Skip to main content
SpringerLink
Log in

Dynamic analysis of a new two-dimensional map in three forms: integer-order, fractional-order and improper fractional-order

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

In this paper, a new 2-dimensional chaotic map with a simple algebraic form is proposed. And the numerical solution of the corresponding fractional-order map is derived. It is novel that the new map still exhibits chaotic behaviors when the new map is expanded to fractional-order and improper fractional-order. The dynamical characteristics of these three forms are detected through the bifurcation diagram, the maximum Lyapunov exponent spectrum, Kolmogorov entropy and attractor portraits. More interestingly, the new map has multiple coexisting attractors, but the multistability of the fractional-order and improper fractional-order is more complicated than the integer-order form. In addition, the Permutation entropy (PE) complexity algorithm and 2-dimensional maximum Lyapunov exponent diagram are used to explore the dynamic changes of three forms when the amplitude changes simultaneously. The analysis shows that under the appropriate order, the chaotic range of the fractional-order and improper fractional-order is larger than that of the integer order. This research provides guidance on the application and teaching of discrete fractional-order systems.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  1. I. Gkolias, J. Daquin, D.K. Skoulidou, K. Tsiganis, C. Efthymiopoulos, Chaotic transport of navigation satellites. Chaos 29, 8 (2019)

    Article  MathSciNet  Google Scholar 

  2. T. Devolder, D. Rontani, S. Petit-Watelot, K. Bouzehouane, S. Andrieu, J. Ltang, M.-W. Yoo, J.-P. Adam, C. Chappert, S. Girod, Chaos in magnetic nanocontact vortex oscillators. Phys. Rev. Lett. 123, 147701 (2019)

    Article  ADS  Google Scholar 

  3. H. Liu, F. Wen, A. Kadir, Construction of a new 2d chebyshev-sine map and its application to color image encryption. Multimed. Tools Appl. 78, 15997 (2019)

    Article  Google Scholar 

  4. S. Takeuchi, M. Hasegawa, K. Kanno, A. Uchida, N. Chauvet, M. Naruse, Dynamic channel selection in wireless communications via a multi-armed bandit algorithm using laser chaos time series. Sci. Rep. 10, 1574 (2020)

    Article  ADS  Google Scholar 

  5. F. Yu, L. Liu, B. He, Y. Huang, Q. Wan, Analysis and fpga realization of a novel 5d hyperchaotic four-wing memristive system, active control synchronization, and secure communication application. Complexity 2019, 1 (2019)

    Google Scholar 

  6. Q. Lai, Z. Wan, P.D.K. Kuate, H. Fotsin, Coexisting attractors, circuit implementation and synchronization control of a new chaotic system evolved from the simplest memristor chaotic circuit. Commun. Nonlinear Numer. Simul. 89, 105341 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. X. Ma, J. Mou, J. Liu, C. Ma, X. Zhao, A novel simple chaotic circuit based on memristor-cmemcapacitor. Nonlinear Dyn. 100, 2859 (2020)

    Article  Google Scholar 

  8. M. Joshi, A. Ranjan, An autonomous simple chaotic jerk system with stable and unstable equilibria using reverse sine hyperbolic functions. Int. J. Bifurcat. Chaos 30, 2050070 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Y. Fei, L. Li, S. Hui et al., Dynamic analysis, circuit design, and synchronization of a novel 6d memristive four-wing hyperchaotic system with multiple coexisting attractors. Complexity 2020, 1 (2020)

    Article  MATH  Google Scholar 

  10. B. Bao, A. Hu, B. Han, Q. Xu, M. Chen, H. Wu, Three-dimensional memristive hindmarshcrose neuron model with hidden coexisting asymmetric behaviors. Complexity 2018, 1 (2018)

    Google Scholar 

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

    Article  Google Scholar 

  12. E.V. Altay, B. Alatas, Bird swarm algorithms with chaotic mapping. Artif. Intell. Rev. 53, 1373 (2019)

    Article  Google Scholar 

  13. Y. Peng, K. Sun, S. He, X. Yang, Parameter estimation of a complex chaotic system with unknown initial values. Eur. Phys. J. Plus 133, 305 (2018)

    Article  Google Scholar 

  14. B. Chen, S. Yu, P. Chen, L. Xiao, J. L, Design and virtex-7-based implementation of video chaotic secure communications. Int. J. Bifurcat. Chaos 30, 2050075 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. A.A. Eshmawi, E.E. Mahmoud, Secure communications via complex phase synchronization of pair complex chaotic structures with a similar structure of linear terms with modifying in nonlinear terms. AEJ 59, 1107 (2020)

    Google Scholar 

  16. H. Liu, A. kadir, C. Xu Cryptanalysis and constructing s-box based on chaotic map and backtracking. Appl. Math. Comput. 376, 125153 (2020)

  17. L. Qiang, N. Benyamin, L. Feng, Dynamic analysis, circuit realization, control design and image encryption application of an extended l system with coexisting attractors. Chaos Solit. Fract. 114, 230 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. A.V. Tutueva, E.G. Nepomuceno, A.I. Karimov, V.S. Andreev, D.N. Butusov, Adaptive chaotic maps and their application to pseudo-random numbers generation. Chaos Solit. Fract. 133, 109615 (2020)

    Article  MathSciNet  Google Scholar 

  19. P.S. Sneha, S. Sankar, A.S. Kumar, A chaotic colour image encryption scheme combining walshchadamard transform and arnoldctent maps. J. Ambient Intell. Hum. Comput. 11, 1289 (2019)

    Article  Google Scholar 

  20. H.R. Shakir, A color-image encryption scheme using a 2d chaotic system and dna coding. Adv. Multimed. 2019, 1 (2019)

    Article  Google Scholar 

  21. Z.H. Gan, X.L. Chai, D.J. Han, Y.R. Chen, A chaotic image encryption algorithm based on 3-d bit-plane permutation. Neural Comput. Appl. 31, 7111 (2018)

    Article  Google Scholar 

  22. F. Yang, J. Mou, K. Sun, R. Chu, Lossless image compression-encryption algorithm based on bp neural network and chaotic system. Multimed. Tools Appl. (1C2), (2020)

  23. H. Liu, A. Kadir, J. Liu, Color pathological image encryption algorithm using arithmetic over galois field and coupled hyper chaotic system. Opt. Lasers Eng. 122, 123 (2019)

    Article  Google Scholar 

  24. A. Girdhar, V. Kumar, A reversible and affine invariant 3d data hiding technique based on difference shifting and logistic map. J. Ambient Intell. Hum. Comput. 10, 4947 (2019)

    Article  Google Scholar 

  25. A. Akgul, I.M. Moroz, A. Durdu, A novel data hiding method by using a chaotic system without equilibrium points. Mod. Phys. Lett. B 33, 195 (2019)

    Article  MathSciNet  Google Scholar 

  26. M. Yan, H. Xu, A chaotic system with a nonlinear term and multiple coexistence attractors. Eur. Phys. J. Plus 135 (2020)

  27. C. Li, J.C. Sprott, M. Yong, An infinite 2-d lattice of strange attractors. Nonlinear Dyn. 89, 2629 (2017)

    Article  MathSciNet  Google Scholar 

  28. G.D. Leutcho, S. Jafari, I.I. Hamarash, K. Jacques, A new megastable nonlinear oscillator with infinite attractors. Chaos Solit. Fract. 134, 109703 (2020)

    Article  MathSciNet  Google Scholar 

  29. N. Wang, G. Zhang, H. Bao, Bursting oscillations and coexisting attractors in a simple memristor-capacitor-based chaotic circuit. Nonlinear Dyn. 97, 1477 (2019)

    Article  MATH  Google Scholar 

  30. Q. Lai, P.D.K. Kuate, H. Pei, H. Fotsin, Infinitely many coexisting attractors in no-equilibrium chaotic system. Complexity 2020, 1 (2020)

    MATH  Google Scholar 

  31. J. Kengne, G.D. Leutcho, A.N.K. Telem, Reversals of period doubling, coexisting multiple attractors, and offset boosting in a novel memristive diode bridge-based hyperjerk circuit. Analog Integr. Circ. Signal Process. 131, 379 (2018)

    Google Scholar 

  32. Z. Gu, C. Li, X. Pei, C. Tao, Z. Liu, A conditional symmetric memristive system with amplitude and frequency control. Eur. Phys. J. Spec. Top. 229, 1007 (2020)

    Article  Google Scholar 

  33. Y. Fei, L. Li et al., Chaos-based application of a novel multistable 5D memristive hyperchaotic system with coexisting multiple attractors. Complexity 2020, 1 (2020)

    Article  Google Scholar 

  34. Q. Lai, A. Akgul, C. Li, G. Xu et al., A new chaotic system with multiple attractors: dynamic analysis, circuit realization and s-box design. Entropy 20, 12 (2018)

    Article  ADS  Google Scholar 

  35. F. Yu, H. Shen, L. Liu, Z. Zhang, Q. Xu, Ccii and fpga realization: a multistable modified fourth-order autonomous chua’s chaotic system with coexisting multiple attractors. Complexity 2020, 1 (2020)

    Article  MATH  Google Scholar 

  36. B. Bao, T. Jiang, G. Wang, P. Jin, H. Bao, M. Chen, Two-memristor-based chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability. Nonlinear Dyn. 89, 1157 (2017)

    Article  Google Scholar 

  37. Q. Lai, P.D.K. Kuate, F. Liu, H.H.-C. Iu, An extremely simple chaotic system with infinitely many coexisting attractors. IEEE Trans. Circ. Syst. II Express Briefs 67, 1129 (2020)

    ADS  Google Scholar 

  38. L. Gervais D, K. Jacques, et al. Multistability control of space magnetization in hyperjerk oscillator: a case study. J. Comput. Nonlinear Dyn. 15, (2020)

  39. J.G. Silva, A.C. Ribeiro, R.F. Camargo, P.F. Mancera, F.L. Santos, Stability analysis and numerical simulations via fractional calculus for tumor dormancy models. Commun. Nonlinear Sci. Numer. Simul. 72, 528 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. A. Persechino, An introduction to fractional calculus numerical methods and application to hf dielectric response. Smart Mater. Struct. 9, 106 (2020)

    Google Scholar 

  41. R. Khalil, M.A. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. F.C.F. Marques Jr., T.P. De Araujo, J.V.M. Sousa, C.C. Nator Jr., A. Saraiva, Recognition of simple handwritten polynomials using segmentation with fractional calculus and convolutional neural networks. In: 2019 8th Brazilian Conference on Intelligent Systems (BRACIS), 245 (2019)

  43. B. Ghanbari, H. Gnerhan, H. Srivastava, An application of the atangana–baleanu fractional derivative in mathematical biology: a three-species predator-prey model. Chaos Solit. Fract. 138, 109910 (2020)

    Article  MathSciNet  Google Scholar 

  44. A. Yousefpour, H. Jahanshahi, J.M. Munoz-Pacheco, S. Bekiros, Z. Wei, A fractional-order hyper-chaotic economic system with transient chaos. Chaos Solit. Fract. 130, 109400 (2019)

    Article  MathSciNet  Google Scholar 

  45. S. Soradi-Zeid, H. Jahanshahi, A. Yousefpour, S. Bekiros, King algorithm: a novel optimization approach based on variable-order fractional calculus with application in chaotic financial systems. Chaos Solit. Fract. 132, 109569 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  46. Y. Peng, K. Sun, S. He, Synchronization for the integer-order and fractional-order chaotic maps based on parameter estimation with jaya-ipso algorithm. Eur. Phys. J. Plus 135, 331 (2020)

    Article  Google Scholar 

  47. C. Ma, J. Mou, Y. Cao, T. Liu, J. Wang, Multistability analysis of a conformable fractional-order chaotic system. Phys. Scr. 95, 75204 (2020)

    Article  Google Scholar 

  48. S. He, K. Sun, H. Wang, Dynamics and synchronization of conformable fractional-order hyperchaotic systems using the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 73, 146 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. M. Dutta, K. Binoy, Roy, A new fractional-order system displaying coexisting multiwing attractors; its synchronisation and circuit simulation. Chaos Solit. Fract. 130, 109414 (2020)

    Article  Google Scholar 

  50. D. Ding, X. Shan, L. Jun, Y. Hu, L. Ding, Initial boosting phenomenon of a fractional-order hyperchaotic system based on dual memristors. Mod. Phys. Lett. B 3, 2050191 (2020)

    Article  MathSciNet  Google Scholar 

  51. Y. Peng, K. Sun, D. Peng, W. Ai, Dynamics of a higher dimensional fractional-order chaotic map. Physica A Stat. Mech. Appl. 525, 96 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  52. S. He, K. Sun, Y. Peng, Detecting chaos in fractional-order nonlinear systems using the smaller alignment index. Phys. Lett. A 383, 2267 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  53. M. Wang, X. Liao, Y. Deng, Z. Li, Y. Zeng, Dynamics, synchronization and circuit implementation of a simple fractional-order chaotic system with hidden attractors. Chaos Solit. Fract. 130, 109406 (2020)

    Article  MathSciNet  Google Scholar 

  54. M. Pacheco, J. M, Infinitely many hidden attractors in a new fractional-order chaotic system based on a fracmemristor. Eur. Phys. J. Spec. Top. 228, 2185 (2020)

    Article  Google Scholar 

  55. F. Chen, X. Luo, Y. Zhou, Existence results for nonlinear fractional difference equation. Adv. Differ. Equ. 2011, 1 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  56. A. Wolf, J.B. Swift, H. Swinney, J.A. Vastano, Determining lyapunov exponents from a time series. Physica D Nonlinear Phenomena 16, 285 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. P. Grassberger, I. Procaccia, Estimation of the kolmogorov entropy from a chaotic signal. Phys. Rev. A 28, 2591 (1983)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by the Basic Scientific Research Projects of Colleges and Universities of Liaoning Province (Grant Nos. 2017J045); the Natural Science Foundation of Liaoning province(2020-MS-274).

Author information

Authors and Affiliations

  1. School of Information Science and Engineering, Dalian polytechnic University, Dalian, 116034, China

    Chenguang Ma, Jun Mou, Peng Li & Tianming Liu

Authors
  1. Chenguang Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Mou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tianming Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Chenguang Ma designed, simulated the experiment and wrote the manuscript. Jun Mou made the theoretical guidance for this paper. Peng Li improved the algorithm. Tianming Liu in charge of the data analyzed.

Corresponding author

Correspondence to Jun Mou.

Ethics declarations

Conflict of interest

The authors declare that we have no conflicts of interest about the publication of this paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ma, C., Mou, J., Li, P. et al. Dynamic analysis of a new two-dimensional map in three forms: integer-order, fractional-order and improper fractional-order. Eur. Phys. J. Spec. Top. 230, 1945–1957 (2021). https://doi.org/10.1140/epjs/s11734-021-00133-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjs/s11734-021-00133-w

Search

Navigation