Skip to main content
SpringerLink
Log in

Explosive behaviors on coupled fractional-order system

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Fractional derivatives provide a prominent platform for various chemical and physical system with memory and hereditary properties, while most of the previous differential systems used to describe dynamic phenomena including oscillation quenching are integer order. Here, effects of fractional derivative on the transition process from oscillatory state to stationary state are illustrated for the first time on mean-filed coupled oscillators. It is found the fractional derivative could induce the emergence of a first-order discrete transition with hysteresis between oscillatory and stationary state. However, if the fractional derivative is smaller than the critical value, the transition will be invertible. Besides, the theoretical conditions for the steady state are calculated via Lyapunov indirect method which probe that, the backward transition point is unrelated to mean-field density. Our result is a step forward in enlightening the control mechanism of explosive phenomenon, which is of great importance to highlight the function of fractional-order derivative in the emergence of collective behaviors on coupled nonlinear model.

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

Similar content being viewed by others

Data availability

Data will be made available on reasonable request.

References

  1. Abrams, D.M., Strogatz, S.H.: Chimera states for coupled oscillators. Phys. Rev. Lett. 93(17), 174102.1-174102.4 (2004)

    Google Scholar 

  2. Bagley, R.L., Calico, R.: Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control. Dyn. 14(2), 304–311 (1991)

    Google Scholar 

  3. Bagley, R.L., Torvik, P.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27(3), 201–210 (1983)

    MATH  Google Scholar 

  4. Bar-Eli, K., Reuveni, S.: Stable stationary states of coupled chemical oscillators. Experimental evidence. J. Phys. Chem. 89(8), 1329–1330 (1985)

    Google Scholar 

  5. Berner, R., Vock, S., Schöll, E., Yanchuk, S.: Desynchronization transitions in adaptive networks. Phys. Rev. Lett. 126(2), 028301 (2021)

    MathSciNet  Google Scholar 

  6. Beuter, A., Glass, L., Mackey, M.C., Titcombe, M.S.: Nonlinear Dynamics in Physiology and Medicine. Springer, Berlin (2003)

    MATH  Google Scholar 

  7. Bi, H., Hu, X., Zhang, X., Zou, Y., Liu, Z., Guan, S.: Explosive oscillation death in coupled Stuart-Landau oscillators. EPL (Europhys. Lett.) 108(5), 50003 (2014)

    Google Scholar 

  8. Boccaletti, S., Almendral, J., Guan, S., Leyva, I., Liu, Z., Sendiña-Nadal, I., Wang, Z., Zou, Y.: Explosive transitions in complex networks’ structure and dynamics: percolation and synchronization. Phys. Rep. 660, 1–94 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Crowley, M.F., Epstein, I.R.: Experimental and theoretical studies of a coupled chemical oscillator: phase death, multistability and in-phase and out-of-phase entrainment. J. Phys. Chem. 93(6), 2496–2502 (1989)

    Google Scholar 

  10. Dai, X., Li, X., Guo, H., Jia, D., Boccaletti, S.: Discontinuous transitions and rhythmic states in the D-dimensional Kuramoto model induced by a positive feedback with the global order parameter. Phys. Rev. Lett. 125(19), 194101 (2020)

    MathSciNet  Google Scholar 

  11. Dixit, S., Chowdhury, S.N., Ghosh, D., Shrimali, M.D.: Dynamic interaction induced explosive death. EPL (Europhys. Lett.) 133(4), 40003 (2021)

    Google Scholar 

  12. Fan, H., Lai, Y.C., Wang, X.: Enhancing network synchronization by phase modulation. Phys. Rev. E 98(1), 012212 (2018)

    Google Scholar 

  13. Gómez-Gardenes, J., Gómez, S., Arenas, A., Moreno, Y.: Explosive synchronization transitions in scale-free networks. Phys. Rev. Lett. 106(12), 128701 (2011)

    Google Scholar 

  14. Karnatak, R., Ansmann, G., Feudel, U., Lehnertz, K.: Route to extreme events in excitable systems. Phys. Rev. E 90(2), 022917 (2014)

    Google Scholar 

  15. Kim, M., Mashour, G.A., Moraes, S.B., Vanini, G., Tarnal, V., Janke, E., Hudetz, A.G., Lee, U.: Functional and topological conditions for explosive synchronization develop in human brain networks with the onset of anesthetic-induced unconsciousness. Front. Comput. Neurosci. 10, 1 (2016)

    Google Scholar 

  16. Kong, L.W., Lai, Y.C.: Scaling law of transient lifetime of chimera states under dimension-augmenting perturbations. Phys. Rev. Res. 2(2), 023196 (2020)

    Google Scholar 

  17. Koseska, A., Volkov, E., Kurths, J.: Oscillation quenching mechanisms: amplitude vs. oscillation death. Phys. Rep. 531(4), 173–199 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Kuramoto, Y.: Springer Series in Synergetics, vol. 19. Springer, Berlin (1984)

    Google Scholar 

  19. Kuramoto, Y., Battogtokh, D.: Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Physics 4, 385 (2002)

    Google Scholar 

  20. Leyva, I., Sevilla-Escoboza, R., Buldú, J., Sendina-Nadal, I., Gómez-Gardeñes, J., Arenas, A., Moreno, Y., Gómez, S., Jaimes-Reátegui, R., Boccaletti, S.: Explosive first-order transition to synchrony in networked chaotic oscillators. Phys. Rev. Lett. 108(16), 168702 (2012)

    Google Scholar 

  21. Li, C., Deng, W.: Remarks on fractional derivatives. Appl. Math. Comput. 187(2), 777–784 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Liu, S., Sun, Z., Zhao, N.: Tuning coupling rate to control oscillation quenching in fractional-order coupled oscillators. Chaos 30(10), 103108 (2020)

    MathSciNet  MATH  Google Scholar 

  23. Liu, S., Sun, Z., Zhao, N., Xu, W.: Explosive death induced by environmental coupling. Commun. Nonlinear Sci. Numer. Simul. 98, 105774 (2021)

    MathSciNet  MATH  Google Scholar 

  24. Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems Applications, vol. 2, pp. 963–968. Citeseer (1996)

  25. Mishra, V., Das, S., Jafari, H., Ong, S.: Study of fractional order van der pol equation. J. King Saud Univ.-Sci. 28(1), 55–60 (2016)

    Google Scholar 

  26. Motter, A.E., Myers, S.A., Anghel, M., Nishikawa, T.: Spontaneous synchrony in power-grid networks. Nat. Phys. 9(3), 191–197 (2013)

    Google Scholar 

  27. Pereira, E., Monje, C., Vinagre, B., Gordillho, F.: Matlab toolbox for the analysis of fractional order systems with hard nonlinearities. In: Proceedings of the First IFAC Workshop on Fractional Differentiation and Its Applications (FDA’04), Bordeaux, France, pp. 214–219 (2004)

  28. Pikovsky, A., Kurths, J., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences, vol. 12. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  29. Ray, A., Mishra, A., Ghosh, D., Kapitaniak, T., Hens, C.: Extreme events in a network of heterogeneous Josephson junctions. Phys. Rev. E 101(3), 032209 (2020)

    Google Scholar 

  30. Rodrigues, F.A., Peron, T.K.D., Ji, P., Kurths, J.: The Kuramoto model in complex networks. Phys. Rep. 610, 1–98 (2016)

    MathSciNet  MATH  Google Scholar 

  31. Rui, X., Sun, Z., Yang, X., Wei, X.: Amplitude death islands in globally delay-coupled fractional-order oscillators. Nonlinear Dyn. 95(3), 2093 (2018)

    MATH  Google Scholar 

  32. Saxena, G., Prasad, A., Ramaswamy, R.: Amplitude death: The emergence of stationarity in coupled nonlinear systems. Phys. Rep. 521(5), 205–228 (2012)

    Google Scholar 

  33. Strogatz, S.H.: Nonlinear Dynamics and Chaos with Student Solutions Manual: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press, Boca Raton (2018)

    Google Scholar 

  34. Sun, Z., Liu, S., Zhao, N.: Explosive and semi-explosive death in coupled oscillators. Chaos Solitons Fractals 142, 110514 (2021)

    MathSciNet  MATH  Google Scholar 

  35. Tavazoei, M.S., Haeri, M.: A note on the stability of fractional order systems. Math. Comput. Simul. 79(5), 1566–1576 (2009)

    MathSciNet  MATH  Google Scholar 

  36. Thorson, J., Biederman-Thorson, M.: Distributed relaxation processes in sensory adaptation: spatial nonuniformity in receptors can explain both the curious dynamics and logarithmic statics of adaptation. Science 183(4121), 161–172 (1974)

    Google Scholar 

  37. Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 294 (1984)

    MATH  Google Scholar 

  38. Turing, A.M.: The chemical basis of morphogenesis. Bull. Math. Biol. 52(1), 153–197 (1990)

    Google Scholar 

  39. Van der Pol, B.: A theory of the amplitude of free and forced triode vibrations. Radio Rev. 1, 701–710 (1920)

    Google Scholar 

  40. Verma, U.K., Chaurasia, S.S., Sinha, S.: Explosive death in nonlinear oscillators coupled by quorum sensing. Phys. Rev. E 100(3), 032203 (2019)

    Google Scholar 

  41. Verma, U.K., Sharma, A., Kamal, N.K., Kurths, J., Shrimali, M.D.: Explosive death induced by mean-field diffusion in identical oscillators. Sci. Rep. 7(1), 1–7 (2017)

  42. Verma, U.K., Sharma, A., Kamal, N.K., Shrimali, M.D.: First order transition to oscillation death through an environment. Phys. Lett. A 382(32), 2122–2126 (2018)

    MathSciNet  Google Scholar 

  43. Xie, F., Lin, X.: Asymptotic solution of the van der pol oscillator with small fractional damping. Phys. Scr. 2009(T136), 014033 (2009)

    Google Scholar 

  44. Yaffe, R.B., Borger, P., Megevand, P., Groppe, D.M., Kramer, M.A., Chu, C.J., Santaniello, S., Meisel, C., Mehta, A.D., Sarma, S.V.: Physiology of functional and effective networks in epilepsy. Clin. Neurophysiol. 126(2), 227–236 (2015)

    Google Scholar 

  45. Zhang, X., Boccaletti, S., Guan, S., Liu, Z.: Explosive synchronization in adaptive and multilayer networks. Phys. Rev. Lett. 114(3), 038701 (2015)

    Google Scholar 

  46. Zhang, X., Hu, X., Kurths, J., Liu, Z.: Explosive synchronization in a general complex network. Phys. Rev. E 88(1), 010802 (2013)

    Google Scholar 

  47. Zhao, N., Sun, Z., Yang, X., Xu, W.: Explosive death of conjugate coupled van der pol oscillators on networks. Phys. Rev. E 97(6), 062203 (2018)

    Google Scholar 

  48. Zou, W., Senthilkumar, D., Zhan, M., Kurths, J.: Quenching, aging, and reviving in coupled dynamical networks. Phys. Rep. 931, 1–72 (2021)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11772254, 11972-288) and by Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant Nos. CX2021035). The authors would like to thank the anonymous referees for their efforts and valuable comments.

Funding

National Natural Science Foundation of China (Grant Nos. 11772254, 11972288) and Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant Nos. CX2021035)

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, People’s Republic of China

    Shutong Liu, Zhongkui Sun, Luyao Yan & Wei Xu

  2. Department of Mathematics and Information Science, Chang’an University, Xi’an, 710064, People’s Republic of China

    Nannan Zhao

Authors
  1. Shutong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhongkui Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luyao Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nannan Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Wei Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhongkui Sun.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest, with respected to the research, authorship and publication.

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

Liu, S., Sun, Z., Yan, L. et al. Explosive behaviors on coupled fractional-order system. Nonlinear Dyn 110, 2741–2751 (2022). https://doi.org/10.1007/s11071-022-07712-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07712-z

Keywords

Search

Navigation