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Josephson junction circuit model and its global bifurcation diagram
Volume 118, number 2
PHYSICS LETTERS A
29 September 1986
J O S E P H S O N J U N C T I O N CIRCUIT M O D E L A N D ITS GLOBAL B I F U R C A T I O N D I A G R A M
Y. YAO Institute of Electrical Engineering and Computer Sciences, Shanghai Jiao-Tong University, Shanghai, P.R. China Received 13 January 1986; revised manuscript received 28 June 1986; accepted for publication 5 August 1986
In order to understand the meaning of a typical Josephson junction, we introduce a circuit model which can be used to describe the dynamical behavior of the junction under the exciting of radio-frequency current. The global bifurcation diagram of the Josephson junction is demonstrated with a computer simulation• Through the diagram we observe the complex and rich dynamics of the junction, e.g. bifurcation, phase-locking, and chaos. The relation between chaotic motion and symmetry breaking is also discussed.
Since Josephson first proposed the idea of Josephson junctions in 1962 [1], junction devices have widely been used, ranging from supersensitive detectors to superfast computers [2,3]. Using Josephson junction devices, a lot of extremely fast processes can be studied because the junction is very sensitive to small fluxes. Recently, with an Upsurge in pursuing chaos, more attention is given to the junction's dynamical behavior [2-5]• This kind of nonlinear devices is imbued with rich dynamics and displays a wide variety of strange phenomena. For a typical Josephson junction, a previous model adopted generally is McCuber's model [6] or Chua's circuit model [7]. But to describe the dynamics of the junction under the exciting of radio-frequency current, a more realistic Josephson junction circuit model is shown in fig. 1, where the basic junction elements are a nonlinear inductor and a nonlinear resistor governed by • /4qre
i((#) = L sln[---~- (#)
the Planck constant, e the electron charge, R 0 indicates the junction resistor corresponding to I c = 0 and c is called the coherence coefficient. From experimental results [7], ~ is related to the frequency tOrf and the amplitude I 0. Under common frequency, the value of c is nearly -0.8. In addition, the third element in the circuit is a linear capacitor named the junction capacitor. It is worth noting that the phenomenon of electromagnetic radiation in Josephson junction devices is due to the existence of these equivalent inductor i(g,) and capacitor C. According to Kirchhoff's laws the equation governing the nonlinear circuit in fig. 1 is given by
I
I,~o~t
(1)
I
Rt*):
iC
R._j_____
* .[
and / 4~re ~]-1 R(q~) = R o 1 + , cos~--h---~b)]
[
(2)
respectively, where q~ denotes the flux linkage, h
_[
Fig. 1. A Josephson junction circuit model driven by a radiofrequency current source.
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
59
V o l u m e 118, n u m b e r 2
29 S e p t e m b e r 1986
PHYSICS LETTERS A
....
•
¢¢**o....*.°°* °
o
• *
{*t
°t ~t °
,,:t.~ .,." t.t
***t
"
;°"
°
:i" :,!l." 'fii'ii ! -**°
,
-1 t[*
-2
"*; *tIt°•
Itlt.:V"
*
* t';*t
***. .
",,, .
.
.
.
, ....
..
- ,
-3 co
v~. 2
0.0
~5.4
~ .6
0 . :5:
i. (~
1,Z
Fig. 2. T h e g l o b a l b i f u r c a t i o n d i a g r a m s o f the J o s e p h s o n j u n c t i o n circuit m o d e l ; c~ = 0.4, ~ = - 0.8, fl = 0.7999.
C d2q~ + [1 + e
d,2
L
= l°sin 2"~
c°st
[4"~e
'
~lR_ldq)
Jlo
d,
wrf/.
(3)
The above equation can easily be transformed into the dimensionless form 2 + a(1 + ~ cos x ) 2 + sin x = fl sin ~0t.
(4)
For the sake of c o m p u t e r simulation, eq. (4) can equivalently be rewritten as
d x J d t = x 2,
(5)
d x z / d t = - a ( 1 + ~ cos x])x 2 - sin xl + f l sin ~0t.
60
(6)
N o w put a = 0 . 4 , ~ = - 0 . 8 , /3=0.7999. The variation of steady-state responses (i.e. the output voltage corresponding to the steady state [8]) of the Josephson junction with the change of the frequency ~orf is shown in fig. 2 *] When ~o varies from 1.2 to 0.0, the corresponding bifurcation process of steady-state solutions [9 is as follows: period-1 solution (0.69 ~< w ~< 1.2) -* period-2 solution (~o = 0.68) ~ period-3 solution (~0 --~ 0.67) ---~ chaos (0.59 ~< w ~< 0.66) ---~ period-1 solution (0.57 ~< w ~< 0.58) ---, period-2 Solution o~ -- 0.56) ---~ chaos (0.51 ~< w <~ 0.55) ---~ period-3 solution (~0 ~ 0.50) ---~ chaos (0.47 ~< ~0 .<<0.49) ---~ period-1 solution (0.0 ~< ~o ~< 0.47). The broken arrow denotes a possible middle ,1 In fact, we c h a n g e the p a r a m e t e r ~a w h e n the c o m p u t e r p r o g r a m is u n d e r w a y . But in the a b o v e d i m e n s i o n l e s s p r o c e d u r e ~,, is t a k e n to be ~rf/o~ o, while w o = ( 4 v e l c / h C ) 1/2 is a c o n s t a n t . T h u s the c h a n g e o f ~a is e q u a l to t h a t o f Wrr
Volume
118, number
2
PHYSICS
LETTERS
A
29 September
1986
3
2
1
0
-1
-2
-3
It.2
0.0
0.4
Fig. 3. Order, symmetry,
0.6 and symmetry
process. It is impossible to identify these middle structures completely although the observation can be made in more detail. In order to study the relation between chaos and symmetry, we replace the sampling distance 2n by 1 arc. The simulation results are given in fig. 3. It is quite unusual and exotic, that the picture consists of one “circular pagoda” and one “basket of flowers”. Within the parameter interval 0.0 G w G 0.47 and 0.67 G o > 1.2, i.e. the period region, the picture possesses symmetry and order structure. Within the chaotic or phase-locking region, however, this kind of symmetry and order structure disapears. In other words, chaos leads to disordering of the output and symmetry breaking of the structure in the Josephson junction device. From (3)-(6), it is easy to see that x2 = i, = (4Te/h)&
= (47re/h)V.
Thus the above symmetry
indicates
(7) the symmetry
0.8 breaking
l.M
1.2
of the solution.
of the voltage V (i.e. the output response of the device) with respect to I’= 0. The output signal is sinusoidal in the periodic region as long as the input radio-frequency current is sinusoidal, but this is not true in the region of chaotic motion and phase-locking. Finally we emphasize that the significance of the Josephson junction circuit model is that it is favorable to the application of Josephson junctions since its phenomena appear without extreme physical conditions. References [l] B.D. Josephson,
Phys. Lett. 1 (1962) 251. [2] SQUID, Proc. Int. Conf. Superconducting quantum devices, Berlin, 1976 (1st Conf.), 1980 (2nd). [3] M. Odyniec and L.O. Chua, IEEE Trans. CAS 30 (1983) 308. [4] D.Z. Wang and X.X. Yao, Acta Phys. Sin. 34 (1985) 1140 (in Chinese).
67
Volume 118, number 2
PHYSICS LETTERS A
[5] Y. Imry, in: Statics and dynamics of nonlinear systems, eds. G. Benedek et al. (Springer, Berlin, 1983). [6] D.E. McCuber, J. Appl. Phys. 39 (1968) 3113. [7] D.N. Langenberg, Rev. Phys. Appl. 9 (1974) 35.
62
29 September 1986
[8] Z.J. Zhang and Y. Yao, Proc. Int. Conf. CAS, Beijing, China (1985) p. 439. [9] Y. Yao, Nature J. 9 (1985) 687 (in Chinese).
PHYSICS LETTERS A
29 September 1986
J O S E P H S O N J U N C T I O N CIRCUIT M O D E L A N D ITS GLOBAL B I F U R C A T I O N D I A G R A M
Y. YAO Institute of Electrical Engineering and Computer Sciences, Shanghai Jiao-Tong University, Shanghai, P.R. China Received 13 January 1986; revised manuscript received 28 June 1986; accepted for publication 5 August 1986
In order to understand the meaning of a typical Josephson junction, we introduce a circuit model which can be used to describe the dynamical behavior of the junction under the exciting of radio-frequency current. The global bifurcation diagram of the Josephson junction is demonstrated with a computer simulation• Through the diagram we observe the complex and rich dynamics of the junction, e.g. bifurcation, phase-locking, and chaos. The relation between chaotic motion and symmetry breaking is also discussed.
Since Josephson first proposed the idea of Josephson junctions in 1962 [1], junction devices have widely been used, ranging from supersensitive detectors to superfast computers [2,3]. Using Josephson junction devices, a lot of extremely fast processes can be studied because the junction is very sensitive to small fluxes. Recently, with an Upsurge in pursuing chaos, more attention is given to the junction's dynamical behavior [2-5]• This kind of nonlinear devices is imbued with rich dynamics and displays a wide variety of strange phenomena. For a typical Josephson junction, a previous model adopted generally is McCuber's model [6] or Chua's circuit model [7]. But to describe the dynamics of the junction under the exciting of radio-frequency current, a more realistic Josephson junction circuit model is shown in fig. 1, where the basic junction elements are a nonlinear inductor and a nonlinear resistor governed by • /4qre
i((#) = L sln[---~- (#)
the Planck constant, e the electron charge, R 0 indicates the junction resistor corresponding to I c = 0 and c is called the coherence coefficient. From experimental results [7], ~ is related to the frequency tOrf and the amplitude I 0. Under common frequency, the value of c is nearly -0.8. In addition, the third element in the circuit is a linear capacitor named the junction capacitor. It is worth noting that the phenomenon of electromagnetic radiation in Josephson junction devices is due to the existence of these equivalent inductor i(g,) and capacitor C. According to Kirchhoff's laws the equation governing the nonlinear circuit in fig. 1 is given by
I
I,~o~t
(1)
I
Rt*):
iC
R._j_____
* .[
and / 4~re ~]-1 R(q~) = R o 1 + , cos~--h---~b)]
[
(2)
respectively, where q~ denotes the flux linkage, h
_[
Fig. 1. A Josephson junction circuit model driven by a radiofrequency current source.
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
59
V o l u m e 118, n u m b e r 2
29 S e p t e m b e r 1986
PHYSICS LETTERS A
....
•
¢¢**o....*.°°* °
o
• *
{*t
°t ~t °
,,:t.~ .,." t.t
***t
"
;°"
°
:i" :,!l." 'fii'ii ! -**°
,
-1 t[*
-2
"*; *tIt°•
Itlt.:V"
*
* t';*t
***. .
",,, .
.
.
.
, ....
..
- ,
-3 co
v~. 2
0.0
~5.4
~ .6
0 . :5:
i. (~
1,Z
Fig. 2. T h e g l o b a l b i f u r c a t i o n d i a g r a m s o f the J o s e p h s o n j u n c t i o n circuit m o d e l ; c~ = 0.4, ~ = - 0.8, fl = 0.7999.
C d2q~ + [1 + e
d,2
L
= l°sin 2"~
c°st
[4"~e
'
~lR_ldq)
Jlo
d,
wrf/.
(3)
The above equation can easily be transformed into the dimensionless form 2 + a(1 + ~ cos x ) 2 + sin x = fl sin ~0t.
(4)
For the sake of c o m p u t e r simulation, eq. (4) can equivalently be rewritten as
d x J d t = x 2,
(5)
d x z / d t = - a ( 1 + ~ cos x])x 2 - sin xl + f l sin ~0t.
60
(6)
N o w put a = 0 . 4 , ~ = - 0 . 8 , /3=0.7999. The variation of steady-state responses (i.e. the output voltage corresponding to the steady state [8]) of the Josephson junction with the change of the frequency ~orf is shown in fig. 2 *] When ~o varies from 1.2 to 0.0, the corresponding bifurcation process of steady-state solutions [9 is as follows: period-1 solution (0.69 ~< w ~< 1.2) -* period-2 solution (~o = 0.68) ~ period-3 solution (~0 --~ 0.67) ---~ chaos (0.59 ~< w ~< 0.66) ---~ period-1 solution (0.57 ~< w ~< 0.58) ---, period-2 Solution o~ -- 0.56) ---~ chaos (0.51 ~< w <~ 0.55) ---~ period-3 solution (~0 ~ 0.50) ---~ chaos (0.47 ~< ~0 .<<0.49) ---~ period-1 solution (0.0 ~< ~o ~< 0.47). The broken arrow denotes a possible middle ,1 In fact, we c h a n g e the p a r a m e t e r ~a w h e n the c o m p u t e r p r o g r a m is u n d e r w a y . But in the a b o v e d i m e n s i o n l e s s p r o c e d u r e ~,, is t a k e n to be ~rf/o~ o, while w o = ( 4 v e l c / h C ) 1/2 is a c o n s t a n t . T h u s the c h a n g e o f ~a is e q u a l to t h a t o f Wrr
Volume
118, number
2
PHYSICS
LETTERS
A
29 September
1986
3
2
1
0
-1
-2
-3
It.2
0.0
0.4
Fig. 3. Order, symmetry,
0.6 and symmetry
process. It is impossible to identify these middle structures completely although the observation can be made in more detail. In order to study the relation between chaos and symmetry, we replace the sampling distance 2n by 1 arc. The simulation results are given in fig. 3. It is quite unusual and exotic, that the picture consists of one “circular pagoda” and one “basket of flowers”. Within the parameter interval 0.0 G w G 0.47 and 0.67 G o > 1.2, i.e. the period region, the picture possesses symmetry and order structure. Within the chaotic or phase-locking region, however, this kind of symmetry and order structure disapears. In other words, chaos leads to disordering of the output and symmetry breaking of the structure in the Josephson junction device. From (3)-(6), it is easy to see that x2 = i, = (4Te/h)&
= (47re/h)V.
Thus the above symmetry
indicates
(7) the symmetry
0.8 breaking
l.M
1.2
of the solution.
of the voltage V (i.e. the output response of the device) with respect to I’= 0. The output signal is sinusoidal in the periodic region as long as the input radio-frequency current is sinusoidal, but this is not true in the region of chaotic motion and phase-locking. Finally we emphasize that the significance of the Josephson junction circuit model is that it is favorable to the application of Josephson junctions since its phenomena appear without extreme physical conditions. References [l] B.D. Josephson,
Phys. Lett. 1 (1962) 251. [2] SQUID, Proc. Int. Conf. Superconducting quantum devices, Berlin, 1976 (1st Conf.), 1980 (2nd). [3] M. Odyniec and L.O. Chua, IEEE Trans. CAS 30 (1983) 308. [4] D.Z. Wang and X.X. Yao, Acta Phys. Sin. 34 (1985) 1140 (in Chinese).
67
Volume 118, number 2
PHYSICS LETTERS A
[5] Y. Imry, in: Statics and dynamics of nonlinear systems, eds. G. Benedek et al. (Springer, Berlin, 1983). [6] D.E. McCuber, J. Appl. Phys. 39 (1968) 3113. [7] D.N. Langenberg, Rev. Phys. Appl. 9 (1974) 35.
62
29 September 1986
[8] Z.J. Zhang and Y. Yao, Proc. Int. Conf. CAS, Beijing, China (1985) p. 439. [9] Y. Yao, Nature J. 9 (1985) 687 (in Chinese).