Skip to main content
SpringerLink
Log in

Non-Smooth Bifurcation in Two Fractional-Order Memristive Circuits

  • Conference paper
  • First Online:
New Trends in Nonlinear Dynamics

  • 593 Accesses

Abstract

Two fractional-order Chua’s memristive circuits named Model 1 and Model 2 are proposed. Model 1 is a fractional-order memristive circuit with only the memristor described by a fractional-order derivative due to the memory loss observed experimentally, while Model 2 is a direct fractional-order generalization of integer-order Chua’s memristive circuit without considering the physical background. Both models are non-smooth systems with a line equilibrium depending on the memristor’s initial state. Numerical simulation shows that both models exhibit multi-stability and different steady states switch via “grazing bifurcation” or “tangent bifurcation,” “intermittent chaos” is found in Model 1 as the fractional order is close to 0 or 1, but no “intermittent chaos” is found in Model 2 as the fractional order is between 0 and 1.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

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

    Article  ADS  Google Scholar 

  2. Chua, L.O.: Memristor-the missing circuit element. IEEE Trans. Circuit Theory 18(5), 507–519 (1971)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Iton M, Chua L.O.: Duality of memristor circuits. Int. J. Bifurcation Chaos 23, 1330001 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  5. Pham, V.T., Volos, C., Jafari, S., Wang, X., Vaidynathan, S.: Hidden hyperchaotic attractor in a novel simple memristive neural network. J. Optoelectron. Adv. Mater. 8, 11–12 (2014).

    Google Scholar 

  6. Bao, B.C., Bao, H., Wang, N., Chen, M., Xu, Q.: Hidden extreme multistability in memristive hyperchaotic system. Chaos, Solitons Fractals 94, 102–111 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  7. Njitacke, Z.T., Kengne, J., Fotsin, H.B., Negou, A.N. Tchiotsop, D.: Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bridge-based Jerk circuit. Chaos, Solitons Fractals 91, 180–197 (2016)

    Article  ADS  Google Scholar 

  8. Li, Q.D., Zeng, H.Z. Li, J.: Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria. Nonlinear Dyn. 79, 2295–2308 (2015)

    Article  MathSciNet  Google Scholar 

  9. Dutta, M., Nusse, H.E., Ott, E., Yorke, J.A., Yuan, G.: Multiple attractor bifurcations: a source of unpredictability in piecewise smooth systems. Phys. Rev. Lett. 83, 4281–4284 (1999)

    Article  ADS  Google Scholar 

  10. Carroll, T., Pecora, L.: Using multiple attractor chaotic systems for communication. Chaos: Interdiscip. J. Nonlinear Sci. 9, 445–451 (1999)

    Article  Google Scholar 

  11. Ascoli, A., Tetzlaff, R., Menzel, S.: Exploring the dynamics of real-world memristors on the basis of circuit theoretic model predictions. IEEE Circuits Syst. Mag. 18, 48–76 (2018)

    Article  Google Scholar 

  12. Yu, Y.J., Wang, Z.H.: A fractional-order memristor model and the fingerprint of the simple series circuits including a fractional-order memristor. Acta Phys. Sin. 64, 0238401 (2015) (in Chinese)

    Google Scholar 

  13. Fouda, M.E., Radwan, A.G.: On the fractional-order memristor model. JFCA 4, 1–7 (2013)

    Google Scholar 

  14. Fouda, M.E., Radwan, A.G.: Fractional-order memristor response under DC and periodic signals. Circuits Syst. Signal Process. 34, 961–970 (2015)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Yang, N.N., Xu, C., Wu, C.J., Jia, R., Liu, C.X.: Modeling and analysis of a fractional-order generalized memristor-based chaotic system and circuit implementation. Int. J. Bifurcation Chaos 27, 1750199 (2017)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  18. Bao, B.C., Xu, Q., Bao, H., Chen, M.: Extreme multistability in a memristive circuit. Electron. Lett. 52, 1008–1010 (2016)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  20. 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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  23. Matignon, D.: Stability properties for generalized fractional differential systems. ESAIM Proc. 5, 145–158 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

This work was supported by Natural Science Foundation of China under Grants 11602035 and 11372354.

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Changzhou University, Changzhou, China

    Yajuan Yu

  2. State Key Lab of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, China

    Zaihua Wang

Authors
  1. Yajuan Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zaihua Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zaihua Wang .

Editor information

Editors and Affiliations

  1. Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Rome, Italy

    Walter Lacarbonara

  2. Department of Mechanical Engineering, University of Maryland, College Park, MD, USA

    Balakumar Balachandran

  3. Department of Physics, Lanzhou University of Technology, Lanzhou, Gansu, China

    Jun Ma

  4. Department of Electrical Engineering, Polytechnic of Porto - School of Engineering (ISEP), Porto, Portugal

    J. A. Tenreiro Machado

  5. Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest, Hungary

    Gabor Stepan

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Yu, Y., Wang, Z. (2020). Non-Smooth Bifurcation in Two Fractional-Order Memristive Circuits. In: Lacarbonara, W., Balachandran, B., Ma, J., Tenreiro Machado, J., Stepan, G. (eds) New Trends in Nonlinear Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-030-34724-6_33

Download citation

Publish with us

Policies and ethics

Search

Navigation