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Complexity
Volume 2017, Article ID 4107358, 12 pages
https://doi.org/10.1155/2017/4107358
Research Article

Analysis of a No Equilibrium Linear Resistive-Capacitive-Inductance Shunted Junction Model, Dynamics, Synchronization, and Application to Digital Cryptography in Its Fractional-Order Form

1Department of Mechanical and Electrical Engineering, Institute of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon
2Department of Physics, Higher Teacher Training College, University of Bamenda, P.O. Box 39, Bamenda, Cameroon
3Research Group on Experimental and Applied Physics for Sustainable Development, Faculty of Science, Department of Physics, University of Dschang, P.O. Box 412, Dschang, Cameroon
4Laboratory of Electronics and Signal Processing Faculty of Science, Department of Physics, University of Dschang, P.O. Box 67, Dschang, Cameroon
5Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes (LaMSEBP) and TWAS Research Unit, Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon

Correspondence should be addressed to Gaetan Fautso Kuiate; moc.oohay@etaiuk_ostuaf

Received 16 June 2017; Accepted 9 August 2017; Published 12 October 2017

Academic Editor: Karthikeyan Rajagopal

Copyright © 2017 Sifeu Takougang Kingni et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. B. Soodchomshom, I.-M. Tang, and R. Hoonsawat, “Josephson effects in MgB2/Thin Insulator/MgB2 tunnel junction,” Solid State Communications, vol. 149, no. 25-26, pp. 1012–1016, 2009. View at Publisher · View at Google Scholar · View at Scopus
  2. B. D. Josephson, “Possible new effects in superconductive tunnelling,” Physics Letters, vol. 1, no. 7, pp. 251–253, 1962. View at Publisher · View at Google Scholar
  3. J. Clarke, “A superconducting galvanometer employing Josephson tunnelling,” Philosophical Magazine, vol. 13, no. 121, pp. 115–127, 1966. View at Publisher · View at Google Scholar
  4. V. K. Kornev and A. V. Arzumanov, “Josephson-junction oscillation spectral linewidth for some phase-locked multijunction systems,” Le Journal de Physique IV, vol. 8, no. 3, pp. Pr3-279–Pr3-282, 1998. View at Publisher · View at Google Scholar
  5. A. Kanasugi, M. Morisue, H. Noguchi, M. Yamadaya, and H. Furukawa, “Oscillation modes in a josephson circuit and its application to digital systems,” IEICE Transactions on Electronics, vol. E79-C, no. 9, pp. 1206–1211, 1996. View at Google Scholar · View at Scopus
  6. J. W. Spargo, “Applied superconductivity conference,” IEEE Transactions on Applied Superconductivity, vol. 13, pp. I–III, 2003. View at Google Scholar
  7. R. Guo, U. E. Vincent, and B. A. Idowu, “Synchronization of chaos in RCL-shunted Josephson junction using a simple adaptive controller,” Physica Scripta, vol. 79, no. 3, Article ID 035801, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. W. C. Stewart, “Current-voltage characteristics of Josephson junctions,” Applied Physics Letters, vol. 12, no. 8, pp. 277–280, 1968. View at Publisher · View at Google Scholar · View at Scopus
  9. D. E. McCumber, “Effect of ac impedance on dc voltage-current characteristics of superconductor weak-link junctions,” Journal of Applied Physics, vol. 39, no. 7, pp. 3113–3118, 1968. View at Publisher · View at Google Scholar · View at Scopus
  10. M. Levi, F. C. Hoppensteadt, and W. L. Miranker, “Dynamics of the Josephson junction,” Quarterly of Applied Mathematics, vol. 36, no. 2, pp. 167–198, 1978/79. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. F. M. Salam and S. S. Sastry, “Dynamics of the forced Josephson junction circuit: the regions of chaos,” Institute of Electrical and Electronics Engineers. Transactions on Circuits and Systems, vol. 32, no. 8, pp. 784–796, 1985. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. C. B. Whan, C. J. Lobb, and M. G. Forrester, “Effect of inductance in externally shunted Josephson tunnel junctions,” Journal of Applied Physics, vol. 77, no. 1, pp. 382–389, 1995. View at Publisher · View at Google Scholar · View at Scopus
  13. C. B. Whan and C. J. Lobb, “Complex dynamical behavior in,” Physical Review E, vol. 53, no. 1, pp. 405–413, 1996. View at Publisher · View at Google Scholar
  14. A. B. Cawthorne, C. B. Whan, and C. J. Lobb, “Complex dynamics of resistively and inductively shunted Josephson junctions,” Journal of Applied Physics, vol. 84, no. 2, pp. 1126–1132, 1998. View at Publisher · View at Google Scholar · View at Scopus
  15. S. K. Dana, D. C. Sengupta, and K. D. Edoh, “Chaotic dynamics in Josephson Junction,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 48, no. 8, pp. 990–996, 2001. View at Publisher · View at Google Scholar · View at Scopus
  16. S. K. Dana, “Spiking and Bursting in Josephson Junction,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 53, no. 10, pp. 1031–1034, 2006. View at Publisher · View at Google Scholar · View at Scopus
  17. A. D. Smith, R. D. Sandell, A. H. Silver, and J. F. Burch, “Chaos and Bifurcation in Josephson Voltage-Controlled Oscillators,” IEEE Transactions on Magnetics, vol. 23, no. 2, pp. 1267–1270, 1987. View at Publisher · View at Google Scholar · View at Scopus
  18. E. Neumann and A. Pikovsky, “Slow-fast dynamics in Josephson junctions,” European Physical Journal B, vol. 34, no. 3, pp. 293–303, 2003. View at Publisher · View at Google Scholar · View at Scopus
  19. G. A. Leonov, N. V. Kuznetsov, and V. I. Vagaitsev, “Localization of hidden Chua's attractors,” Physics Letters. A, vol. 375, no. 23, pp. 2230–2233, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. G. A. Leonov, N. V. Kuznetsov, and V. I. Vagaitsev, “Hidden attractor in smooth Chua systems,” Physica D. Nonlinear Phenomena, vol. 241, no. 18, pp. 1482–1486, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. G. A. Leonov and N. V. Kuznetsov, “Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and KALman problems to hidden chaotic attractor in Chua circuits,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 23, no. 1, Article ID 1330002, 69 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  22. G. A. Leonov, N. V. Kuznetsov, M. A. Kiseleva, E. P. Solovyeva, and A. M. Zaretskiy, “Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor,” Nonlinear Dynamics, vol. 77, no. 1-2, pp. 277–288, 2014. View at Publisher · View at Google Scholar · View at Scopus
  23. G. A. Leonov, N. V. Kuznetsov, and T. N. Mokaev, “Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity,” Communications in Nonlinear Science and Numerical Simulation, vol. 28, no. 1-3, pp. 166–174, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. G. A. Leonov, N. V. Kuznetsov, and T. N. Mokaev, “Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion,” European Physical Journal: Special Topics, vol. 224, no. 8, pp. 1421–1458, 2015. View at Publisher · View at Google Scholar · View at Scopus
  25. P. R. Sharma, M. D. Shrimali, A. Prasad, N. V. Kuznetsov, and G. A. Leonov, “Controlling dynamics of hidden attractors,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 25, no. 4, Article ID 1550061, 1550061, 7 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. D. Dudkowski, S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A. Leonov, and A. Prasad, “Hidden attractors in dynamical systems,” Physics Reports. A Review Section of Physics Letters, vol. 637, pp. 1–50, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. E. M. Izhikevich, “Simple model of spiking neurons,” IEEE Transactions on Neural Networks, vol. 14, no. 6, pp. 1569–1572, 2003. View at Publisher · View at Google Scholar · View at Scopus
  28. C. Masoller, “Coexistence of attractors in a laser diode with optical feedback from a large external cavity,” Physical Review A, vol. 50, no. 3, pp. 2569–2578, 1994. View at Publisher · View at Google Scholar · View at Scopus
  29. S. T. Kingni, G. V. D. Sande, I. V. Ermakov, and J. Danckaert, “Theoretical analysis of semiconductor ring lasers with short and long time-delayed optoelectronic and incoherent feedback,” Optics Communications, vol. 341, pp. 147–154, 2015. View at Publisher · View at Google Scholar · View at Scopus
  30. J. M. Cushing, S. M. Henson, and C. . Blackburn, “Multiple mixed-type attractors in a competition model,” Journal of Biological Dynamics, vol. 1, no. 4, pp. 347–362, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  31. A. Massoudi, M. G. Mahjani, and M. Jafarian, “Multiple attractors in Koper-Gaspard model of electrochemical periodic and chaotic oscillations,” Journal of Electroanalytical Chemistry, vol. 647, no. 1, pp. 74–86, 2010. View at Publisher · View at Google Scholar · View at Scopus
  32. C. Li, J. C. Sprott, and H. Xing, “Constructing chaotic systems with conditional symmetry,” Nonlinear Dynamics, vol. 87, no. 2, pp. 1351–1358, 2017. View at Publisher · View at Google Scholar · View at Scopus
  33. C. Li, J. C. Sprott, and H. Xing, “Hypogenetic chaotic jerk flows,” Physics Letters. A, vol. 380, no. 11-12, pp. 1172–1177, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. C. Li, W. Hu, J. C. Sprott, and X. Wang, “Multistability in symmetric chaotic systems,” European Physical Journal: Special Topics, vol. 224, no. 8, pp. 1493–1506, 2015. View at Publisher · View at Google Scholar · View at Scopus
  35. C. Li, J. C. Sprott, and H. Xing, “Crisis in amplitude control hides in multistability,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 26, no. 14, Article ID 1650233, 1650233, 11 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. J. Kengne, “Coexistence of chaos with hyperchaos, period-3 doubling bifurcation, and transient chaos in the hyperchaotic oscillator with gyrators,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 25, no. 4, Article ID 1550052, 17 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  37. A. P. Kuznetsov, S. P. Kuznetsov, E. Mosekilde, and N. V. Stankevich, “Co-existing hidden attractors in a radio-physical oscillator system,” Journal of Physics A: Mathematical and Theoretical, vol. 48, no. 12, 12 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  38. K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 29, no. 1-4, pp. 3–22, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  39. R. Caponetto, R. Dongola, L. Fortuna, I. Petra, and I. Petraš, “Fractional-order system: modelling and control applications. World scientific series on nonlinear science,” series A, vol. 72, 2010. View at Google Scholar
  40. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  41. H. B. Fotsin and J. Daafouz, “Adaptive synchronization of uncertain chaotic colpitts oscillators based on parameter identification,” Physics Letters, Section A: General, Atomic and Solid State Physics, vol. 339, no. 3-5, pp. 304–315, 2005. View at Publisher · View at Google Scholar · View at Scopus
  42. R. Kengne, R. Tchitnga, A. Mezatio, A. Fomethe, and G. Litak, “Finite-time synchronization of fractional-order simplest two-component chaotic oscillators,” European Physical Journal B, vol. 90, pp. 88–96, 2017. View at Publisher · View at Google Scholar
  43. P. Muthukumar and P. Balasubramaniam, “Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 74, no. 4, pp. 1169–1181, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  44. P. Muthukumar, P. Balasubramaniam, and K. Ratnavelu, “Fast projective synchronization of fractional order chaotic and reverse chaotic systems with its application to an affine cipher using date of birth (DOB),” Nonlinear Dynamics, vol. 80, no. 4, pp. 1883–1897, 2015. View at Publisher · View at Google Scholar · View at Scopus
  45. W. W. Yu and J. D. Cao, “Adaptive synchronization and lag synchronization of uncertain dynamical system with time delay based on parameter identification,” Physica A, vol. 375, no. 2, pp. 467–482, 2007. View at Publisher · View at Google Scholar · View at Scopus