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Effects of Variable-Order Passive Circuit Element in Chua Circuit

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Abstract

In this paper, the behaviour of a variable-order passive circuit element which is used in Chua chaotic circuit is analysed. Firstly, the behaviour of a circuit with a variable-order memristor is presented. In the generalized Ohm’s law for a memory element, the order of a passive circuit element is defined as a kind of the element, so the circuit shows unusual behaviour. Finally, the chaotic circuit is designed with a variable-order memristor, and the effect of the variable-order element is shown in the Chua circuit. The chaotic circuit model with the novel memristor shows limit-cycle behaviour.

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References

  1. M.-S. Abdelouahab, R. Lozi, L. Chua, Memfractance: a mathematical paradigm for circuit elements with memory. Int. J. Bifurc. Chaos 24, 1430023 (2014)

    Article  Google Scholar 

  2. O. Atan, D. Chen, M. Türk, Fractional order PID and application of its circuit model. J. Chin. Inst. Eng. 2016, 1–9 (2016)

    Google Scholar 

  3. O. Atan, Implementation and simulation of fractional order chaotic circuits with time-delay. Analog. Integr. Circuits Signal Process. 96, 485–494 (2018)

    Article  Google Scholar 

  4. B.-C. Bao, J.-P. Xu, G.-H. Zhou, Z.-H. Ma, L. Zou, Chaotic memristive circuit: equivalent circuit realization and dynamical analysis. Chin. Phys. B 20, 120502 (2011)

    Article  Google Scholar 

  5. D. Biolek, Z. Biolek, V. Biolkova, SPICE modeling of memristive, memcapacitative and meminductive systems. IEEE Eur. Conf. Circuit Theory Des. 2009, 249–252 (2009)

    MATH  Google Scholar 

  6. M. Borah, B.K. Roy, Dynamics of the fractional-order chaotic PMSG, its stabilisation using predictive control and circuit validation. IET Electr. Power Appl. 11, 707–716 (2017)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. D. Cafagna, G. Grassi, Elegant Chaos in fractional-order system without equilibria. Math. Probl. Eng. 2013, 1–7 (2013)

    Article  MathSciNet  Google Scholar 

  9. D. Cafagna, G. Grassi, Chaos in a new fractional-order system without equilibrium points. Commun. Nonlinear Sci. Numer. Simul. 19, 2919–2927 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. L. Chua, Device modeling via nonlinear circuit elements. IEEE Trans. Circuits Syst. 27, 1014–1044 (1980)

    Article  MathSciNet  Google Scholar 

  13. L. Chua, Nonlinear circuit foundations for nanodevices, part I: the four-element torus. Proc. IEEE 9, 1830–1859 (2003)

    Article  Google Scholar 

  14. S. Jameel, C. Korasli, A. Nacaroglu, Realization of biquadratic filter by using memristor. 2013 IEEE Int. Conf. Technol. Adv. Electr. Electron. Comput. Eng. 2013, 52–56 (2013)

    Google Scholar 

  15. H.Y. Jia, Z.Q. Chen, G.Y. Qi, Chaotic characteristics analysis and circuit implementation for a fractional-order system. IEEE Trans. Circuits Syst. I Regul. Pap. 61, 845–53 (2014)

    Article  Google Scholar 

  16. H. Kim, M.P. Sah, C. Yang, S. Cho, L.O. Chua, Memristor emulator for memristor circuit applications. IEEE Trans. Circuits Syst. I Regul. Pap. 59, 2422–2431 (2012)

    Article  MathSciNet  Google Scholar 

  17. L. Li, Z.J. Chew, Printed circuit board based memristor in adaptive lowpass filter. Electron. Lett. 48, 1610–1611 (2012)

    Article  Google Scholar 

  18. A.E. Matouk, Chaos, feedback control and synchronization of a fractional-order modified Autonomous Van der Pol–Duffing circuit. Commun. Nonlinear Sci. Numer. Simul. 16, 975–986 (2011)

    Article  MathSciNet  Google Scholar 

  19. T. Matsumoto, A chaotic attractor from Chua’s circuit. IEEE Trans. Circuits Syst. 31, 1055–1058 (1984)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  22. V.-T. Pham, A. Ouannas, C. Volos, T. Kapitaniak, A simple fractional-order chaotic system without equilibrium and its synchronization. AEU Int. J. Electron. Commun. 86, 69–76 (2018)

    Article  Google Scholar 

  23. Y.-F. Pu, X. Yuan, Fracmemristor: fractional-order memristor. IEEE Access 4, 1872–1888 (2016)

    Article  Google Scholar 

  24. A.G. Radwan, K.N. Salama, Fractional-order RC and RL circuits. Circuits Syst. Signal Process. 31, 1901–1915 (2012)

    Article  MathSciNet  Google Scholar 

  25. C. Sánchez-López, M.A. Carrasco-Aguilar, C. Muñiz-Montero, A 16Hz–160kHz memristor emulator circuit. AEU Int. J. Electron. Commun. 69, 1208–1219 (2015)

    Article  Google Scholar 

  26. D. Sierociuk, W. Malesza, M. Macias, Derivation, interpretation, and analog modelling of fractional variable order derivative definition. Appl. Math. Model. 39, 3876–3888 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  28. D. Tavares, R. Almeida, D.F.M. Torres, Caputo derivatives of fractional variable order: numerical approximations. Commun. Nonlinear Sci. Numer. Simul. 35, 69–87 (2016)

    Article  MathSciNet  Google Scholar 

  29. M.S. Tavazoei, M. Haeri, Chaotic attractors in incommensurate fractional order systems. Phys. D Nonlinear Phenom. 237, 2628–2637 (2008)

    Article  MathSciNet  Google Scholar 

  30. M.S. Tavazoei, M. Haeri, A necessary condition for double scroll attractor existence in fractional-order systems. Phys. Lett. A 367, 102–113 (2007)

    Article  Google Scholar 

  31. A. Thomas, Memristor-based neural networks. J. Phys. D Appl. Phys. 46, 093001 (2013)

    Article  Google Scholar 

  32. X.-B. Tian, H. Xu, The design and simulation of a titanium oxide memristor-based programmable analog filter in a simulation program with integrated circuit emphasis. Chin. Phys. B 22, 088501 (2013)

    Article  Google Scholar 

  33. G. Tsirimokou, C. Psychalinos, A. Elwakil, Design of CMOS Analog Integrated Fractional-Order Circuits (Springer, Cham, 2017), pp. 41–54

    Book  Google Scholar 

  34. A. Wu, S. Wen, Z. Zeng, Synchronization control of a class of memristor-based recurrent neural networks. Inf. Sci. (Ny) 183, 106–116 (2012)

    Article  MathSciNet  Google Scholar 

  35. A. Yeşil, Y. Babacan, F. Kaçar, A new DDCC based memristor emulator circuit and its applications. Microelectron. J. 45, 282–287 (2014)

    Article  Google Scholar 

  36. G. Zhang, Y. Shen, Exponential synchronization of delayed memristor-based chaotic neural networks via periodically intermittent control. Neural Netw. 55, 1–10 (2014)

    Article  Google Scholar 

  37. X. Zhang, Y. Qi, Design of an assemble-type fractional-order unit circuit and its application in Lorenz system. IET Circuits Devices Syst. 11, 437–445 (2017)

    Article  Google Scholar 

  38. X. Zhao, Z. Sun, G.E. Karniadakis, Second-order approximations for variable order fractional derivatives: algorithms and applications. J. Comput. Phys. 293, 184–200 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I would like to thank the reviewers and editors for their thoughtful comments and efforts towards improving our manuscript.

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  1. Department of Electrical and Electronics Engineering, Van Yüzüncü Yıl Üniversity, Van, Turkey

    Özkan Atan

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Appendix

Appendix

Theorem

A system is asymptotically stable if all eigenvalues have negative real parts. For chaotic behaviour, at least one eigenvalue of the Jacobian matrix is in the unstable region as follows:

$$\begin{aligned} \left| \arg (eig(J)) \right| =\left| \arg ({{\lambda }_{k}}) \right| <\alpha \frac{\pi }{2}\quad k=1,2,3\ldots ,n. \end{aligned}$$

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Atan, Ö. Effects of Variable-Order Passive Circuit Element in Chua Circuit. Circuits Syst Signal Process 39, 2293–2306 (2020). https://doi.org/10.1007/s00034-019-01271-2

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