We report an interesting bow-tie shaped bursting behaviour in a certain parameter regime of two resistive-capacitative shunted Josephson junctions, one in the oscillatory and the other in the excitable mode and coupled together resistively. The burst emerges in both the junctions and they show near-complete synchronization for strong enough couplings. We discuss a possible bifurcation scenario to explain the origin of the burst. An exhaustive study on the parameter space of the system is performed, demarcating the regions of bursting from other solutions.
References
1.
V. N.
Belykh
, N. F.
Pedersen
, and O. H.
Soerensen
, “ Shunted-Josephson-junction model. I. The autonomous case
,” Phys. Rev. B
16
, 4853
–4859
(1977
).2.
J. A.
Blackburn
and H. J. T.
Smith
, “ Dynamics of double-Josephson-junction interferometers
,” J. Appl. Phys.
49
, 2452
–2455
(1978
).3.
M.
Levi
, F. C.
Hoppensteadt
, and W. L.
Miranker
, “ Dynamics of the Josephson junction
,” Q. Appl. Math.
36
, 167
–199
(1978
).4.
T.
Bohr
, P.
Bak
, and M. H.
Jensen
, “ Transition to chaos by interaction of resonances in dissipative systems. II. Josephson junctions, charge-density waves, and standard maps
,” Phys. Rev. A
30
, 1970
–1981
(1984
).5.
K. K.
Likharev
, Dynamics of Josephson Junction and Circuits
(Gordon and Breach
, New York
, 1986
).6.
P.
Hadley
, M. R.
Beasley
, and K.
Wiesenfeld
, “ Phase locking of Josephson-junction series arrays
,” Phys. Rev. B
38
, 8712
–8719
(1988
).7.
K.
Wiesenfeld
and J. W.
Swift
, “ Averaged equations for Josephson junction series arrays
,” Phys. Rev. E
51
, 1020
–1025
(1995
).8.
A. A.
Chernikov
and G.
Schmidt
, “ Conditions for synchronization in Josephson-junction arrays
,” Phys. Rev. E
52
, 3415
–3419
(1995
).9.
K.
Wiesenfeld
, P.
Colet
, and S. H.
Strogatz
, “ Synchronization transitions in a disordered Josephson series array
,” Phys. Rev. Lett.
76
, 404
–407
(1996
).10.
K.
Wiesenfeld
, P.
Colet
, and S. H.
Strogatz
, “ Frequency locking in Josephson arrays: Connection with the Kuramoto model
,” Phys. Rev. E
57
, 1563
–1569
(1998
).11.
F.
Xie
and H.
Cerdeira
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
8
, 1713
(1998
).12.
A. B.
Cawthorne
, C. B.
Whan
, and C. J.
Lobb
, “ Complex dynamics of resistively and inductively shunted Josephson junctions
,” J. App. Phys.
84
, 1126
–1132
(1998
).13.
B. D.
Josephson
, “ Possible new effects in superconductive tunneling
,” Phys. Lett.
1
, 251
–253
(1962
).14.
M.
Octavio
, “ Bifurcating, chaotic, and intermittent solutions in the RF-biased Josephson junction
,” Phys. Rev. B
29
, 1231
–1242
(1984
).15.
T. P.
Valkering
, C. L. A.
Hooijer
, and M. F.
Kroon
, “ Dynamics of two capacitively coupled Josephson junctions in the overdamped limit
,” Physica D
135
, 137
–153
(2000
).16.
M.
Qian
, J.-Z.
Wang
, and X.-J.
Zhang
, “ Resonant regions of Josephson junction equation in case of large damping
,” Phys. Lett. A
372
, 3640
–3644
(2008
).17.
B. P.
Koch
and B.
Bruhn
, “ Transient chaos in weakly coupled Josephson junctions
,” J. Phys. France
49
, 35
–40
(1988
).18.
S. K.
Dana
, D. C.
Sengupta
, and K. D.
Edoh
, “ Chaotic Dynamics in Josephson Junction
,” IEEE Trans. Circuits Syst.
48
,(8
) 990
–996
(2001
).19.
P.
Crotty
, D.
Schult
, and K.
Segall
, “ Josephson junction simulation of neurons
,” Phys. Rev. E
82
, 011914
(2010
).20.
R. P.
Giffard
, J. C.
Gallop
, and B. W.
Petley
, “ Applications of the Josephson effects
,” Prog. Quantum Electron.
4
, 301
–402
(1976
).21.
Y.
Nakamura
, Yu. A.
Pashkin
, and J. S.
Tsai
, “ Coherent control of macroscopic quantum states in a single-Cooper-pair box
,” Nature
398
, 786
–788
(1999
).22.
I.
Giaever
, “ Energy gap in superconductors measured by electron tunneling
,” Phys. Rev. Lett.
5
(4
), 147
–148
(1960
).23.
T. A.
Fulton
, R. C.
Dynes
, and P. W.
Anderson
, “ The flux shuttle—A Josephson junction shift register employing single flux quanta
,” Proc. IEEE
61
, 28
–35
(1973
).24.
T. A.
Fulton
, S. S.
Pei
, and L. N.
Dunkleberger
, “ Josephson junction current-switched logic circuits
,” IEEE Trans. Magn.
15
(6
), 1876
–1879
(1979
).25.
S.
Pagano
and A.
Barone
, “ Josephson junctions
,” Supercond. Sci. Technol.
10
, 904
–908
(1997
).26.
A.
Barone
and G.
Paterno
, Physics and Applications of the Josephson Effect
(Wiley
, New York
, 1982
).27.
Superconducting Electronics
, edited by H.
Weinstock
and M.
Nisenoff
(Springer
, Berlin
, 1989
).28.
See http://www.dwavesys.com/ for integrated circuit processors realized with Josephson junction qubits.
29.
P.
Coullet
, J. M.
Gilli
, M.
Monticelli
, and N.
Vandenberghe
, “ A damped pendulum forced with a constant torque
,” Am. J. Phys.
73
(12
), 1122
(2005
).30.
G. B.
Ermentrout
and N.
Kopell
, “ Parabolic bursting in an excitable system coupled with a slow oscillation
,” SIAM J. Appl. Math.
46
, 233
–253
(1986
).31.
E. M.
Izhikevich
, “ Neural excitability, spiking and bursting
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
10
, 1171
(2000
).32.
N. F.
Rulkov
, “ Modeling of spiking-bursting neural behavior using two-dimensional map
,” Phys. Rev. E
65
, 041922
(2002
).33.
S. K.
Dana
, D. C.
Sengupta
, and C.-K.
Hu
, “ Spiking and bursting in Josephson junction
,” IEEE Trans. Circuits Syst.
53
(10
), 1031
(2006
).34.
A.
Sherman
and J.
Rinzel
, “ Model for synchronization of pancreatic β-cells by gap junction coupling
,” Biophys. J
59
, 547
(1991
).35.
F. R.
Gantmacher
, Applications of the Theory of Matrices
(Interscience Publishers
, 1959
).36.
T.
Hongray
, J.
Balakrishnan
, and S. K.
Dana
, “ Complex dynamics in coupled Josephson junctions
,” talk by J. Balakrishnan, and poster presentation by T. Hongray on “Studies of coupled excitable Josephson junctions,” at the International Symposium on Complex Dynamical Systems and Applications (CDSA-2009), Digha Science Centre, West Bengal, India
, 4–6 December 2009
.37.
C.
Van den Broeck
, M. M.
Mansour
, and F.
Baras
, “ Asymptotic properties of coupled nonlinear Langevin equations in the limit of weak noise. I: Cusp bifurcation
,” J. Stat. Phys.
28
, 557
(1982
).38.
E.
Knobloch
and K. A.
Wiesenfeld
, “ Bifurcations in fluctuating systems: The center manifold approach
,” J. Stat. Phys.
33
, 611
(1983
).39.
K. A.
Wiesenfeld
, “ Noisy precursors of nonlinear instabilities
,” J. Stat. Phys.
38
, 1071
(1985
).40.
C.
Meunier
and A. D.
Verga
, “ Noise and bifurcations
,” J. Stat. Phys.
50
, 345
(1988
).41.
C
. Hens
, P
. Pal
, and S. K
. Dana
, Phys. Rev. E
92
, 022915
(2015
).© 2015 AIP Publishing LLC.
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