Skip to main content
SpringerLink
Log in

Transformation of Master-Slave Systems with Harmonic Terms for Improved Stability in Numerical Continuation

  • Conference paper
  • First Online:
15th Chaotic Modeling and Simulation International Conference (CHAOS 2022)

Part of the book series: Springer Proceedings in Complexity ((SPCOM))

Included in the following conference series:

Abstract

Nonlinear problems with forced oscillations occur in many applications in physics, neuroscience, epidemiology, or physiology. Forced oscillations are usually modeled as non-autonomous master-slave systems with harmonic driving. The aim of this work is to provide a transformation suitable for analyzing such systems via standard numerical continuation packages such as MATCONT and AUTO. We transform the original system into a structurally stable generalized system by replacing each harmonic term in the original system by a supercritical Hopf bifurcation normal form subsystem. Our method is general; being applicable to an important class of nonlinear equations in which the driving or coupling occurs via harmonic terms involving phases. Here we apply it to analyze the dynamics of a driven single Josephson junction, shunted by an inductor-resistor-capacitor resonant circuit.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence. Chemistry Series (Dover, New York, 1984)

    Google Scholar 

  2. A.T. Winfree, The Geometry of Biological Time (Springer, New York, 2001)

    Book  MATH  Google Scholar 

  3. H. Ju, A.B. Neiman, A.L. Shilnikov, Bottom-up approach to torus bifurcation in neuron models. Chaos 28(10), 106317 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. S.A. Campbell, Z. Wang, Phase models and clustering in networks of oscillators with delayed coupling. Physica D 363, 44–55 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Alinejad, D.-P. Yang, P.A. Robinson, Mode-locking dynamics of corticothalamic system response to periodic external stimuli. Physica D 402, 132231 (2020)

    Google Scholar 

  6. U. Welp, K. Kadowaki, R. Kleiner, Superconducting emitters of THz radiation. Nat. Photonics 7, 702 (2013)

    Article  Google Scholar 

  7. A.I. Braginski, Superconductor electronics: status and outlook. J. Supercond. Nov. Magn. 32, 23–44 (2019)

    Article  Google Scholar 

  8. B. Sturgis-Jensen, P.-L. Buono, A. Palacios, J. Turtle, V. In, P. Longhini, On the synchronization phenomenon of a parallel array of spin torque nano-oscillators. Physica D 396, 71–81 (2019)

    Google Scholar 

  9. A. Mallick et al., Using synchronized oscillators to compute the maximum independent set. Nat Commun. 11, 4689 (2020)

    Article  Google Scholar 

  10. J. Yan, C. Beck, Nonlinear dynamics of coupled axion-Josephson junction systems. Physica D 403, 132294 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Yu.M. Shukrinov, M. Hamdipour, M.R. Kolahchi, A.E. Botha, M. Suzuki, Manifestation of resonance-related chaos in coupled Josephson junctions. Phys. Lett. A 376, 3609–3619 (2012)

    Article  Google Scholar 

  12. B.D. Josephson, Possible new effects in superconductive tunnelling. Phys. Lett. 1, 251 (1962)

    Article  MATH  Google Scholar 

  13. A. Barone, G. Paterno, Physics and Applications of the Josephson Effect (Wiley, New York, 1982)

    Book  Google Scholar 

  14. K.K. Likharev, Dynamics of Josephson Junctions and Circuits (Gordon and Breach Science Publishers, New York, 1986)

    Google Scholar 

  15. P. Crotty, D. Schult, K. Segall, Josephson junction simulation of neurons. Phys. Rev. E 82, 011914 (2010)

    Article  Google Scholar 

  16. M. Jun, H. Long, X. Zhen-Bo, C. Wang, Simulated test of electric activity of neurons by using Josephson junction based on synchronization scheme. Commun. Nonlinear Sci. Numer. Simul. 17, 2659 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Segall, S. Guo, P. Crotty, D. Schult, M. Millera, Phase-flip bifurcation in a coupled Josephson junction neuron system. Physica B Condensed Matter 455, 71 (2014)

    Article  Google Scholar 

  18. R. Kleiner, X. Zhou, E. Dorsch, X. Zhang, D. Koelle, D. Jin, Space-time crystalline order of a high-critical-temperature superconductor with intrinsic Josephson junctions. Nat. Commun. 12, 6038 (2021)

    Article  Google Scholar 

  19. S. Saha, S.K. Dana, Smallest chimeras under repulsive interactions. Front. Netw. Physiol. 1, 17 (2021)

    Google Scholar 

  20. A. Mishra, S. Ghosh, S. Kumar, T. Kapitaniak, Ch. Hens, Neuron-like spiking and bursting in Josephson junctions: a review. Chaos 31, 052102 (2021)

    Article  MathSciNet  Google Scholar 

  21. D. Chalkiadakis, J. Hizanidis, Dynamical properties of neuromorphic Josephson junctions. Phys. Rev. E 106, 044206 (2022)

    Google Scholar 

  22. Yu.M. Shukrinov, I.R. Rahmonov, K.V. Kulikov, P. Seidel, Effects of LC shunting on the Shapiro steps features of Josephson junction. Europhys. Lett. 110, 47001 (2015)

    Article  Google Scholar 

  23. Yu.M. Shukrinov, I.R. Rahmonov, K.V. Kulikov, Double resonance in the system of coupled Josephson junctions. JETP Lett. 96, 588 (2012)

    Article  Google Scholar 

  24. M.R. Roussel, Nonlinear Dynamics: A Hands-on Introductory Survey (Morgan & Claypool Publishers, San Rafael, U.S.A., 2019)

    Google Scholar 

  25. X. Li, J. Ren, S.A. Campbell, G.S.K. Wolkowicz, H. Zhu, How seasonal forcing influences the complexity of a predator-prey system. Discrete Continuous Dyn. Syst. B 23(2), 785 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, MatCont: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29(2), 141–164 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, H.G.E. Meijer, B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems. Math. Comput. Model. Dyn. Syst. 14(2), 147–175 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. (Springer, New York, 2003)

    Google Scholar 

  29. V. Eclerová, L. Přibylová, A.E. Botha, Embedding nonlinear systems with two or more harmonic phase terms near the Hopf-Hopf bifurcation. Nonlinear Dyn 111, 1537–1551 (2022)

    Google Scholar 

  30. Yu.A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, vol. 112, 2nd edn. (Springer, New York, 1998)

    Google Scholar 

  31. J. Dushoff, J.B. Plotkin, S.A. Levin, D.J.D. Earn, Dynamical resonance can account for seasonality of influenza epidemics. Proc. Nat. Acad. Sci. 101(48), 16915–16916 (2004)

    Article  Google Scholar 

  32. R.B. Simpson, B. Zhou, E.N. Naumova, Seasonal synchronization of foodborne outbreaks in the United States, 1996–2017. Sci. Rep. 10(1) (2020)

    Google Scholar 

Download references

Acknowledgements

The work has received financial support from Mathematical and Statistical Modelling Project MUNI/A/1342/2021 and the National Research Foundation of South Africa Grant Number JINR190408428424.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlarska 2, Brno, Czech Republic

    Veronika Eclerová & Lenka Přibylová

  2. Department of Physics, Unisa Science Campus, University of South Africa, Johannesburg, 1710, South Africa

    André E. Botha

Authors
  1. Veronika Eclerová
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lenka Přibylová
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. André E. Botha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Veronika Eclerová .

Editor information

Editors and Affiliations

  1. ISAST, Athens, Greece

    Christos H. Skiadas

  2. Department of Management Science and Technology, Hellenic Mediterranean University, Crete, Lasithi, Greece

    Yiannis Dimotikalis

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Eclerová, V., Přibylová, L., Botha, A.E. (2023). Transformation of Master-Slave Systems with Harmonic Terms for Improved Stability in Numerical Continuation. In: Skiadas, C.H., Dimotikalis, Y. (eds) 15th Chaotic Modeling and Simulation International Conference. CHAOS 2022. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-031-27082-6_7

Download citation

Publish with us

Policies and ethics

Search

Navigation