Skip to main content
SpringerLink
Log in

Research on cascading high-dimensional isomorphic chaotic maps

  • Research Article
  • Published:
Cognitive Neurodynamics Aims and scope Submit manuscript

Abstract

In order to overcome the security weakness of the discrete chaotic sequence caused by small Lyapunov exponent and keyspace, a general chaotic construction method by cascading multiple high-dimensional isomorphic maps is presented in this paper. Compared with the original map, the parameter space of the resulting chaotic map is enlarged many times. Moreover, the cascaded system has larger chaotic domain and bigger Lyapunov exponents with proper parameters. In order to evaluate the effectiveness of the presented method, the generalized 3-D Hénon map is utilized as an example to analyze the dynamical behaviors under various cascade modes. Diverse maps are obtained by cascading 3-D Hénon maps with different parameters or different permutations. It is worth noting that some new dynamical behaviors, such as coexisting attractors and hyperchaotic attractors are also discovered in cascaded systems. Finally, an application of image encryption is delivered to demonstrate the excellent performance of the obtained chaotic sequences.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Aihara K, Takabe T, Toyoda M (1990) Chaotic neural networks. Phys Lett A 144(6–7):333–340

    Article  Google Scholar 

  • Baier G, Klein M (1990) Maximum hyperchaos in generalized Hénon maps. Phys Lett A 151(6–7):281–284

    Article  Google Scholar 

  • Balasubramaniam P, Chandran R, Theesar SJS (2011) Synchronization of chaotic nonlinear continuous neural networks with time-varying delay. Cogn Neurodyn 5(4):361–371

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Bao B, Qian H, Wang J, Xu Q, Chen M, Wu H, Yu Y (2017a) Numerical analyses and experimental validations of coexisting multiple attractors in Hopfield neural network. Nonlinear Dyn 90(4):2359–2369

    Article  Google Scholar 

  • Bao B, Qian H, Xu Q, Chen M, Wang J, Yu Y (2017b) Coexisting behaviors of asymmetric attractors in hyperbolic-type memristor based Hopfield neural network. Front Comput Neurosci 11:81

    Article  PubMed  PubMed Central  Google Scholar 

  • Chen M, Sun M, Bao B, Wu H, Xu Q, Wang J (2018) Controlling extreme multistability of memristor emulator-based dynamical circuit in flux-charge domain. Nonlinear Dyn 91(2):1395–1412

    Article  Google Scholar 

  • Chen C, Chen J, Bao H, Chen M, Bao B (2019) Coexisting multi-stable patterns in memristor synapse-coupled Hopfield neural network with two neurons. Nonlinear Dyn 95(4):3385–3399

    Article  Google Scholar 

  • Daniel C, Ion T (2014) On the security of a new image encryption scheme based on a chaotic function. Signal Image Video Process 8(4):641–646

    Article  Google Scholar 

  • Dong J, Zhang G, Xie Y, Yao H, Wang J (2014) Dynamic behavior analysis of fractional-order Hindmarsh–Rose neuronal model. Cogn Neurodyn 8(2):167–175

    Article  Google Scholar 

  • Du Y, Wang R, Han F, Lu Q (2015) Parameter-dependent synchronization of coupled neurons in cold receptor model. Int J Non-Linear Mech 70:95–104

    Article  Google Scholar 

  • Dudkowski D, Jafari S, Kapitaniak T, Kuznetsov NV, Leonov GA, Prasad A (2016) Hidden attractors in dynamical systems. Phys Rep 637:1–50

    Article  Google Scholar 

  • Hong Q, Li Y, Wang X, Zeng Z (2018) A versatile pulse control method to generate arbitrary multi-direction multi-butterfly chaotic attractors. IEEE Trans Comput-Aided Des Integr 38(8):1480–1492

    Article  Google Scholar 

  • Hong Q, Wu Q, Wang X (2019) Novel nonlinear function shift method for generating multiscroll attractors using memristor-based control circuit. IEEE Trans Very Large Scale Integr (VLSI) Syst 27(5):1174–1185

    Google Scholar 

  • Hua Z, Zhou Y (2015) Dynamic parameter-control chaotic system. IEEE Trans Cybern 46(12):3330–3341

    Article  PubMed  Google Scholar 

  • Hua Z, Zhou Y (2017) One-dimensional nonlinear model for producing chaos. IEEE Trans Circuits Syst I-Regul Pap 65(1):235–246

    Article  Google Scholar 

  • Hua Z, Zhou B, Zhou Y (2017) Sine-transform-based chaotic system with FPGA implementation. IEEE Trans Ind Electron 65(3):2557–2566

    Article  Google Scholar 

  • Irak M, Soylu C, Turan G, Çapan D (2019) Neurobiological basis of feeling of knowing in episodic memory. Cogn Neurodyn 13(3):239–256

    Article  PubMed  PubMed Central  Google Scholar 

  • Jia B, Gu H, Li L, Zhao X (2012) Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns. Cogn Neurodyn 6(1):89–106

    Article  PubMed  Google Scholar 

  • Leonov GA, Kuznetsov NV (2013) Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int J Bifurc Chaos 23(1):1330002

    Article  Google Scholar 

  • Li MW, Geng J, Han DF, Zheng TJ (2016) Ship motion prediction using dynamic seasonal RvSVR with phase space reconstruction and the chaos adaptive efficient FOA. Neurocomputing 174:661–680

    Article  Google Scholar 

  • Li C, Sprott JC, Kapitaniak T, Lu T (2018) Infinite lattice of hyperchaotic strange attractors. Chaos Solitons Fractals 109:76–82

    Article  Google Scholar 

  • Liu Y, Li S, Liu Z, Wang R (2016) High codimensional bifurcation analysis to a six-neuron BAM neural network. Cogn Neurodyn 10(2):149–164

    Article  PubMed  Google Scholar 

  • Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20(2):130–141

    Article  Google Scholar 

  • Mora-Sánchez A, Dreyfus G, Vialatte FB (2019) Scale-free behaviour and metastable brain-state switching driven by human cognition, an empirical approach. Cogn Neurodyn 13(5):437–452

    Article  PubMed  PubMed Central  Google Scholar 

  • Muñoz-Pacheco JM, Guevara-Flores DK, Félix-Beltrán OG, Tlelo-Cuautle E, Barradas-Guevara JE, Volos CK (2018) Experimental verification of optimized multiscroll chaotic oscillators based on irregular saturated functions. Complexity 2018:3151840

    Article  Google Scholar 

  • Puanhvuan D, Khemmachotikun S, Wechakarn P, Wijarn B, Wongsawat Y (2017) Navigation-synchronized multimodal control wheelchair from brain to alternative assistive technologies for persons with severe disabilities. Cogn Neurodyn 11(2):117–134

    Article  PubMed  PubMed Central  Google Scholar 

  • Qu J, Wang R, Du Y, Cao J (2012) Synchronization study in ring-like and grid-like neuronal networks. Cogn Neurodyn 6(1):21–31

    Article  PubMed  Google Scholar 

  • Sambas A, Vaidyanathan S, Tlelo-Cuautle E, Zhang S, Guillen-Fernandez O, Hidayat Y, Gundara G et al (2019a) A novel chaotic system with two circles of equilibrium points: multistability, electronic circuit and FPGA realization. Electronics 8(11):1211

    Article  Google Scholar 

  • Sambas A, Vaidyanathan S, Zhang S, Zeng Y, Mohamed MA, Mamat M (2019b) A new double-wing chaotic system with coexisting attractors and line equilibrium: bifurcation analysis and electronic circuit simulation. IEEE Access 7:115454–115462

    Article  Google Scholar 

  • Vaidyanathan S, Dolvis LG, Jacques K, Lien CH, Sambas A (2019a) A new five-dimensional four-wing hyperchaotic system with hidden attractor, its electronic circuit realisation and synchronisation via integral sliding mode control. Int J Model Identif Control 32(1):30–45

    Article  Google Scholar 

  • Vaidyanathan S, Sambas A, Zhang S, Zeng Y, Mohamed MA, Mamat M (2019b) A new two-scroll chaotic system with two nonlinearities: dynamical analysis and circuit simulation. Telkomnika 17(5):2465–2474

    Article  Google Scholar 

  • Wang L (2007) Interactions between neural networks: a mechanism for tuning chaos and oscillations. Cogn Neurodyn 1(2):185–188

    Article  PubMed  PubMed Central  Google Scholar 

  • Wang G, Yuan F (2013) Cascade chaos and its dynamic characteristics. Acta Phys Sin 62(2):020506

    Google Scholar 

  • Wang H, Wang Q, Lu Q, Zheng Y (2013) Equilibrium analysis and phase synchronization of two coupled hr neurons with gap junction. Cogn Neurodyn 7(2):121–131

    Article  PubMed  Google Scholar 

  • Wang G, Yuan F, Chen G, Zhang Y (2018) Coexisting multiple attractors and riddled basins of a memristive system. Chaos 28(1):013125

    Article  PubMed  Google Scholar 

  • Wu Y, Zhou Y, Bao L (2014) Discrete wheel-switching chaotic system and applications. IEEE Trans Circuits Syst I-Regul Pap 61(12):3469–3477

    Article  Google Scholar 

  • Xiao M, Cao J (2010) Approximate expressions of the bifurcating periodic solutions in a neuron model with delay-dependent parameters by perturbation approach. Cogn Neurodyn 4(3):241–250

    Article  PubMed  PubMed Central  Google Scholar 

  • Zhang S, Zeng Y, Li Z, Wang M, Xiong L (2018a) Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability. Chaos 28(1):013113

    Article  PubMed  Google Scholar 

  • Zhang S, Zeng YC, Li ZJ (2018b) A novel four-dimensional no-equilibrium hyper-chaotic system with grid multiwing hyper-chaotic hidden attractors. J Comput Nonlinear Dyn 13(9):090908

    Article  Google Scholar 

  • Zheng G, Tonnelier A (2009) Chaotic solutions in the quadratic integrate-and-fire neuron with adaptation. Cogn Neurodyn 3(3):197–204

    Article  PubMed  Google Scholar 

  • Zommara NM, Takahashi M, Ounjai K, Lauwereyns J (2018) A gaze bias with coarse spatial indexing during a gambling task. Cogn Neurodyn 12(2):171–181

    Article  PubMed  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Key Research and Development Program of China under Grant 2016YFB0800402, the Natural Science Foundation of China under Grants 61936004 and 61673188 and 61673188, the Innovation Group Project of the National Natural Science Foundation of China under Grant 61821003, the Foundation for Innovative Research Groups of Hubei Province of China under Grant 2017CFA005 and the 111 Project on Computational Intelligence and Intelligent Control under Grant B18024.

Author information

Authors and Affiliations

  1. School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, 430074, China

    Qiujie Wu, Fanghai Zhang, Xiaoping Wang & Zhigang Zeng

  2. Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan, 430074, China

    Qiujie Wu, Fanghai Zhang, Xiaoping Wang & Zhigang Zeng

  3. College of Computer Science and Electronic Engineering, Hunan University, Changsha, 410082, China

    Qinghui Hong

Authors
  1. Qiujie Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fanghai Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qinghui Hong
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiaoping Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Zhigang Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhigang Zeng.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict 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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, Q., Zhang, F., Hong, Q. et al. Research on cascading high-dimensional isomorphic chaotic maps. Cogn Neurodyn 15, 157–167 (2021). https://doi.org/10.1007/s11571-020-09583-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11571-020-09583-9

Keywords

Search

Navigation