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Analysis of a new coupled hyperchaotic model and its topological types

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Abstract

In order to construct the high-dimensional discrete hyperchaotic systems systemically, this paper proposes a new coupled chaotic model. It has a wide chaotic parameter range and can relax the election of the coupling coefficient. Sufficient conditions are derived to prove the existence of Li–Yorke chaos in the proposed model. Meanwhile, the existence of hyperchaos is also demonstrated. To discern the effects of different coupling types on the chaotic dynamics more comprehensively, we further explore the dynamical behaviors with various coupling structures by using Lyapunov spectrum, bifurcation analysis, and phase portraits. We investigate the interaction relationship between coupled units and give suggestions for selecting coupling types. The results indicate that the coupled model has more complex and more stable chaotic performance when there exists a loop in its topological structure. Further, synchronization is also discussed in this work; analysis results illustrate that the proposed model cannot be suppressed to periodic points at sufficiently high coupling strengths. This paper suggests an effective method that may contribute to studying hyperchaos design and coupled chaotic systems.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Wu, G.C., Baleanu, D.: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75, 283–287 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Guegan, D.: Chaos in economics and finance. Annu. Rev. Control 33, 89–93 (2009)

    Article  Google Scholar 

  3. Kocarev, L.: Chaos-based cryptography: a brief overview. IEEE Circuits Syst. Mag. 1, 6–21 (2002)

    Article  Google Scholar 

  4. Zheng, J., Hu, H., Xiang, X.: Applications of symbolic dynamics in counteracting the dynamical degradation of digital chaos. Nonlinear Dyn. 94, 1535–1546 (2018)

    Article  Google Scholar 

  5. Bao, H., Hu, A., Liu, W., Bao, B.: Hidden bursting firings and bifurcation mechanisms in memristive neuron model with threshold electromagnetic induction. IEEE Trans. Neural Netw. Learn. Syst. 31, 502–511 (2020)

    Article  Google Scholar 

  6. Wang, X., Qin, X.: A new pseudo-random number generator based on cml and chaotic iteration. Nonlinear Dyn. 70, 1589–1592 (2012)

    Article  MathSciNet  Google Scholar 

  7. Hu, H.P., Liu, L.F., Ding, N.D.: Pseudorandom sequence generator based on the chen chaotic system. Comput. Phys. Commun. 184, 765–768 (2013)

    Article  MathSciNet  Google Scholar 

  8. Broumandnia, A.: The 3d modular chaotic map to digital color image encryption. Future Gener. Comput. Syst. 99, 489–499 (2019)

    Article  Google Scholar 

  9. Hua, Z., Zhou, Y., Huang, H.: Cosine-transform-based chaotic system for image encryption. Inf. Sci. 480, 403–419 (2019)

    Article  Google Scholar 

  10. Hua, Z., Zhou, Y., Pun, C.M., Chen, C.: 2d sine logistic modulation map for image encryption. Inf. Sci. 297, 80–94 (2015)

    Article  Google Scholar 

  11. Zhang, Y.Q., Wang, X.Y.: A symmetric image encryption algorithm based on mixed linear–nonlinear coupled map lattice. Inf. Sci. 273, 329–351 (2014)

    Article  Google Scholar 

  12. Özkaynak, F.: Brief review on application of nonlinear dynamics in image encryption. Nonlinear Dyn. 92, 305–313 (2018)

    Article  Google Scholar 

  13. Li, Y., Tang, W.K., Chen, G.: Hyperchaos evolved from the generalized Lorenz equation. Int. J. Circuit Theory Appl. 33, 235–251 (2005)

    Article  MATH  Google Scholar 

  14. Shen, C., Yu, S., Lu, J., Chen, G.: Designing hyperchaotic systems with any desired number of positive lyapunov exponents via a simple model. IEEE Trans. Circuits Syst. I Regul. Pap. 61, 2380–2389 (2014)

    Article  Google Scholar 

  15. Li, Y., Chen, G., Tang, W.K.S.: Controlling a unified chaotic system to hyperchaotic. IEEE Trans. Circuits Syst. II Express Briefs 52, 204–207 (2005)

    Article  Google Scholar 

  16. Sun, K., Xuan, L., Zhu, C., Sprott, J.C.: Hyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system. Nonlinear Dyn. 69, 1383–1391 (2012)

    Article  MathSciNet  Google Scholar 

  17. Cafagna, D., Grassi, G.: Hyperchaotic coupled Chua circuits: an approach for generating new n \(\times \) m-scroll attractors. Int. J. Bifurc. Chaos 13, 2537–2550 (2003)

    Article  MATH  Google Scholar 

  18. Wang, J., Chen, Z., Chen, G., Yuan, Z.: A novel hyperchaotic system and its complex dynamics. Int. J. Bifurc. Chaos 18, 3309–3324 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Shen, C., Yu, S., Lu, J., Chen, G.: A systematic methodology for constructing hyperchaotic systems with multiple positive Lyapunov exponents and circuit implementation. IEEE Trans. Circuits Syst. I Regul. Pap. 61, 854–864 (2014)

    Article  Google Scholar 

  20. Kong, S., Li, C., Jiang, H., Lai, Q., Jiang, X.: A 2d hyperchaotic map with conditional symmetry and attractor growth. Chaos 31, 043121 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Natiq, H., Banerjee, S., Said, M.: Cosine chaotification technique to enhance chaos and complexity of discrete systems. Eur. Phys. J. Spec. Top. 228, 185–194 (2019)

    Article  Google Scholar 

  22. Peng, Y., He, S., Sun, K.: A higher dimensional chaotic map with discrete memristor. AEU Int. J. Electron. Commun. 129, 153539 (2021)

    Article  Google Scholar 

  23. Bao, B.C., Li, H., Wu, H., Zhang, X., Chen, M.: Hyperchaos in a second-order discrete memristor-based map model. Electron. Lett. 56(15), 769–770 (2020)

    Article  Google Scholar 

  24. Chen, G., Lai, D.: Feedback control of Lyapunov exponents for discrete-time dynamical systems. Int. J. Bifurc. Chaos 6, 1341–1349 (1996)

    Article  MATH  Google Scholar 

  25. Wang, X.F., Chen, G.: On feedback anticontrol of discrete chaos. Int. J. Bifurc. Chaos 9, 1435–1441 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chen, C., Sun, K., He, S.: A class of higher-dimensional hyperchaotic maps. Eur. Phys. J. Plus 134, 410 (2019)

    Article  Google Scholar 

  27. Liu, W., Sun, K., He, S.: Sf-simm high-dimensional hyperchaotic map and its performance analysis. Nonlinear Dyn. 89, 2521–2532 (2017)

    Article  MathSciNet  Google Scholar 

  28. Shakiba, A.: A novel 2d cascade modulation couple hyperchaotic mapping for randomized image encryption. Multimed. Tools Appl. 80, 17983–18006 (2021)

    Article  Google Scholar 

  29. Singha, J., Gupte, N.: Chimera states in globally coupled sine circle map lattices: spatiotemporal intermittency and hyperchaos. Phys. Lett. A 384, 126225 (2020)

    Article  MathSciNet  Google Scholar 

  30. Wang, X., Zhao, H., Feng, L., Ye, X., Zhang, H.: High-sensitivity image encryption algorithm with random diffusion based on dynamic-coupled map lattices. Opt. Lasers Eng. 122, 225–238 (2019)

    Article  Google Scholar 

  31. Batista, C.A., Viana, R.L.: Chaotic maps with nonlocal coupling: Lyapunov exponents, synchronization of chaos, and characterization of chimeras. Chaos Solitons Fract. 131, 109501 (2020)

    Article  MathSciNet  Google Scholar 

  32. Chazottes, J.R., Fernandez, B.: Dynamics of Coupled Map Lattices and Related Spatially Extended Systems. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  33. Rajvaidya, B.P., Deshmukh, A.D., Gade, P.M., Sahasrabudhe, G.G.: Transition to coarse-grained order in coupled logistic maps: effect of delay and asymmetry. Chaos Solitons Fract. 139, 110301 (2020)

    Article  MathSciNet  Google Scholar 

  34. Wang, S., Hu, G., Zhou, H.: A one-way coupled chaotic map lattice based self-synchronizing stream cipher. Commun. Nonlinear Sci. Numer. Simul. 19, 905–913 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Nag, M., Poria, S.: Li–Yorke chaos in globally coupled map lattice with delays. Int. J. Bifurc. Chaos 29, 1950183 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nag, M., Poria, S.: Effects of time delay on the synchronized states of globally coupled network. Chaos 30, 093122 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  37. Chaurasia, S.S., Sinha, S.: Suppression of chaos through coupling to an external chaotic system. Nonlinear Dyn. 87, 159–167 (2017)

    Article  Google Scholar 

  38. Collet, P., Eckmann, J.P.: Iterated Maps on the Interval as Dynamical Systems. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  39. Pincus, S.M.: Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. U. S. A. 88, 2297–2301 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  40. Marotto, F.R.: On redefining a snap-back repeller. Chaos Solitons Fract. 25, 25–28 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Chen, G., Shi, Y.: Introduction to anti-control of discrete chaos: theory and applications. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 364, 2433–2447 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. Eckmann, J., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57(3), 1115–1115 (2008)

    MathSciNet  MATH  Google Scholar 

  43. Vasegh, N.: Spatiotemporal and synchronous chaos in accumulated coupled map lattice. Nonlinear Dyn. 89, 1089–1097 (2017)

    Article  MATH  Google Scholar 

  44. Batista, C.A.S., Batista, A.M., de Pontes, J.A.C., Viana, R.L., Lopes, S.R.: Chaotic phase synchronization in scale-free networks of bursting neurons. Phys. Rev. E 76, 016218 (2007)

    Article  MathSciNet  Google Scholar 

  45. Perego, A., Lamperti, M.: Collective excitability, synchronization, and array-enhanced coherence resonance in a population of lasers with a saturable absorber. Phys. Rev. A 94, 033839 (2016)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Key R&D Program of China [Grant Number 2017YFB0802000]; and the Cryptography Theoretical Research of National Cryptography Development Fund [Grant Number MMJJ20170109]; and the Key R&D Program of Hubei Province [Grant Number 2020BAB104].

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Authors and Affiliations

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

    Hao Ming, Hanping Hu & Jun Zheng

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

    Hanping Hu

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  1. Hao Ming
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  2. Hanping Hu
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  3. Jun Zheng
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Correspondence to Hanping Hu.

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Ming, H., Hu, H. & Zheng, J. Analysis of a new coupled hyperchaotic model and its topological types. Nonlinear Dyn 105, 1937–1952 (2021). https://doi.org/10.1007/s11071-021-06692-w

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