Izvestiya of Saratov University.

Physics

ISSN 1817-3020 (Print)
ISSN 2542-193X (Online)

For citation:

Kuznetsov A. P., Sataev I. R., Sedova Y. V. Analysis of three non-identical Josephson junctions by the method of Lyapunov exponent charts. Izvestiya of Saratov University. Physics , 2023, vol. 23, iss. 1, pp. 4-13. DOI: 10.18500/1817-3020-2023-23-1-4-13, EDN: AKMGFG

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
01.03.2023
Full text:
download
(downloads: 262)
Language: 
Russian
Heading: 
Radiophysics, electronics, acoustics
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/1817-3020-2023-23-1-4-13
EDN: 
AKMGFG

Analysis of three non-identical Josephson junctions by the method of Lyapunov exponent charts

Autors: 
Kuznetsov Alexander Petrovich, Saratov Branch of the Institute of RadioEngineering and Electronics of Russian Academy of Sciences
Sataev Igor R., Saratov Branch of the Institute of RadioEngineering and Electronics of Russian Academy of Sciences
Sedova Yuliya Viktorovna, Saratov Branch of the Institute of RadioEngineering and Electronics of Russian Academy of Sciences
Abstract: 

Background and Objectives: The Josephson effect is widely used for both generating and receiving very high frequency signals. The approaches and methods of nonlinear dynamics are commonly used: construction of phase portraits, bifurcation analysis, Kuramoto model approach, etc. Usually large arrays of identical junctions are considered. However, the non-identity of junctions leads to new and interesting effects. A very popular model is a chain of junctions connected via an RLC circuit. In this case, two types of non-identity are possible – in terms of critical currents IN through the junctions and in terms of the value rN of junction resistances. An increase in the number of junctions from two to three leads to the possibility of quasiperiodic dynamics with invariant tori of dimensions both two and three and to a complex structure of the parameter space. Materials and Methods: In this paper, we will mainly consider three junctions that are not identical in terms of resistance. As the main research tool, we will use the method of construction of Lyapunov exponent charts. Within the framework of this method, the type of dynamics of the system is determined by the signature of the spectrum of Lyapunov exponents. The parameter plane is scanned and the types of modes are identified at each point. The method is effective in that it allows one to study all types of possible regimes and fine details of the parameter space arrangement. Results: For the analysis of the dynamics of non-identical Josephson junctions, the method of Lyapunov exponent charts is effective. With its help, the regions of periodic regimes, regimes of two-frequency and three-frequency quasiperiodicity, chaos have been revealed. As a rule, the regions of two-frequency quasiperiodicity have the form of bands of different widths immersed in the region of three-frequency quasiperiodicity, which form the structure of the Arnold’s resonant web. Conclusion: The boundaries of the regions of two-frequency quasiperiodicity are the lines of saddle-node bifurcations of invariant tori. As the area of chaos increases, the Arnold’s resonant web can collapse. Changing the type of external circuit coupling the junctions does not fundamentally affect the dynamics of the system. 

Key words: 
Josephson junction
Lyapunov exponent chart
invariant torus
quasiperiodicity
Arnold’s resonant web
Acknowledgments: 
This study was supported by the Russian Science Foundation (project no. 21-12-00121, https://rscf.ru/en/project/21-12-00121/).
Reference: 
  1. Pikovsky A., Rosenblum M., Kurths J. Synchronization: A universal concept in nonlinear sciences. Cambridge University Press, 2001. 432 p.
  2. Askerzade I., Bozbey A. Cantürk M. Modern aspects of Josephson dynamics and superconductivity electronics. Springer International Publishing, 2017. 186 p.
  3. Likharev K. K. Dynamics of Josephson Junctions and Circuits. Gordon and Breach, New York, 1986. 614 p.
  4. Wiesenfeld K., Colet P., Strogatz S. H. Synchronization Transitions in a Disordered Josephson Series Array. Phys. Rev. Lett., 1996, vol. 76, no. 3, pp. 404–407. https://doi.org/10.1103/PhysRevLett.76.404
  5. Wiesenfeld K., Colet P., Strogatz S. H. Frequency locking in Josephson arrays: Connection with the Kuramoto model. Phys. Rev. E, 1998, vol. 57, pp. 1563–1569. https://doi.org/10.1103/PhysRevE.57.1563
  6. Wiesenfeld K., Swift J. W. Averaged equations for Josephson junction series arrays. Phys. Rev. E, 1995, vol. 51, iss. 2, pp. 1020–1025. https://doi.org/10.1103/PhysRevE.51.1020
  7. Nichols S., Wiesenfeld K. Ubiquitous neutral stability of splay-phase states. Physical Review A, 1992, vol. 45, no. 12, pp. 8430–8435. https://doi.org/10.1103/PhysRevA.45.8430
  8. Filatrella G., Pedersen N. F., Wiesenfeld K. High-Q cavity-induced synchronization in oscillator arrays. Phys. Rev. E, 2000, vol. 61, iss. 3, pp. 2513–2518. https://doi.org/10.1103/PhysRevE.61.2513
  9. Jain A. K., Likharev K. K., Lukens J. E., Sauvageau J. E. Mutual phase-locking in Josephson junction arrays. Phys. Rep., 1984, vol. 109, pp. 310–426. https://doi.org/10.1016/0370-1573(84)90002-4
  10. Dana S. K., Sengupta D. C., Edoh K. D. Chaotic dynamics in Josephson junction. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2001, vol. 48, no. 8, pp. 990–996. https://doi.org/10.1109/81.940189
  11. Valkering T. P., Hooijer C. L. A., Kroon M. F. Dynamics of two capacitively coupled Josephson junctions in the overdamped limit. Physica D, 2009, vol. 135, iss. 1–2, pp. 137–153. https://doi.org/10.1016/S0167-2789(99)00116-5
  12. Abdullaev F. Kh., Abdumalikov A. A., Jr., Buisson O., Tsoy E. N. Phase-locked states in the system of two capacitively coupled Josephson junctions. Phys. Rev. B, 2000, vol. 62, iss. 10, pp. 6766–6773. https://doi.org/10.1103/PhysRevB.62.6766
  13. Kurt E., Canturk M. Chaotic dynamics of resistively coupled DC-driven distinct Josephson junctions and the effects of circuit parameters. Physica D, 2009, vol. 238, no. 22, pp. 2229–2237. https://doi.org/10.1016/j.physd.2009.09.005
  14. Kurt E., Canturk M. Bifurcations and Hyperchaos from a DC Driven Nonidentical Josephson Junction System. Int. J. Bifur. Chaos, 2010, vol. 20, no. 11, pp. 3725–374 https://doi.org/10.1142/S021812741002801X
  15. Stork M., Kurt E. Control system approach to the dynamics of nonidentical Josephson junction systems. 2015 International Conference on Applied Electronics (AE). IEEE, 2015, pp. 233–238.
  16. Celík K., Kurt E., Stork M. Can non-identical josephson junctions be synchronized? 2017 IEEE 58th International Scientific Conference on Power and Electrical Engineering of Riga Technical University (RTUCON). IEEE, 2017, pp. 1–5. https://doi.org/10.1109/RTUCON.2017.8124771
  17. Ojo K. S., Njah A. N., Olusola O. I., Omeike M. O. Generalized reduced-order hybrid combination synchronization of three Josephson junctions via backstepping technique. Nonlinear Dynamics, 2014, vol. 77, no. 3, pp. 583–595. https://doi.org/10.1007/s11071-014-1319-z
  18. Vlasov V., Pikovsky A. Synchronization of a Josephson junction array in terms of global variables. Phys. Rev. E, 2013, vol. 88, iss. 2, pp. 022908 (5 pages). https://doi.org/10.1103/PhysRevE.88.022908
  19. Kuznetsov A. P., Sataev I. R., Sedova Yu. V. Dynamics of three and four non-identical Josephson junctions. Journal of Applied Nonlinear Dynamics, 2018, vol. 7, no. 1, pp. 105–110. https://doi.org/10.5890/JAND.2018.03.009
  20. Baesens C., Guckenheimer J., Kim S., MacKay R. S. Three coupled oscillators: Mode locking, global bifurcations and toroidal chaos. Physica D, 1991, vol. 49, pp. 387–475. https://doi.org/10.1016/0167-2789(91)90155-3
  21. Broer H. Simó C., Vitolo R. The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: The Arnol’d resonance web. Reprint from the Belgian Mathematical Society, 2008, pp. 769–787. https://doi.org/10.36045/bbms/1228486406
  22. Inaba N., Kamiyama K., Kousaka T., Endo T. Numerical and experimental observation of Arnol’d resonance webs in an electrical circuit. Physica D, 2015, vol. 311, pp. 17–24. https://doi.org/10.1016/j.physd.2015.08.008
  23. Truong T. Q., Tsubone T., Sekikawa M., Inaba N., Endo T. Arnol’d resonance webs and Chenciner bubbles from a three-dimensional piecewise-constant hysteresis oscillator. Progress of Theoretical and Experimental Physics, 2017, vol. 2017, iss. 5, pp. 053A04 (15 pages). https://doi.org/10.1093/ptep/ptx058
  24. Kuznetsov A. P., Sedova Y. V. Low-dimensional discrete Kuramoto model: Hierarchy of multifrequency quasiperiodicity regimes. Int. J. Bifurcation Chaos, 2014, vol. 24, iss. 07, pp. 1430022 (10 pages). https://doi.org/10.1142/S0218127414300225
  25. Emelianova Y. P., Kuznetsov A. P., Turukina L. V., Sataev I. R., Chernyshov N. Yu. A structure of the oscillation frequencies parameter space for the system of dissipatively coupled oscillators. Communications in Nonlinear Science and Numerical Simulation, 2014, vol. 19, iss. 4, pp. 1203–1212. https://doi.org/10.1016/j.cnsns.2013.08.004
  26. Emelianova Yu. P., Kuznetsov A. P., Sataev I. R., Turukina L. V. Synchronization and multi-frequency oscillations in the low-dimensional chain of the self-oscillators. Physica D, 2013, vol. 244, no. 1, pp. 36–49. https://doi.org/10.1016/j.physd.2012.10.012
  27. Broer H., Simó C., Vitolo R. Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems. Regular and Chaotic Dynamics, 2011, vol. 16, no. 1–2, pp. 154–184. https://doi.org/10.1134/S1560354711010060
  28. Landa P. S. Avtokolebaniya v sistemakh s konechnym chislom stepeney svobody [Self-oscillations in systems with finite degrees of freedom]. Moscow, LIBROKOM Publ., 2010. 360 p. (in Russian).
  29. Balanov A. G., Janson N. B., Postnov D. E., Sosnovtseva O. Synchronization: From simple to complex. Springer, 2009. 425 p.
Received: 
10.10.2022
Accepted: 
30.11.2022
Published: 
01.03.2023
  • 692 reads