Abstract
The design and analysis of a simple autonomous memristive chaotic circuit are important in theoretical, numerical, and experimental demonstrations of complex dynamics. In this paper, a simple autonomous memristive circuit is implemented, which only consists of an active second-order memristive diode bridge and a capacitor. Based on the available circuit, the mathematical model is established and its symmetry, dissipativity, and equilibrium stability are analyzed. Numerical simulations show that the proposed circuit exhibits complex behaviors of unipolar periodic and chaotic bursting oscillations along with coexisting attractors. It is worth noting that the circuit exhibits such a special bursting behavior previously unobserved in third-order autonomous memristive circuits. Moreover, spectral entropy complexities are calculated to provide an intuitive and effectual method for the circuit parameter configurations. The circuit simulations and hardware experiments verify the theoretical analyses and numerical simulations.
Similar content being viewed by others
References
Vaidyanathan, S., Volos, C.: Advances and Applications in Chaotic Systems. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-30279-9
Wang, X.Y., Teng, L., Zhou, Y.C.: A novel colour image encryption algorithm based on chaos. Signal Process. 92(4), 1101–1108 (2012)
Wang, X.Y., Zhang, H.L.: A novel image encryption algorithm based on genetic recombination and hyper-chaotic systems. Nonlinear Dyn. 83(1–2), 333–346 (2016)
Hua, Z.Y., Zhou, B.H., Zhou, Y.C.: Sine chaotification model for enhancing chaos and its hardware implementation. IEEE Trans. Ind. Electron. 66(2), 1273–1284 (2019)
Jafari, S., Sprott, J.C., Mohammad Reza Hashemi Golpayegani, S.: Elementary quadratic chaotic flows with no equilibria. Phys. Lett. A 377(9), 699–702 (2013)
Wang, X., Chen, G.R.: A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17(3), 1264–1272 (2012)
Molaie, M., Jafari, S., Sprott, J.C.: Simple chaotic flows with one stable equilibrium. Int. J. Bifurc. Chaos 23(11), 1350188 (2013)
Jafari, S., Sprott, J.C.: Simple chaotic flows with a line equilibrium. Chaos Solitons Fractals 42(4), 2246–2253 (2013)
Barati, K., Jafari, S., Sprott, J.C., Pham, V.T.: Simple chaotic flows with a curve of equilibria. Int. J. Bifurc. Chaos 26(12), 1630034 (2016)
Gotthans, T., Sprott, J.C., Petrzela, J.: Simple chaotic flow with circle and square equilibrium. Int. J. Bifurc. Chaos 26(08), 1650137 (2016)
Jafari, S., Sprott, J.C., Molaie, M.: A simple chaotic flow with a plane of equilibria. Int. J. Bifurc. Chaos 26(06), 1650098 (2016)
Li, C.B., Sprott, J.C., Xing, H.Y.: Constructing chaotic systems with conditional symmetry. Nonlinear Dyn. 87(2), 1351–1358 (2016)
Li, C.B., Sprott, J.C.: Variable-boostable chaotic flows. Optik 127(22), 10389–10398 (2016)
Li, C.B., Wang, X., Chen, G.R.: Diagnosing multistability by offset boosting. Nonlinear Dyn. 90(2), 1335–1341 (2017)
Murali, K., Lakshmanan, M., Chua, L.O.: The simplest dissipative nonautonomous chaotic circuit. IEEE Trans. Circuits Syst. I 41(6), 462–463 (1994)
Sprott, J.C.: Simple chaotic systems and circuits. Am. J. Phys. 68(8), 758–763 (2000)
Piper, J.R., Sprott, J.C.: Simple autonomous chaotic circuits. IEEE Trans. Circuits Syst. II 57(9), 730–734 (2010)
Tchitnga, R., Fotsin, H.B., Nana, B., Fotso, P.H.L., Woafo, P.: Hartley’s oscillator: the simplest chaotic two-component circuit. Chaos Solitons Fractals 45(3), 306–313 (2012)
Tchitnga, R., Nguazon, T., Fotso, P.H.L., Gallas, J.A.C.: Chaos in a single op-amp-based jerk circuit: experiments and simulations. IEEE Trans. Circuits Syst. II 63(3), 239–243 (2017)
Chua, L.O.: Memristor-the missing circuit element. IEEE Trans. Circuit Theory 18, 507–519 (1971)
Gupta, I., Serb, A., Khiat, A., Zeitler, R., Vassanelli, S., Prodromakis, T.: Real-time encoding and compression of neuronal spikes by metal-oxide memristors. Nat. Commun. 7, 12805 (2016)
Xu, Y., Jia, Y., Ma, J., Alsaedi, A., Ahmad, B.: Synchronization between neurons coupled by memristor. Chaos Solitons Fractals 104, 435–442 (2017)
Ren, G.D., Zhou, P., Ma, J., Cai, N., Alsaedi, A., Ahmad, B.: Dynamical response of electrical activities in digital neuron circuit driven by autapse. Int. J. Bifurc. Chaos 27(12), 1750187 (2017)
Bao, H., Liu, W.B., Hu, A.H.: Coexisting multiple firing patterns in two adjacent neurons coupled by memristive electromagnetic induction. Nonlinear Dyn. 95, 43–56 (2019)
Ostrovskii, V.Y., Butusov, D.N., Belkin, D.A., Okoli, G.: Studying the dynamics of memristive synapses in spiking neuromorphic systems. In: IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering, pp. 225–230 (2018)
Pham, V.T., Jafari, S., Vaidyanathan, S., Volos, C.K., Wang, X.: A novel memristive neural network with hidden attractors and its circuitry implementation. Sci. China Technol. Sci. 59(3), 358–363 (2016)
Chen, C.J., Chen, J.Q., Bao, H., Chen, M., Bao, B.C.: Coexisting multi-stable patterns in memristor synapse-coupled Hopfield neural network with two neurons. Nonlinear Dyn. 95(4), 3385–3399 (2019)
Zhou, L., Wang, C.H., Zhou, L.L.: Generating hyperchaotic multi-wing attractor in a 4D memristive circuit. Nonlinear Dyn. 85(4), 1–11 (2016)
Tan, Q.W., Zeng, Y.C., Li, Z.Z.: A simple inductor-free memristive circuit with three line equilibria. Nonlinear Dyn. 94(3), 1585–1602 (2018)
Bao, B.C., Wang, N., Xu, Q., Wu, H.G., Hu, Y.H.: A simple third-order memristive band pass filter chaotic circuit. IEEE Trans. Circuits Syst. II 64(8), 977–981 (2017)
Galias, Z.: Study of amplitude control and dynamical behaviors of a memristive band pass filter circuit. IEEE Trans. Circuits Syst. II 65(5), 637–641 (2018)
Zhou, L., Wang, C.H., Zhang, X., Yao, W.: Various attractors, coexisting attractors and antimonotonicity in a simple fourth-order memristive twin-T oscillator. Int. J. Bifurc. Chaos 28(4), 1850050 (2018)
Peng, G.Y., Min, F.H.: Multistability analysis, circuit implementations and application in image encryption of a novel memristive chaotic circuit. Nonlinear Dyn. 90(3), 1607–1625 (2017)
Ma, J., Wu, F.Q., Ren, G.D., Tan, J.: A class of initials-dependent dynamical systems. Appl. Math. Comput. 298, 65–76 (2017)
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)
Bao, H., Wang, N., Chen, M., Jin, P.P., Wang, G.Y., Bao, B.C.: Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria. Commun. Nonlinear Sci. Numer. Simul. 57, 264–275 (2018)
Wang, X.Y., Fitch, A.L.F., Iu, H.H.C., Screeram, V., Qi, W.G.: Implementation of an analogue model of a memristor based on a light-dependent resistor. Chin. Phys. B 21(10), 108501 (2012)
Muthuswamy, B.: Implementing memristor based chaotic circuits. Int. J. Bifurc. Chaos 20(5), 1335–1350 (2010)
Corinto, F., Ascoli, A.: Memristive diode bridge with LCR filter. Electron. Lett. 48(14), 824–825 (2012)
Chua, L.O.: Everything you wish to know about memristors but are afraid to ask. Radioengineering 24(2), 319–368 (2015)
Bao, B.C., Yu, J.J., Wu, H.G., Liu, Z.: Generalized memristor consisting of diode bridge with first order parallel RC filter. Int. J. Bifurc. Chaos 24(11), 1450143 (2014)
Wu, H.G., Bao, B.C., Liu, Z., Xu, Q., Jiang, P.: Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator. Nonlinear Dyn. 83(1–2), 893–903 (2016)
Bao, B.C., Wu, P.Y., Bao, H., Chen, M., Xu, Q.: Chaotic bursting in memristive diode bridge coupled Sallen–Key low-pass filter. Electron. Lett. 53(16), 1104–1105 (2017)
Kengne, J., Njitacke, T.Z., Kamdoum, T.V., Nguomkam, N.A.: Periodicity, chaos, and multiple attractors in a memristor-based Shinriki’s circuit. Chaos 25(10), 103126 (2015)
Volos, C.K., Akgul, A., Pham, V.T., Baptista, M.S.: Antimonotonicity, crisis and multiple attractors in a simple memristive circuit. J. Circuits Syst. Comput. 27(2), 1850026 (2018)
Muthuswamy, B., Chua, L.O.: Simplest chaotic circuit. Int. J. Bifurc. Chaos 20(5), 1567–1580 (2010)
Pham, V.T., Buscarino, A., Fortuna, L., Frasca, M.: Simple memristive time-delay chaotic systems. Int. J. Bifurc. Chaos 23(04), 1350073 (2013)
Sprott, J.C.: A proposed standard for the publication of new chaotic systems. Int. J. Bifurc. Chaos 21(9), 2391–2394 (2011)
Silva, C.P.: Shil’nikov’s theorem—a tutorial. IEEE Trans. Circuits Syst. I 40(10), 675–682 (1993)
Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16(3), 285–317 (1985)
Kingni, S.T., Keuninckx, L., Woafo, P., Sande, G., Danckaert, J.: Dissipative chaos, shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electronic implementation. Nonlinear Dyn. 73(1–2), 1111–1123 (2013)
Wu, R.P., Wang, C.H.: A new simple chaotic circuit based on memristor. Int. J. Bifurc. Chaos 26(9), 1650145 (2016)
Bao, B.C., Xu, L., Wang, N., Bao, H., Xu, Q., Chen, M.: Third-order RLCM-four-elements-based chaotic circuit and its coexisting bubbles. AEU Int. J. Electron. Commun. 94, 26–35 (2018)
Ostrovskii, V.Y., Karimov, A.I., Rybin, V.G., Kopets, E.E., Butusov, D.N.: Comparing the finite-difference scheme in the simulation of shunted Josephson junctions. In: Proceedings of the 23rd Conference of Open Innovations Association FRUCT, pp. 300–305 (2018)
Bao, H., Hu, A.H., Liu, W.B., Bao, B.C.: Hidden bursting firings and bifurcation mechanisms in memristive neuron model with threshold electromagnetic induction. IEEE Trans. Neural Netw. Learn. Syst. (2019). https://doi.org/10.1109/TNNLS.2019.2905137
He, S.B., Sun, K.H., Banerjee, S.: Dynamical properties and complexity in fractional-order diffusionless Lorenz system. Eur. Phys. J. Plus 131, 254 (2016)
Ye, X.L., Mou, J., Luo, C.F., Wang, Z.S.: Dynamics analysis of Wien-bridge hyperchaotic memristive circuit system. Nonlinear Dyn. 92(3), 923–933 (2018)
Zhang, S., Zeng, Y.C., Li, Z.J., Wang, M.J., Zhang, X., Chang, D.: A novel simple no-equilibrium chaotic system with complex hidden dynamics. Int. J. Dyn. Control 6(4), 1465–1476 (2018)
Li, C.Q., Lin, D.D., Lü, J.H., Hao, F.: Cryptanalyzing an image encryption algorithm based on autoblocking and electrocardiography. IEEE MultiMed. 25(4), 46–56 (2018)
Li, C.Q., Feng, B.B., Li, S.J., Kurths, J., Chen, G.R.: Dynamics analysis of digital chaotic maps via state-mapping networks. IEEE Trans. Circuits Syst. I 66(6), 2322–2335 (2019)
Nazaré, T.E., Nepomuceno, E.G., Paiva, B.P.O.: On the construction bifurcation diagram of the quadratic map with floating-point arithmetic (2017). arXiv: 1711.10084
Silva, M.R., Nepomuceno, E.G., Amaral, G.F.V., Martins, S.A.M., Nardo, L.G.: Exploiting the rounding mode of floating-point in the simulation of Chua’s circuit. Discontinuity Nonlinearity Complex. 7(2), 185–193 (2018)
Chen, M., Xu, Q., Lin, Y., Bao, B.C.: Multistability induced by two symmetric stable node-foci in modified canonical Chua’s circuit. Nonlinear Dyn. 87(2), 789–802 (2017)
Acknowledgements
This work was supported by the grants from the National Natural Science Foundations of China (Grant No. 61473202).
Author information
Authors and Affiliations
Corresponding author
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
About this article
Cite this article
Wang, N., Zhang, G. & Bao, H. Bursting oscillations and coexisting attractors in a simple memristor-capacitor-based chaotic circuit. Nonlinear Dyn 97, 1477–1494 (2019). https://doi.org/10.1007/s11071-019-05067-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-019-05067-6