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Analytical analysis of periodic solution and its stability in Josephson junction

Zhang Li-Sen Cai Li Feng Chao-Wen

Analytical analysis of periodic solution and its stability in Josephson junction

Zhang Li-Sen, Cai Li, Feng Chao-Wen
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  • Analytical expressions of periodic solutions in rf-biased resistively-capacitively-shunted Josephson junction were derived by incremental harmonic balance method, and the stability of the periodic solutions was investigated using Floquet theory. We fownd that while the system is in stable periodic states, plentiful unstable periodic orbits still exist in the system. Critical parameter values for which the stable periodic solutions of the system lose their stability are obtained and the type of bifurcation is determined by computing the Floquet multipliers. We have also theoretically confirmed the period-doubling-route to chaos with increasing amplitude of driving current, which acts as the control parameter in the system. The results from analytical analysis coincide with that from numerical simulation.
    • Funds:
    [1]

    Winkler D 2003 Supercond. Sci. Technol. 16 1583

    [2]

    Bens S P, Hamilton C A 2004 Proc. of the IEEE 92 1617

    [3]

    Mao B, Dai Y D, Wang F R 2005 Chin. Phys. 14 301

    [4]

    Osborn K D, Strong J A, Sirois A J, Simmonds R W 2007 IEEE Trans. Appl. Supercond. 17 166

    [5]

    Forn-Diaz P, Schouten R N, den Braver W A, Mooij J E, Harmans C J P M 2009 Appl. Phys. Lett. 95 042505

    [6]

    Hu J P, Wu C J, Dai X 2007 Phys. Rev. Lett. 99 067004

    [7]

    Liang B L, Wang J S, Meng X G, Su J 2010 Chin. Phys. B 19 010315

    [8]

    Yue H W, Yan S L, Zhou T G, Xie Q L, You F, Wang Z, He M, Zhou X J, Fang L, Yang Y, Wang F Y, Tao W W 2010 Acta Phys. Sin. 59 1282 (in Chinese) [岳宏卫、阎少林、周铁戈、谢清连、游 峰、王 争、何 明、赵新杰、方 兰、杨 扬、王福音、陶薇薇 2010 物理学报 59 1282]

    [9]

    van der Wal C H, ter Haar A C J, Wilhelm F K, Schouten R N, Harmans C J P M, Orlando T P, Lloyd S, Mooij J E 2000 Science 290 773

    [10]

    Chiorescu I, Nakamura Y, Harmans C J P M, Mooij J E 2003 Science 299 1869

    [11]

    Cui D J, Lin D H, Yu H F, Peng Z H, Zhu X B, Zheng D N, Jing X N, Lu L, Zhao S P 2008 Acta Phys. Sin. 57 5933 (in Chinese) [崔大健、林德华、于海峰、彭智慧、朱晓波、郑东宁、景秀年、吕 力、赵士平 2008 物理学报 57 5933]

    [12]

    Huberman B A, Crutchfield J P, Packard N H 1980 Appl. Phys. Lett. 37 750

    [13]

    Octavio M 1984 Phys. Rev. B 29 1231

    [14]

    Jensen H D, Larsen A, Mygind J 1990 Physica B 165-166 1661

    [15]

    Wang Z Y, Liao H Y, Zhou S P 2001 Acta Phys. Sin. 50 1996 (in Chinese) [王震宇、廖红印、周世平 2001 物理学报 50 1996]

    [16]

    Lei Y M, Xu W 2008 Acta Phys. Sin. 57 3342 (in Chinese) [雷佑铭、徐 伟 2008 物理学报 57 3342]

    [17]

    Liu C X 2007 Nonlinear Circuit Theory and Its Applications (Xi’an: Xi’an Jiaotong University Press) p106 (in Chinese) [刘崇新 2007 非线性电路理论及应用(西安: 西安交通大学出版社) 第106页]

    [18]

    Lau S L, Cheung Y K 1981 ASME J. Appl. Mech. 48 959

    [19]

    Shen J H, Lin K C, Chen S H, Sze K Y 2008 Nonlinear Dyn. 52 403

    [20]

    Xu L, Lu M W, Cao Q 2002 Phys. Lett. A 301 65

    [21]

    Raghothama A, Narayanan S 1999 J. Sound Vib. 226 469

    [22]

    Raghothama A, Narayanan S 2000 Ocean Eng. 27 1087

    [23]

    Wang H L, Zhang Q C 2002 Nonlinear Dynamics Theory and Its Applications (Tianjin: Tianjin Science and Technology Press) p241 (in Chinese) [王洪礼、张琪昌 2002 非线性动力学理论及其应用(天津: 天津科学技术出版社) 第241页]

  • [1]

    Winkler D 2003 Supercond. Sci. Technol. 16 1583

    [2]

    Bens S P, Hamilton C A 2004 Proc. of the IEEE 92 1617

    [3]

    Mao B, Dai Y D, Wang F R 2005 Chin. Phys. 14 301

    [4]

    Osborn K D, Strong J A, Sirois A J, Simmonds R W 2007 IEEE Trans. Appl. Supercond. 17 166

    [5]

    Forn-Diaz P, Schouten R N, den Braver W A, Mooij J E, Harmans C J P M 2009 Appl. Phys. Lett. 95 042505

    [6]

    Hu J P, Wu C J, Dai X 2007 Phys. Rev. Lett. 99 067004

    [7]

    Liang B L, Wang J S, Meng X G, Su J 2010 Chin. Phys. B 19 010315

    [8]

    Yue H W, Yan S L, Zhou T G, Xie Q L, You F, Wang Z, He M, Zhou X J, Fang L, Yang Y, Wang F Y, Tao W W 2010 Acta Phys. Sin. 59 1282 (in Chinese) [岳宏卫、阎少林、周铁戈、谢清连、游 峰、王 争、何 明、赵新杰、方 兰、杨 扬、王福音、陶薇薇 2010 物理学报 59 1282]

    [9]

    van der Wal C H, ter Haar A C J, Wilhelm F K, Schouten R N, Harmans C J P M, Orlando T P, Lloyd S, Mooij J E 2000 Science 290 773

    [10]

    Chiorescu I, Nakamura Y, Harmans C J P M, Mooij J E 2003 Science 299 1869

    [11]

    Cui D J, Lin D H, Yu H F, Peng Z H, Zhu X B, Zheng D N, Jing X N, Lu L, Zhao S P 2008 Acta Phys. Sin. 57 5933 (in Chinese) [崔大健、林德华、于海峰、彭智慧、朱晓波、郑东宁、景秀年、吕 力、赵士平 2008 物理学报 57 5933]

    [12]

    Huberman B A, Crutchfield J P, Packard N H 1980 Appl. Phys. Lett. 37 750

    [13]

    Octavio M 1984 Phys. Rev. B 29 1231

    [14]

    Jensen H D, Larsen A, Mygind J 1990 Physica B 165-166 1661

    [15]

    Wang Z Y, Liao H Y, Zhou S P 2001 Acta Phys. Sin. 50 1996 (in Chinese) [王震宇、廖红印、周世平 2001 物理学报 50 1996]

    [16]

    Lei Y M, Xu W 2008 Acta Phys. Sin. 57 3342 (in Chinese) [雷佑铭、徐 伟 2008 物理学报 57 3342]

    [17]

    Liu C X 2007 Nonlinear Circuit Theory and Its Applications (Xi’an: Xi’an Jiaotong University Press) p106 (in Chinese) [刘崇新 2007 非线性电路理论及应用(西安: 西安交通大学出版社) 第106页]

    [18]

    Lau S L, Cheung Y K 1981 ASME J. Appl. Mech. 48 959

    [19]

    Shen J H, Lin K C, Chen S H, Sze K Y 2008 Nonlinear Dyn. 52 403

    [20]

    Xu L, Lu M W, Cao Q 2002 Phys. Lett. A 301 65

    [21]

    Raghothama A, Narayanan S 1999 J. Sound Vib. 226 469

    [22]

    Raghothama A, Narayanan S 2000 Ocean Eng. 27 1087

    [23]

    Wang H L, Zhang Q C 2002 Nonlinear Dynamics Theory and Its Applications (Tianjin: Tianjin Science and Technology Press) p241 (in Chinese) [王洪礼、张琪昌 2002 非线性动力学理论及其应用(天津: 天津科学技术出版社) 第241页]

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  • Received Date:  08 May 2010
  • Accepted Date:  02 September 2010
  • Published Online:  15 March 2011

Analytical analysis of periodic solution and its stability in Josephson junction

  • 1. School of Science, Air Force Engineering University, Xi’ an 710051, China

Abstract: Analytical expressions of periodic solutions in rf-biased resistively-capacitively-shunted Josephson junction were derived by incremental harmonic balance method, and the stability of the periodic solutions was investigated using Floquet theory. We fownd that while the system is in stable periodic states, plentiful unstable periodic orbits still exist in the system. Critical parameter values for which the stable periodic solutions of the system lose their stability are obtained and the type of bifurcation is determined by computing the Floquet multipliers. We have also theoretically confirmed the period-doubling-route to chaos with increasing amplitude of driving current, which acts as the control parameter in the system. The results from analytical analysis coincide with that from numerical simulation.

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