Skip to main content
SpringerLink
Log in

Chaotic system dynamics analysis and synchronization circuit realization of fractional-order memristor

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

This paper studies the dynamic analysis, circuit implementation and circuit synchronization of a class of fractional-order memristor chaotic system based on Chua’s circuit. First, the dynamic characteristics of the system was analyzed with the common chaotic description methods, such as Lyapunov exponent spectrum(LEs), bifurcation diagram, Poincaré diagram, phase trajectory diagram and Complexity spectral entropy (Complexity SE). It can be found that the small change of the initial value can make the dynamic behavior of the system change between the periodic window and the chaotic region. When fractional order q and system parameters are used as a variable, the dynamic behavior of the system can be changed. Second, the system circuit model was modeled and simulated, and then the consistency with the numerical simulation was verified through the implementation of the circuit. Finally, through the design of fractional-order memristor circuit, the circuit synchronization of the system was realized. Therefore, this paper not only studies the dynamic behavior of the system through numerical analysis. Moreover, the effectiveness of the system design is further illustrated by the implementation of analog circuit.

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

Similar content being viewed by others

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article. All the program data included in this manuscript are available upon request by contacting with the corresponding author.

References

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

    Article  Google Scholar 

  2. M. Itoh, L.O. Chua, Memristor oscillators. Int. J. Bifurc. Chaos 18(11), 3183–3206 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. B. Muthuswamy, Implementing memristor based chaotic circuits. Int. J. Bifurc. Chaos 20(05), 1335–1350 (2010)

    Article  MATH  Google Scholar 

  4. M. Yildirim, Dna encoding for rgb image encryption with memristor based neuron model and chaos phenomenon. Microelectron. J. 104, 104878 (2020)

    Article  Google Scholar 

  5. J. Sun, Q. Yang, Y. Wang, Dynamical analysis of novel memristor chaotic system and DNA encryption application. Iran. J. Sci. Technol. Trans. Electr. Eng. 44(1), 449–460 (2020)

    Article  Google Scholar 

  6. F. Yuan, G. Wang, X. Wang, Dynamical characteristics of an HP memristor based on an equivalent circuit model in a chaotic oscillator. Chin. Phys. B 24(6), 060506 (2015)

    Article  ADS  Google Scholar 

  7. A. Buscarino, L. Fortuna, M. Frasca, L.V. Gambuzza, A chaotic circuit based on Hewlett-Packard memristor. Chaos 22(2), 023136 (2012)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  8. Y. Yang, L. Wang, S. Duan, L. Luo, Dynamical analysis and image encryption application of a novel memristive hyperchaotic system. Opt. Laser Technol. 133, 106553 (2021)

    Article  Google Scholar 

  9. S. Wen, Z. Zeng, T. Huang, Adaptive synchronization of memristor-based CHUA-+s circuits. Phys. Lett. A 376(44), 2775–2780 (2012)

    Article  MATH  ADS  Google Scholar 

  10. Z. Li, Y. Zeng, A memristor oscillator based on a twin-t network. Chin. Phys. B 22(4), 040502 (2013)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  MATH  ADS  Google Scholar 

  12. P. Jin, G. Wang, H.H.-C. Iu, T. Fernando, 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Hadi, O. Orozco-López, J.M. Munoz-Pacheco, N.D. Alotaibi, C. Volos, Z. Wang, R. Sevilla-Escoboza, C. Yu-Ming, Simulation and experimental validation of a non-equilibrium chaotic system. Chaos Solitons Fractals 143, 110539 (2021)

    Article  MathSciNet  Google Scholar 

  15. Z. Wang, Z. Wei, K. Sun, S. He, H. Wang, Q. Xu, M. Chen, Chaotic flows with special equilibria. Eur Phys J Spec Top 229(6), 905–919 (2020)

    Article  Google Scholar 

  16. S. Zhou, X. Wang, Z. Wang, C. Zhang, A novel method based on the pseudo-orbits to calculate the largest lyapunov exponent from chaotic equations. Chaos 29(3), 033125 (2019)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  17. Z. Wang, I. Moroz, Z. Wei, H. Ren, Dynamics at infinity and a HOPF bifurcation arising in a quadratic system with coexisting attractors. Pramana 90(1), 1–10 (2018)

    Article  Google Scholar 

  18. D. Peng, K. Sun, S. He, A.O.A. Alamodi, What is the lowest order of the fractional-order chaotic systems to behave chaotically? Chaos Solitons Fractals 119, 163–170 (2019)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  19. H. Wang, K. Sun, S. He, Characteristic analysis and DSP realization of fractional-order simplified Lorenz system based on adomian decomposition method. Int J. Bifurc. Chaos 25(06), 1550085 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. H. Liang, Z. Wang, Z. Yue, R. Lu, Generalized synchronization and control for incommensurate fractional unified chaotic system and applications in secure communication. Kybernetika 48(2), 190–205 (2012)

    MathSciNet  MATH  Google Scholar 

  21. A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series. Physica D 16(3), 285–317 (1985)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  22. T. Liu, H. Yan, S. Banerjee, J. Mou, A fractional-order chaotic system with hidden attractor and self-excited attractor and its DSP implementation. Chaos Solitons Fractals 145, 110791 (2021)

    Article  MathSciNet  Google Scholar 

  23. A. Akgül, K. Rajagopal, A. Durdu, M.A. Pala, Ö.F. Boyraz, M.Z. Yildiz, A simple fractional-order chaotic system based on memristor and memcapacitor and its synchronization application. Chaos Solitons Fractals 152, 111306 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. C. Ma, J. Mou, J. Liu, F. Yang, H. Yan, X. Zhao, Coexistence of multiple attractors for an incommensurate fractional-order chaotic system. Eur. Phys. J. Plus 135(1), 1–21 (2020)

    Article  Google Scholar 

  25. Z. Hammouch, T. Mekkaoui, Circuit design and simulation for the fractional-order chaotic behavior in a new dynamical system. Complex Intell. Syst. 4(4), 251–260 (2018)

    Article  MATH  Google Scholar 

  26. W. Yu, Y. Luo, Y.Q. Chen, Y.G. Pi, Frequency domain modelling and control of fractional-order system for permanent magnet synchronous motor velocity servo system. IET Control Theory Appl. 10(2), 136–143 (2016)

    Article  MathSciNet  Google Scholar 

  27. F. Ozkaynak, A novel random number generator based on fractional order chaotic chua system. Elektronika ir Elektrotechnika 26(1), 52–57 (2020)

    Article  Google Scholar 

  28. J. Hao, H. Li, H. Yan, J. Mou, A new fractional chaotic system and its application in image encryption with DNA mutation. IEEE Access 9, 52364–52377 (2021)

    Article  Google Scholar 

  29. G. Wu, Z. Deng, D. Baleanu, D. Zeng, New variable-order fractional chaotic systems for fast image encryption. Chaos 29(8), 083103 (2019)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  30. Z. Li, D. Chen, J. Zhu, Y. Liu, Nonlinear dynamics of fractional order duffing system. Chaos Solitons Fractals 81, 111–116 (2015)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  31. N. Yang, C. Xu, C. Wu, R. Jia, C. Liu, Modeling and analysis of a fractional-order generalized memristor-based chaotic system and circuit implementation. Int. J. Bifurc. Chaos 27(13), 1750199 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Ouannas, A. Aicha Khennaoui, X. Wang, V.-T. Pham, S. Boulaaras, S. Momani, Bifurcation and chaos in the fractional form of hénon-lozi type map. Eur. Phys. J. Spec. Top. 229(12), 2261–2273 (2020)

    Article  Google Scholar 

  34. A. Ouannas, A. Aicha Khennaoui, T.-E. Oussaeif, V.-T. Pham, G. Grassi, Z. Dibi, Hyperchaotic fractional Grassi–Miller map and its hardware implementation. Integration 80, 13–19 (2021)

    Article  Google Scholar 

  35. A. Aicha Khennaoui, A. Ouannas, S. Boulaaras, V.-T. Pham, A. Taher Azar, A fractional map with hidden attractors: chaos and control. Eur. Phys. J. Spec. Top. 229(6), 1083–1093 (2020)

    Article  Google Scholar 

  36. D. Ding, Y. Weng, N. Wang et al., Dynamics analysis of a fractional-order delayed SBT memristive chaotic system without equilibrium points. Eur. Phys. J. Plus 134(9), 444 (2019)

    Article  Google Scholar 

  37. S. Zhang, J. Zheng, X. Wang, Z. Zeng, X. Peng, A novel nonideal flux-controlled memristor model for generating arbitrary multi-double-scroll and multi-double-wing attractors. Int. J. Bifurc. Chaos 31(06), 2150086 (2021)

    Article  MathSciNet  Google Scholar 

  38. M.A. Khan, The dynamics of a new chaotic system through the Caputo-Fabrizio and Atanagan-Baleanu fractional operators. Adv. Mech. Eng. 11(7), 1687814019866540 (2019)

    Article  Google Scholar 

  39. D. Baleanu, H. Mohammadi, S. Rezapour, A fractional differential equation model for the COVID-19 transmission by using the Caputo-Fabrizio derivative. Adv. Differ. Equ. 2020(1), 1–27 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  40. A.-M. Wazwaz, A reliable modification of adomian decomposition method. Appl. Math. Comput. 102(1), 77–86 (1999)

    MathSciNet  MATH  Google Scholar 

  41. H. Li, X. Liao, M. Luo, A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation. Nonlinear Dyn. 68(1), 137–149 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. H. Tian, Z. Wang, P. Zhang, M. Chen, Y. Wang, Dynamic analysis and robust control of a chaotic system with hidden attractor. Complexity (2021). https://doi.org/10.1155/2021/8865522

    Article  Google Scholar 

  43. Z. Wang, W. Sun, Z. Wei, S. Zhang, Dynamics and delayed feedback control for a 3D jerk system with hidden attractor. Nonlinear Dyn. 82(1), 577–588 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Z. Wang, F. Parastesh, K. Rajagopal, I.I. Hamarash, I. Hussain, Delay-induced synchronization in two coupled chaotic memristive hopfield neural networks. Chaos Solitons Fractals 134, 109702 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author acknowledges the referees and the editor for carefully reading this paper and giving many helpful comments. This work is supported by the Natural Science Basic Research Program of Shaanxi (2021JM-533, 2021JQ-880, 2020JM-646, 2022JM-029), the Innovation Capability Support Program of Shaanxi (2018GHJD-21), the Science and Technology Program of Xi’an (2019218414GXRC020CG021-GXYD20.3), the Support Plan for Sanqin Scholars Innovation Team in Shaanxi Province of China, the Scientific Research Program Funded by Shaanxi Provincial Education Department (21JK0960), the Youth Innovation Team of Shaanxi Universities, the Scientific Research Foundation of Xijing University (XJ21B01), and the Scientific Research Foundation of Xijing University (XJ210203, XJ200202). The authors also express their gratitude to the reviewers for their insightful comments.

Author information

Authors and Affiliations

  1. Xi’an Key Laboratory of Human–Machine Integration and Control Technology for Intelligent Rehabilitation, School of Computer Science, Xijing University, Xi’an, 710123, People’s Republic of China

    Jindong Liu, Zhen Wang, Mingshu Chen, Peijun Zhang, Rui Yang & Baonan Yang

  2. Xi’an Key Laboratory of Advanced Photo-Electronics Materials and Energy Conversion Device, School of Electronic Information, Xijing University, Xi’an, 710123, People’s Republic of China

    Zhen Wang

Authors
  1. Jindong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mingshu Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Peijun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Rui Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Baonan Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhen Wang.

Ethics declarations

Conflict of interest

All authors declare that they have no conflicts of interests about the publication of this paper.

Appendix A

Appendix A

The iterative coefficients of the numerical solution of the system are as follows:

\(c_{1}^{1}=ac_{1}^{0}+bc_{2}^{0}-cc_{1}^{0}\left( c_{4}^{0} \right) ^2\)

\(c_{2}^{1}=c_{1}^{0}-c_{2}^{0}+c_{3}^{0}\)

\({c_{3}^{1}=dc_{2}^{0}+ec_{3}^{0}}\)

\({ c_{4}^{1}=c_{1}^{0}}\)

\({c_{1}^{2}=ac_{1}^{1}+bc_{2}^{1}-2cc_{1}^{0}c_{4}^{0}c_{4}^{1}-cc_{1}^{1}\left( c_{4}^{0} \right) ^2}\)

\({ c_{2}^{2}=c_{1}^{1}-c_{2}^{1}+c_{3}^{1}}\)

\({c_{3}^{2}=dc_{2}^{1}+ec_{3}^{1}}\)

\({c_{4}^{2}=c_{1}^{1}}\)

\(\begin{aligned}c_{1}^{3}&=ac_{1}^{2}+bc_{2}^{2}-2cc_{1}^{0}c_{4}^{0}c_{4}^{2}-cc_{1}^{2}\left( c_{4}^{0} \right) ^2 \\ & \quad -\left( cc_{1}^{0}\left( c_{4}^{1} \right) ^2-2cc_{1}^{1}c_{4}^{0}c_{4}^{1} \right) \frac{\Gamma \left( 2q+1 \right) }{\Gamma ^2\left( q+1 \right) }\end{aligned}\)

\({c_{2}^{3}=c_{1}^{2}-c_{2}^{2}+c_{3}^{2}}\)

\({c_{3}^{3}=dc_{2}^{2}+ec_{3}^{2}}\)

\({c_{4}^{3}=c_{1}^{2}}\)

\(\begin{aligned}c_{1}^{4}&=ac_{1}^{3}+bc_{2}^{3}-2cc_{1}^{0}c_{4}^{0}c_{4}^{3}-cc_{1}^{3}\left( c_{4}^{0} \right) ^2 \\ &\quad -\left( 2cc_{1}^{0}c_{4}^{1}c_{4}^{2}+2cc_{1}^{1}c_{4}^{0}c_{4}^{2}+2cc_{1}^{2}c_{4}^{0}c_{4}^{1} \right) \\ &\quad \frac{\Gamma \left( 3q+1 \right) }{\Gamma \left( q+1 \right) \Gamma \left( 2q+1 \right) }\end{aligned}\)

\({-cc_{1}^{1}c_{4}^{1}c_{4}^{1}\frac{\Gamma \left( 3q+1 \right) }{\Gamma ^3\left( q+1 \right) }}\)

\({c_{2}^{4}=c_{1}^{3}-c_{2}^{3}+c_{3}^{3}}\)

\({c_{3}^{4}=dc_{2}^{3}+ec_{3}^{3}}\)

\({c_{4}^{4}=c_{1}^{3}}\)

\(\begin{aligned}c_{1}^{5}&=ac_{1}^{4}+bc_{2}^{4}-2cc_{1}^{0}c_{4}^{0}c_{4}^{4}-cc_{1}^{4}\left( c_{4}^{0} \right) ^2 \\ &\quad -\left( 2cc_{1}^{0}c_{4}^{1}c_{4}^{3}+2cc_{1}^{1}c_{4}^{0}c_{4}^{3}+2cc_{1}^{3}c_{4}^{0}c_{4}^{1} \right) \\ &\quad \frac{\Gamma \left( 4q+1 \right) }{\Gamma \left( q+1 \right) \Gamma \left( 3q+1 \right) }\end{aligned}\)

\(\begin{aligned}&\quad -\left( cc_{1}^{0}c_{4}^{2}c_{4}^{2}+2cc_{1}^{0}c_{4}^{2}c_{4}^{2} \right) \frac{\Gamma \left( 4q+1 \right) }{\Gamma ^2\left( 2q+1 \right) } \\ &\quad -\left( 2cc_{1}^{1}c_{4}^{1}c_{4}^{2}+2cc_{1}^{2}c_{4}^{1}c_{4}^{1} \right) \frac{\Gamma \left( 4q+1 \right) }{\Gamma \left( 2q+1 \right) \Gamma \left( 2q+1 \right) }\end{aligned}\)

\({c_{2}^{5}=c_{1}^{4}-c_{2}^{4}+c_{3}^{4}}\)

\({c_{3}^{5}=dc_{2}^{4}+ec_{3}^{4}}\)

\({c_{4}^{5}=c_{1}^{4}}\)

\(\begin{aligned}c_{1}^{6}&=ac_{1}^{5}+bc_{2}^{5}-2cc_{1}^{0}c_{4}^{0}c_{4}^{5}-cc_{1}^{5}\left( c_{4}^{0} \right) ^2 \\ &\quad -\left( 2cc_{1}^{0}c_{4}^{1}c_{4}^{4}+2cc_{1}^{1}c_{4}^{0}c_{4}^{4}+2cc_{1}^{4}c_{4}^{0}c_{4}^{1} \right)\\ &\quad \frac{\Gamma \left( 5q+1 \right) }{\Gamma \left( q+1 \right) \Gamma \left( 4q+1 \right) }-\end{aligned}\)

\(\begin{aligned} &\quad \left( 2cc_{1}^{0}c_{4}^{2}c_{4}^{3}+2cc_{1}^{2}c_{4}^{0}c_{4}^{3}+2cc_{1}^{3}c_{4}^{0}c_{4}^{2} \right) \\ &\quad \frac{\Gamma \left( 5q+1 \right) }{\Gamma \left( 3q+1 \right) \Gamma \left( 2q+1 \right) } \\ &\quad -\left( 2cc_{1}^{1}c_{4}^{1}c_{4}^{3}+cc_{1}^{3}c_{4}^{1}c_{4}^{1} \right) \end{aligned}\)

\({-\left( 2cc_{1}^{2}c_{4}^{1}c_{4}^{2}+cc_{1}^{1}c_{4}^{2}c_{4}^{2} \right) \frac{\Gamma \left( 5q+1 \right) }{\Gamma \left( q+1 \right) \Gamma ^2\left( 2q+1 \right) }}\)

\({ c_{2}^{6}=c_{1}^{5}-c_{2}^{5}+c_{3}^{5}}\)

\({c_{3}^{6}=dc_{2}^{5}+ec_{3}^{5}}\)

\({c_{4}^{6}=c_{1}^{5}}\)

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, J., Wang, Z., Chen, M. et al. Chaotic system dynamics analysis and synchronization circuit realization of fractional-order memristor. Eur. Phys. J. Spec. Top. 231, 3095–3107 (2022). https://doi.org/10.1140/epjs/s11734-022-00640-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjs/s11734-022-00640-4

Search

Navigation