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Nonlinear Dynamics

, Volume 95, Issue 3, pp 2093–2102 | Cite as

Amplitude death islands in globally delay-coupled fractional-order oscillators

  • Rui Xiao
  • Zhongkui SunEmail author
  • Xiaoli Yang
  • Wei Xu
Original Paper
  • 191 Downloads

Abstract

In this paper, amplitude death (AD) is investigated theoretically and numerically in N globally delay-coupled fractional-order oscillators. Due to the presence of fractional-order derivative and coupling delay, Laplace transform method has been utilized to obtain the characteristic equations. Then, based on Lyapunov stability, we theoretically get the boundaries and number of death islands. It has been found that with the introduction of the fractional-order derivative, many more death islands emerge, and the oscillation quenching dynamics are facilitated. We find AD only occurs between two critical fractional-order derivatives \(\alpha _c^ - \) (lower-bounded value) and \(\alpha _c^ + \) (upper-bounded value) which are affected by natural frequency and system size. With the increment of system size, the oscillation quenching dynamics are weakened. The number of death islands is closely geared to the fractional-order derivative and the system size. Furthermore, the results from numerical simulations best confirm the theoretical analyses.

Keywords

Coupled oscillators Fractional-order derivative Coupling delay Amplitude death islands 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11772254, 11742013) and the NPU Foundation for Fundamental Research.

Compliances with ethical standards

Conflict of interest

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

References

  1. 1.
    Pikovsky, A., Rosenblum, M., Kurths, J., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences, vol. 12. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  2. 2.
    Sun, Z., Yang, X.: Generating and enhancing lag synchronization of chaotic systems by white noise. Chaos Interdiscip. J. Nonlinear Sci. 21(3), 033114 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Saxena, G., Prasad, A., Ramaswamy, R.: Amplitude death: the emergence of stationarity in coupled nonlinear systems. Phys. Rep. 521(5), 205 (2012)CrossRefGoogle Scholar
  4. 4.
    Koseska, A., Volkov, E., Kurths, J.: Oscillation quenching mechanisms: amplitude vs. oscillation death. Phys. Rep. 531(4), 173 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Daido, H., Nakanishi, K.: Aging transition and universal scaling in oscillator networks. Phys. Rev. Lett. 93(10), 104101 (2004)CrossRefGoogle Scholar
  6. 6.
    Sun, Z., Ma, N., Xu, W.: Aging transition by random errors. Sci. Rep. 7, 42715 (2017)CrossRefGoogle Scholar
  7. 7.
    Reddy, D.R., Sen, A., Johnston, G.L.: Time delay induced death in coupled limit cycle oscillators. Phys. Rev. Lett. 80(23), 5109 (1998)CrossRefGoogle Scholar
  8. 8.
    Reddy, D.R., Sen, A., Johnston, G.L.: Time delay effects on coupled limit cycle oscillators at Hopf bifurcation. Physica D: Nonlinear Phenom. 129(1–2), 15 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Atay, F.M.: Distributed delays facilitate amplitude death of coupled oscillators. Phys. Rev. Lett. 91(9), 094101 (2003)CrossRefGoogle Scholar
  10. 10.
    Kyrychko, Y., Blyuss, K., Schöll, E.: Amplitude and phase dynamics in oscillators with distributed-delay coupling. Phil. Trans. R. Soc. A 371(1999), 20120466 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Saxena, G., Prasad, A., Ramaswamy, R.: Dynamical effects of integrative time-delay coupling. Phys. Rev. E 82(1), 017201 (2010)CrossRefGoogle Scholar
  12. 12.
    Konishi, K., Kokame, H., Hara, N.: Stability analysis and design of amplitude death induced by a time-varying delay connection. Phys. Lett. A 374(5), 733 (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Gjurchinovski, A., Zakharova, A., Schöll, E.: Amplitude death in oscillator networks with variable-delay coupling. Phys. Rev. E 89(3), 032915 (2014)CrossRefGoogle Scholar
  14. 14.
    Aronson, D.G., Ermentrout, G.B., Kopell, N.: Amplitude response of coupled oscillators. Physica D: Nonlinear Phenom. 41(3), 403 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Matthews, P.C., Strogatz, S.H.: Phase diagram for the collective behavior of limit-cycle oscillators. Phys. Rev. Lett. 65(14), 1701 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sun, Z., Zhao, N., Yang, X., Xu, W.: Inducing amplitude death via discontinuous coupling. Nonlinear Dyn. 92(3), 1185 (2018)CrossRefGoogle Scholar
  17. 17.
    Karnatak, R., Ramaswamy, R., Prasad, A.: Amplitude death in the absence of time delays in identical coupled oscillators. Phys. Rev. E 76(3), 035201 (2007)CrossRefGoogle Scholar
  18. 18.
    Sun, Z., Xiao, R., Yang, X., Xu, W.: Quenching oscillating behaviors in fractional coupled Stuart-Landau oscillators. Chaos: Interdiscip. J. Nonlinear Sci. 28(3), 033109 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Konishi, K.: Amplitude death induced by dynamic coupling. Phys. Rev. E 68(6), 067202 (2003)CrossRefGoogle Scholar
  20. 20.
    Konishi, K., Hara, N.: Topology-free stability of a steady state in network systems with dynamic connections. Phys. Rev. E 83(3), 036204 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Prasad, A., Dhamala, M., Adhikari, B.M., Ramaswamy, R.: Amplitude death in nonlinear oscillators with nonlinear coupling. Phys. Rev. E 81(2), 027201 (2010)CrossRefGoogle Scholar
  22. 22.
    Prasad, A., Lai, Y.C., Gavrielides, A., Kovanis, V.: Amplitude modulation in a pair of time-delay coupled external-cavity semiconductor lasers. Phys. Lett. A 318(1–2), 71 (2003)CrossRefzbMATHGoogle Scholar
  23. 23.
    Reddy, D.R., Sen, A., Johnston, G.L.: Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators. Phys. Rev. Lett. 85(16), 3381 (2000)CrossRefGoogle Scholar
  24. 24.
    Ermentrout, G., Kopell, N.: Oscillator death in systems of coupled neural oscillators. SIAM J. Appl. Math. 50(1), 125 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, Z., Huang, X., Shi, G.: Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay. Comput. Math. Appl. 62(3), 1531 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional-order neural networks. Neural Netw. 32, 245 (2012)CrossRefzbMATHGoogle Scholar
  27. 27.
    Yang, X.J., Machado, J.T., Hristov, J.: Nonlinear dynamics for local fractional Burgers equation arising in fractal flow. Nonlinear Dyn. 84(1), 3 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Huang, C., Cao, J., Xiao, M., Alsaedi, A., Alsaadi, F.E.: Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders. Appl. Math. Comput. 293, 293 (2017)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sun, Z., Zhang, J., Yang, X., Xu, W.: Taming stochastic bifurcations in fractional-order systems via noise and delayed feedback. Chaos: Interdiscip. J. Nonlinear Sci. 27(8), 083102 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Chen, L., Chai, Y., Wu, R., Ma, T., Zhai, H.: Dynamic analysis of a class of fractional-order neural networks with delay. Neurocomputing 111, 190 (2013)CrossRefGoogle Scholar
  31. 31.
    Bao, H., Park, J.H., Cao, J.: Adaptive synchronization of fractional-order memristor-based neural networks with time delay. Nonlinear Dyn. 82(3), 1343 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Huang, C., Cao, J., Xiao, M., Alsaedi, A., Hayat, T.: Bifurcations in a delayed fractional complex-valued neural network. Appl. Math. Comput. 292, 210 (2017)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1–4), 3 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Li, C., Deng, W.: Remarks on fractional derivatives. Appl. Math. Comput. 187(2), 777 (2007)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Ngueuteu, G., Yamapi, R., Woafo, P.: Fractional derivation stabilizing virtue-induced quenching phenomena in coupled oscillators. EPL (Europhys. Lett.) 112(3), 30004 (2015)CrossRefGoogle Scholar
  36. 36.
    Deng, W., Li, C., Lü, J.: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn. 48(4), 409 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China
  2. 2.College of Mathematics and Information ScienceShaan’xi Normal UniversityXi’anPeople’s Republic of China

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