Skip to main content
SpringerLink
Log in

Global and local performance metric with inertia effects

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

Abstract

A complex system’s structural–dynamical interplay plays a profound role in determining its collective behavior. Irregular behavior in the form of macroscopic chaos, for instance, can be potentially exhibited by the Kuramoto model of coupled phase oscillators at intermediate coupling strength with frequency assortativity and this behavior is theoretically interesting. In practice, however, such irregular behavior is often not under control and is undesired for the system’s functioning. How the underlying structural and oscillators’ dynamical interplay affects a collective phenomenon (and its corresponding stability) after being subjected to disturbances, attracts great attention. Here, we exploit the concept of a coherency performance metric, as a sum of phase differences and frequency displacements, to evaluate the response to perturbations on network-coupled oscillators. We derive the performance metric as a quadratic form of the eigenvalues and eigenmodes corresponding to the unperturbed system and the perturbation vector, and analyze the influences of perturbation direction as well as strength on the metric. We further apply a computational approach to obtain the performance metric’s derivative with respect to the oscillators’ inertia. We finally extend the metric to a local definition which reflects the pairwise casual effects between any two oscillators. These results deepen the understanding of the combined effects of the structural (eigenmodes) and dynamical (inertia) effects on the system stability.

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

Similar content being viewed by others

References

  1. Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D: Nonlinear Phenom. 143(1–4), 1–20 (2000)

    MathSciNet  MATH  Google Scholar 

  2. Acebrón, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77(1), 137 (2005)

    Google Scholar 

  3. Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469(3), 93–153 (2008)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. Skardal, P.S., Restrepo, J.G., Ott, E.: Frequency assortativity can induce chaos in oscillator networks. Phys. Rev. E 91(6), 060902 (2015)

    MathSciNet  Google Scholar 

  6. So, P., Barreto, E.: Generating macroscopic chaos in a network of globally coupled phase oscillators. Chaos: Interdiscip. J. Nonlinear Sci. 21(3), 033127 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Komarov, M., Pikovsky, A.: Dynamics of multifrequency oscillator communities. Phys. Rev. Lett. 110(13), 134101 (2013)

    Google Scholar 

  8. Popovych, O.V., Maistrenko, Y.L., Tass, P.A.: Phase chaos in coupled oscillators. Phys. Rev. E 71(6), 065201 (2005)

    MathSciNet  Google Scholar 

  9. Maistrenko, Y.L., Popovych, O.V., Tass, P.A.: Chaotic attractor in the Kuramoto model. Int. J. Bifurc. Chaos 15(11), 3457–3466 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Wolfrum, M., Omelchenko, O.E., Yanchuk, S., Maistrenko, Y.L.: Spectral properties of chimera states. Chaos: Interdiscip. J. Nonlinear Sci. 21(1), 013112 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Olmi, S., Navas, A., Boccaletti, S., Torcini, A.: Hysteretic transitions in the Kuramoto model with inertia. Phys. Rev. E 90(4), 042905 (2014)

    Google Scholar 

  12. Acebrón, J.A., Bonilla, L.L., Spigler, R.: Synchronization in populations of globally coupled oscillators with inertial effects. Phys. Rev. E 62(3), 3437 (2000)

    MathSciNet  Google Scholar 

  13. Ha, S.-Y., Noh, S.E., Park, J.: Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators. Anal. Appl. 15(06), 837–861 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Choi, Y.-P., Ha, S.-Y., Morales, J.: Emergent dynamics of the Kuramoto ensemble under the effect of inertia. ArXiv preprint arXiv:1707.07164 (2017)

  15. Hsia, C.-H., Jung, C.-Y., Kwon, B.: On the synchronization theory of Kuramoto oscillators under the effect of inertia. J. Differ. Equ. 267(2), 742–775 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Barre, J., Métivier, D.: Bifurcations and singularities for coupled oscillators with inertia and frustration. Phys. Rev. Lett. 117(21), 214102 (2016)

    Google Scholar 

  17. Tanaka, H.-A., Lichtenberg, A.J., Oishi, S.: First order phase transition resulting from finite inertia in coupled oscillator systems. Phys. Rev. Lett. 78(11), 2104 (1997)

    Google Scholar 

  18. Maistrenko, Y., Brezetsky, S., Jaros, P., Levchenko, R., Kapitaniak, T.: Smallest chimera states. Phys. Rev. E 95(1), 010203 (2017)

    Google Scholar 

  19. Hellmann, F., Schultz, P., Jaros, P., Levchenko, R., Kapitaniak, T., Kurths, J., Maistrenko, Y.: Network-induced multistability through lossy coupling and exotic solitary states. Nat. Commun. 11(1), 1–9 (2020)

    Google Scholar 

  20. Taher, H., Olmi, S., Schöll, E.: Enhancing power grid synchronization and stability through time-delayed feedback control. Phys. Rev. E 100(6), 062306 (2019)

    Google Scholar 

  21. Dai, F., Zhou, S., Peron, T., Lin, W., Ji, P.: Interplay among inertia, time delay, and frustration on synchronization dynamics. Phys. Rev. E 98(5), 052218 (2018)

    Google Scholar 

  22. Chen, L., Ji, P., Waxman, D., Lin, W., Kurths, J.: Effects of dynamical and structural modifications on synchronization. Chaos: Interdiscip. J. Nonlinear Sci. 29(8), 083131 (2019)

    MathSciNet  MATH  Google Scholar 

  23. Ji, P., Zhu, L., Chao, L., Lin, W., Kurths, J.: How price-based frequency regulation impacts stability in power grids: a complex network perspective. Complexity 2020, 1–10 (2020). https://doi.org/10.1155/2020/6297134

    Article  Google Scholar 

  24. Rohden, M., Sorge, A., Timme, M., Witthaut, D.: Self-organized synchronization in decentralized power grids. Phys. Rev. Lett. 109(6), 064101 (2012)

    Google Scholar 

  25. Pasqualetti, F., Dörfler, F., Bullo, F.: Attack detection and identification in cyber-physical systems. IEEE Trans. Autom. Control 58(11), 2715–2729 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Li, K., Ma, S., Li, H., Yang, J.: Transition to synchronization in a Kuramoto model with the first- and second-order interaction terms. Phys. Rev. E 89(3), 032917 (2014)

    Google Scholar 

  27. Pecora, L.M., Carroll, T.L.: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80(10), 2109 (1998)

    Google Scholar 

  28. Menck, P.J., Heitzig, J., Marwan, N., Kurths, J.: How basin stability complements the linear-stability paradigm. Nat. Phys. 9(2), 89 (2013)

    Google Scholar 

  29. Schröder, M., Timme, M., Witthaut, D.: A universal order parameter for synchrony in networks of limit cycle oscillators. Chaos: Interdiscip. J. Nonlinear Sci. 27(7), 073119 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Skardal, P.S., Taylor, D., Sun, J.: Optimal synchronization of complex networks. Phys. Rev. Lett. 113(14), 144101 (2014)

    Google Scholar 

  31. Grabow, C., Hill, S.M., Grosskinsky, S., Timme, M.: Do small worlds synchronize fastest? Europhys. Lett. 90(4), 48002 (2010)

    Google Scholar 

  32. Jörg, D.J., Morelli, L.G., Ares, S., Jülicher, F.: Synchronization dynamics in the presence of coupling delays and phase shifts. Phys. Rev. Lett. 112(17), 174101 (2014)

    Google Scholar 

  33. Schultz, P., Hellmann, F., Webster, K.N., Kurths, J.: Bounding the first exit from the basin: independence times and finite-time basin stability. Chaos: Interdiscip. J. Nonlinear Sci. 28(4), 043102 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Poolla, B.K., Bolognani, S., Dörfler, F.: Placing rotational inertia in power grids. Proc. Am. Control Conf. 2314–2320, 2016 (2016)

    Google Scholar 

  35. Tyloo, M., Coletta, T., Jacquod, P.: Robustness of synchrony in complex networks and generalized Kirchhoff indices. Phys. Rev. Lett. 120(8), 084101 (2018)

    MathSciNet  Google Scholar 

  36. Plietzsch, A., Auer, S., Kurths, J., Hellmann, F.: A generalized linear response theory of complex networks with an application to renewable fluctuations in microgrids. ArXiv preprint arXiv:1903.09585 (2019)

  37. Aljadeff, J., Stern, M., Sharpee, T.: Transition to chaos in random networks with cell-type-specific connectivity. Phys. Rev. Lett. 114(8), 088101 (2015)

    Google Scholar 

  38. Olmi, S.: Chimera states in coupled Kuramoto oscillators with inertia. Chaos: Interdiscip. J. Nonlinear Sci. 25(12), 123125 (2015)

    MathSciNet  MATH  Google Scholar 

  39. Bergen, A.R., Hill, D.J.: A structure preserving model for power system stability analysis. IEEE Trans. Power Appar. Syst. 1, 25–35 (1981)

    Google Scholar 

  40. Gelfand, I.M., Shilov, G.E.: Generalized Functions, Vol. 4: Applications of Harmonic Analysis. Academic Press, New York (1964)

    Google Scholar 

  41. Hewitt, E., Stromberg, K.: Real and Abstract Analysis: A Modern Treatment of the Theory of Functions of a Real Variable. Springer, New York (2013)

    MATH  Google Scholar 

  42. Schultz, P., Heitzig, J., Kurths, J.: Detours around basin stability in power networks. New J. Phys. 16(12), 125001 (2014)

    Google Scholar 

  43. Ji, P., Kurths, J.: Basin stability of the Kuramoto-like model in small networks. Eur. Phys. J. Spec. Top. 223(12), 2483–2491 (2014)

    Google Scholar 

  44. Nitzbon, J., Schultz, P., Heitzig, J., Kurths, J., Hellmann, F.: Deciphering the imprint of topology on nonlinear dynamical network stability. New J. Phys. 19(3), 033029 (2017)

    Google Scholar 

Download references

Acknowledgements

We acknowledge National Key R&D Program of China (2018YFB0904500), the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, NSFC (under Grants 61932008, 61772368, 61773125), Shanghai Municipal Science and Technology Major Project (2018SHZDZX01) and ZJLab. Schultz was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–KU 837/39-1 / RA 516/13-1. J. Kurth was supported by the project RF Goverment Grant No. 075-15-2019-1885. We appreciate Rico Berner for constructive suggestions.

Author information

Authors and Affiliations

  1. Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai, 200433, China

    Qiang Li, Wei Lin & Peng Ji

  2. Research Institute of Intelligent Complex Systems, Fudan University, Shanghai, 200433, China

    Qiang Li, Wei Lin & Peng Ji

  3. LCNBI and LMNS (Fudan University), Ministry of Education, Shanghai, 200433, China

    Qiang Li, Wei Lin & Peng Ji

  4. Potsdam Institute for Climate Impact Research (PIK), 14473, Potsdam, Germany

    Paul Schultz

  5. Department of Physics, Humboldt University, 12489, Berlin, Germany

    Paul Schultz

  6. Saratov State University, 410012, Saratov, Russia

    Jürgen Kurths

Authors
  1. Qiang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul Schultz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jürgen Kurths
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Peng Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peng Ji.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest regarding the publication of this paper.

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

Li, Q., Schultz, P., Lin, W. et al. Global and local performance metric with inertia effects. Nonlinear Dyn 102, 653–665 (2020). https://doi.org/10.1007/s11071-020-05872-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-05872-4

Keywords

Search

Navigation