Abstract
Due to the nonlinearity with inner state, memristors have been applied in vast continuous dynamical systems. However, the application of memristors in discrete dynamical systems has not received enough attention, yet. Toward this end, this paper presents a three-dimensional (3D) memristive Hénon map by coupling a memristor to the classical Hénon map. Using numerical measures, the memristor effects on the presented map are exhibited and the complex dynamical behaviors with multistability are disclosed therein. Particularly, since the presented map has no invariant points, a dimension-reduction conversion method is proposed to investigate its properties and its hidden Neimark–Sacker bifurcations are effectively interpreted. The results demonstrate that the introduction of discrete memristor makes the presented map own complex hidden dynamical behaviors, which greatly enhances the fractal structure complexity of the chaotic attractors. In addition, a digital hardware set is exploited to implement the 3D memristive Hénon map and the chaotic attractors are physically acquired thereby.
Similar content being viewed by others
Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
References
Zhang, G., Ma, J., Alsaedi, A., Ahmad, B., Alzahrani, F.: Dynamical behavior and application in Josephson Junction coupled by memristor. Appl. Math. Comput. 321, 290–299 (2018)
Chen, M., Sun, M.X., Bao, H., Hu, Y.H., Bao, B.C.: Flux-charge analysis of two-memristor-based Chua’s circuit: Dimensionality decreasing model for detecting extreme multistability. IEEE Trans. Ind. Electron. 67(3), 2197–2206 (2020)
Chua, L.: If it’s pinched it’s a memristor. Semicond. Sci. Technol. 29, 104001 (2014)
Bao, H., Wang, N., Bao, B.C., Chen, M., Jin, P.P., Wang, G.Y.: 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)
Ma, J., Wu, F.Q., Ren, G.D., Tang, J.: A class of initials-dependent dynamical systems. Appl. Math. Comput. 298, 65–76 (2017)
El-Sayed, A.M.A., Nour, H.M., Elsaid, A., Matouk, A.E., Elsonbaty, A.: Dynamical behaviors, circuit realization, chaos control and synchronization of a new fractional order hyperchaotic system. Appl. Math. Modell. 40(5–6), 3516–3534 (2016)
Bao, H., Liu, W.B., Chen, M.: Hidden extreme multistability and dimensionality reduction analysis for an improved non-autonomous memristive FitzHugh-Nagumo circuit. Nonlinear Dyn. 96, 1879–1894 (2019)
Ishaq, A.A., Lakshmanan, M.: Discontinuity induced Hopf and Neimark–Sacker bifurcations in a memristive Murali-Lakshmanan-Chua circuit. Int. J. Bifurc. Chaos. 27(6), 1730021 (2017)
Li, H.Z., Hua, Z.Y., Bao, H., Zhu, L., Chen, M., Bao, B.C.: Two-dimensional memristive hyperchaotic maps and application in secure communication. IEEE Trans. Ind. Electron. 68(10), 9931–9940 (2021)
Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., Prasad, A.: Hidden attractors in dynamical systems. Phys. Rep. 637, 1–50 (2016)
Kuznetsov, N.V., Leonov, G.A., Yuldashev, M.V., Yuldashev, R.V.: Hidden attractors in dynamical models of phase-locked loop circuits: Limitations of simulation in MATLAB and SPICE. Commun. Nonlinear Sci. Numer. Simul. 51, 39–49 (2017)
Kapitaniak, T., Leonov, G.A.: Multistability: Uncovering hidden attractors. Eur. Phys. J. Spec. Top. 224(8), 1405–1408 (2015)
Wang, N., Zhang, G., Kuznetsov, N.V., Bao, H.: Hidden attractors and multistability in a modified Chua’s circuit. Commun. Nonlinear Sci. Numer. Simul. 92, 105494 (2021)
Wang, X., Chen, G.R.: A chaotic system with only one stable equilibrium. Commun. Nonlinear Sci. Numer. Simul. 17(3), 1264–1272 (2012)
Marius-F, D., Michal, F.: Hidden chaotic attractors and chaos suppression in an impulsive discrete economical supply and demand dynamical system. Commun. Nonlinear Sci. Numer. Simul. 74, 1–13 (2019)
Yang, Y.J., Qi, G.Y., Hu, J.B., Faradja, P.: Finding method and analysis of hidden chaotic attractors for plasma chaotic system from physical and mechanistic perspectives. Int. J. Bifurc. Chaos 30(5), 2050072 (2020)
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)
Pham, V.-T., Jafari, S., Volos, C., Kapitaniak, T.: Different families of hidden attractors in a new chaotic system with variable equilibrium. Int. J. Bifurc. Chaos. 27(9), 1750138 (2017)
Xu, L., Qi, G.Y., Ma, J.: Modeling of memristor-based Hindmarsh-Rose neuron and its dynamical analyses using energy method. Appl. Math. Modell. 101, 503–516 (2016)
Jafari, S., Sprott, J.C., Nazarimehr, F.: Recent new examples of hidden attractors. Eur. Phys. J. Spl. Top. 224(8), 1469–1476 (2015)
Bao, B.C., Li, H.Z., Zhu, L., Zhang, X., Chen, M.: Initial-switched boosting bifurcations in 2D hyperchaotic map. Chaos 30(3), 033107 (2020)
Jafari, S., Pham, V.-T., Golpayegani, S.M.R.H., Moghtadaei, M., Kingni, S.T.: The relationship between chaotic maps and some chaotic systems with hidden attractors. Int. J. Bifurc. Chaos. 26(13), 1650211 (2016)
Zhang, X., Chen, G.R.: Polynomial maps with hidden complex dynamics. Discr. Contin. Dyn. Syst. Ser. B. 24(6), 2941–2954 (2019)
Panahi, S., Sprott, J.C., Jafari, S.: Two simplest quadratic chaotic maps without equilibrium. Int. J. Bifurc. Chaos. 28(12), 1850144 (2018)
Wang, C.F., Ding, Q.: A new two-dimensional map with hidden attractors. Entropy 20(5), 322 (2018)
Khennaoui, A.A., Ouannas, A., Boulaaras, S., Pham, V.-T., Azar, A.T.: A fractional map with hidden attractors: chaos and control. Eur. Phys. J. Spl Topics. 229, 1083–1093 (2020)
Peng, Y.X., Sun, K.H., He, S.B.: A discrete memristor model and its application in Hénon map. Chaos Solitons Fractals. 137, 109873 (2020)
Bao, H., Hua, Z.Y., Li, H.Z., Chen, M., Bao, B.C.: Discrete memristor hyperchaotic maps. IEEE Trans. Circuits Syst. I. 68(11), 4534–4544 (2021)
Deng, Y., Li, Y.X.: Nonparametric bifurcation mechanism in 2-D hyperchaotic discrete memristor-based map. Nonlinear Dyn. 104, 4601–4614 (2021)
Bao, H., Hua, Z.Y., Wang, N., Zhu, L., Chen, M., Bao, B.C.: Initials-boosted coexisting chaos in a 2-D Sine map and its hardware implementation. IEEE Trans. Ind. Inform. 17(2), 1132–1140 (2021)
Li, K.X., Bao, H., Li, H.Z., Ma, J., Hua, Z.Y., Bao, B.C.: Memristive Rulkov neuron model with magnetic induction effects. IEEE Trans. Ind. Inform. 18(3), 1726–1736 (2022)
Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50(1), 69–77 (1976)
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. 31(2), 502–511 (2020)
Sacker, R.: On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations. Report IMM-NYU 333, New York University (1964)
Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd edn. CRC Press, Boca Raton (2015)
Kangalgil, F.: Neimark–Sacker bifurcation and stability analysis of a discrete-time prey-predator model with Allee effect in prey. Adv. Differ. Equ. 2019, 92 (2019)
Li, B., He, Q., Chen, R.: Neimark–Sacker bifurcation and the generate cases of Kopel oligopoly model with different adjustment speed. Adv. Differ. Equ. 2020, 72 (2020)
Elhadj, Z., Sprott, J.C.: A minimal 2-D quadratic map with quasiperiodic route to chaos. Int. J. Bifurc. Chaos. 18(5), 1567–1577 (2008)
Pisarchik, A.N., Feudel, U.: Control of multistability. Phys. Rep. 540(4), 167–218 (2014)
Natiq, H., Banerjee, S., Ariffin, M.R.K., Said, M.R.M.: Can hyperchaotic maps with high complexity produce multistability? Chaos. 29(1), 011103 (2019)
Zhou, X.J., Li, C.B., Li, Y.X., Lu, X., Lei, T.F.: An amplitude-controllable 3-D hyperchaotic map with homogenous multistability. Nonlinear Dyn. 105, 1843–1857 (2021)
Funding
This work was supported by the National Natural Science Foundations of China under Grant Nos. 51777016 and 62071142 and the Postgraduate Research and Practice Innovation Program of Jiangsu Province, China, under Grant No. KYCX21_2816.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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
Rong, K., Bao, H., Li, H. et al. Memristive Hénon map with hidden Neimark–Sacker bifurcations. Nonlinear Dyn 108, 4459–4470 (2022). https://doi.org/10.1007/s11071-022-07380-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-022-07380-z