article  Google Scholar 

  27. Yu, Y., Wang, Z.: Initial state dependent nonsmooth bifurcations in a fractional-order memristive circuit. Int. J. Bifurc. Chaos 28(07), 1850091 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Chua, L.: If it’s pinched it’s a memristor. Semicond. Sci. Technol. 29(10), 104001 (2014)

  29. Chua, L.: Local activity is the origin of complexity. Int. J. Bifurc. Chaos 15(11), 3435–3456 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Corinto, F., Ascoli, A., Gilli, M.: Nonlinear dynamics of memristor oscillators. IEEE Trans. Circuits Syst. I Reg. Pap. 58(6), 1323–1336 (2011)

    Article  MathSciNet  Google Scholar 

  31. Gibson, G.A., Musunuru, S., Zhang, J., et al.: An accurate locally active memristor model for S-type negative differential resistance in NbOx. Phys. Lett. A 108(2), 023505 (2016)

    Article  Google Scholar 

  32. Weiher, M., Herzig, M., Tetzlaff, R., et al.: Pattern formation with locally active S-type NbOx memristors. IEEE Trans. Circuits Syst. I Reg. Pap. 66(7), 2627–2638 (2019)

    Article  Google Scholar 

  33. Chua, L.: Everything you wish to know about memristors but are afraid to ask. Radioengineering 24(2), 319–368 (2015)

    Article  Google Scholar 

  34. Mannan, Z.I., Choi, H., Kim, H.: Chua corsage memristor oscillator via hopf bifurcation. Int. J. Bifurc. Chaos 26(04), 1630009 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Jin, P., Wang, G., Iu, H.H., Fernando, T.: A locally active memristor and its application in a chaotic circuit. IEEE Trans. Circuits Syst. II Exp. Briefs 65(2), 246–250 (2017)

    Google Scholar 

  36. Chang, H., Wang, Z., Li, Y., Chen, G.: Dynamic analysis of a bistable bi-local active memristor and its associated oscillator system. Int. J. Bifurc. Chaos 28(08), 1850105 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mannan, Z.I., Yang, C., Kim, H.: Oscillation with 4-lobe Chua corsage memristor. IEEE Circuits Syst. Mag. 18(2), 14–27 (2018)

    Article  Google Scholar 

  38. Dong, Y., Wang, G., Chen, G., et al.: A bistable nonvolatile locally-active memristor and its complex dynamics. Commun. Nonlinear Sci. Numer. Simul. 84, 105203 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  39. Tan, Y., Wang, C.: A simple locally active memristor and its application in HR neurons. Chaos 30(5), 053118 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lin, H., Wang, C., Sun, Y., Yao, W.: Firing multistability in a locally active memristive neuron model. Nonlinear Dyn. 100(4), 3667–3683 (2020)

    Article  Google Scholar 

  41. Zhu, M., Wang, C., Deng, Q., Hong, Q.: Locally active memristor with three coexisting pinched hysteresis loops and its emulator circuit. Int. J. Bifurc. Chaos 30(13), 2050184 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lin, H., Wang, C., Hong, Q., Sun, Y.: A multi-stable memristor and its application in a neural network. IEEE Trans. Circuits Syst. II Exp. Briefs 67(12), 3472–3476 (2020)

    Google Scholar 

  43. Gorenflo, R., Mainardi, F.: Fractional Calculus. Fractals and Fractional Calculus in Continuum Mechanics. Springer, Vienna (1997)

    MATH  Google Scholar 

  44. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  45. Kengne, J., Njitacke, Z.T., Fotsin, H.: Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn. 83(1), 751–765 (2016)

    Article  MathSciNet  Google Scholar 

  46. Ahmed, E., El-Sayed, A., El-Saka, H.: Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J. Math. Anal. Appl. 325(1), 542–553 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  47. El-Saka, H., Ahmed, E., Shehata, M., El-Sayed, A.: On stability, persistence, and Hopf bifurcation in fractional order dynamical systems. Nonlinear Dyn. 56(1), 121–126 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. Tavazoei, M., Haeri, M.: A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica 45(8), 1886–1890 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  49. Kaslik, E., Sivasundaram, S.: Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions. Nonlinear Anal. Real World Appl. 13(3), 1489–1497 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  50. Danca, M.-F., Fečkan, M., Kuznetsov, N.V., Chen, G.: Complex dynamics, hidden attractors and continuous approximation of a fractional-order hyperchaotic PWC system. Nonlinear Dyn. 91(4), 2523–2540 (2018)

    Article  MATH  Google Scholar 

  51. Danca, M.-F., Fečkan, M., Chen, G.: Impulsive stabilization of chaos in fractional-order systems. Nonlinear Dyn. 89(3), 1889–1903 (2017)

    Article  MathSciNet  Google Scholar 

  52. Kang, Y.-M., Xie, Y., Lu, J.-C., Jiang, J.: On the nonexistence of non-constant exact periodic solutions in a class of the Caputo fractional-order dynamical systems. Nonlinear Dyn. 82(3), 1259–1267 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  53. Danca, M.-F., Kuznetsov, N.V.: Matlab code for Lyapunov exponents of fractional-order systems. Int. J. Bifurc. Chaos 28(5), 1850067 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  54. 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 

  55. Li, C., Sprott, J.C.: Multistability in the Lorenz system:a broken butterfly. Int. J. Bifurc. Chaos 24(10), 1450131 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  56. Yu, Y., Bao, H., Shi, M., et al.: Complex dynamical behaviors of a fractional-order system based on a locally active memristor. Complexity 2019, 2051053 (2019)

    Article  MATH  Google Scholar 

  57. Wu, J., Wang, G., Iu, H.H., et al.: A nonvolatile fractional order memristor model and its complex dynamics. Entropy 21(10), 955 (2019)

    Article  MathSciNet  Google Scholar 

  58. Pham, V.T., Kingni, S.T., et al.: A simple three-dimensional fractional-order chaotic system without equilibrium: Dynamics, circuitry implementation, chaos control and synchronization. AEU Int. J. Electron. Commun. 78, 220–227 (2017)

    Article  Google Scholar 

  59. Min, F., Shao, S., Huang, W., Wang, E.: Circuit implementations, bifurcations and chaos of a novel fractional-order dynamical system. Chin. Phys. Lett. 32(3), 030503 (2015)

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by the Major Research Plan of the National Natural Science Foundation of China (No.91964108), the National Natural Science Foundation of China (No.61971185) and Natural Science Foundation of Hunan Province (2020JJ4218)

Author information

Authors and Affiliations

  1. College of Computer Science and Electronic Engineering Hunan University, Changsha, 410082, China

    Wenli Xie, Chunhua Wang & Hairong Lin

Authors
  1. Wenli Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunhua Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hairong Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chunhua Wang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflicts 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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xie, W., Wang, C. & Lin, H. A fractional-order multistable locally active memristor and its chaotic system with transient transition, state jump. Nonlinear Dyn 104, 4523–4541 (2021). https://doi.org/10.1007/s11071-021-06476-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06476-2

Keywords

Search

Navigation