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The bifurcations in the rf-SQUID system
Physica B 159 (1989) 399-401 North-Holland, Amsterdam
THE BIFURCATIONS IN THE rf-SQUID SYSTEM Zbigniew
J. KOWALIK’
and Jan STANKOWSKI
Institute of Molecular Physics, Polish Academy Received 14 December 1987 Revised manuscript received 30 September
Measurements of the response points have been found
of Sciences,
0921-4526/89/$03.50 0 (North-Holland Physics
17119, PL-60-179
Poznari, Poland
1988
of the rf-SQUID
system
Many practical devices exhibit universal behaviour which was predicted theoretically by Feigenbaum for nonlinear dissipative structures [l]. The period doubling scenario in the route to chaos with increasing of the nonlinear parameter or the amplitude of an external driving force is characteristic and universal for a wide class of dynamical systems. This scenario describes the creation of the modes of motion which implicates new components in the Fourier spectrum with frequencies being the subharmonical of the fundamental frequency w,,. These new components appear successively with growth of the amplitude of the nonlinear term in the equation of motion and they have frequencies equal to 4 oO, $ oO, . . . and so on. For this reason that mechanism was called period doubling bifurcation. The limit of the bifurcation cascade is the chaotic behaviour. After this limit many phenomena like periodic windows, intermittencies, crises [2], selfreanimating chaotic attractors [3], and others can occur. It is known that chaotic motion (or deterministic chaos) occurs also in the Josephson devices [4,5]. Moreover, it was recently studied in the Phase-Locked-Loop (PLL) model of the Josephson junction [6]. We expected such behaviour also in the rf-SQUID system treated as a whole, ’ Present address: Institute of Applied Miinster, Corrensstr. 2-4, D-4400 Germany.
Smoluchowskiego
Physics, University of Miinster, Fed. Rep.
Elsevier Science Publishers Publishing Division)
on the external
driving
force have been made.
Two bifurcation
i.e., the rf-SQUID sensor coupled with a detection circuit in the flux-locked-loop operation mode as this system has properties similar to the PLL device [7]. Figure 1 presents the experimental setup. Here, the typical rf-SQUID system [7,8] is periodically disturbed by the low-frequency (If) oscillator. (Except for the oscilloscope and lfoscillator, all elements are substantial parts of the SQUID measurement system which was constructed in our laboratory.) The frequency of this signal was chosen as w,,/2n-4.5 Hz, which is close to the value of the characteristic frequency of the whole system related to the time constant of the integrator and cut-off frequency of the detector filter. The driving current is supplied to the SQUID sensor through the tank coil L, together with the feedback signal and audiofrequency bias. We observed the response of the system on the external driving force using the technique of the Lissajous-figures. The driving signal was put on the X-axis of the screen and the total response of the SQUID system on the Y-axis. The experimental results are presented in fig. 2. The regular motion is characterized by a simple limit cycle (fig. 2a). Here the influence of the nonlinearity appears only in the asymmetry of the phase portrait shape. Next, the trajectory in the phase space presented in fig. 2b splits with growing excitation amplitude. The phenomenon shown in this picture, consisting of doubling of B.V.
400
Fig. I. Observation of the nonlinear rcsponse of the rf-SQUID system. The SQUID (characteristic regime ot Integration (T = 100 ms). External w,,i2n = 31 MHz, w,,i2n = 9 kHz) works in the “middle” The sen SOT is of the Zimmermann type
Fig. 2. Phase portraits of the periodically 9.05 V; (c) 9.40 V: (d) 9.64 V.
disturbed
SQUID
system.
The amplitude
of excitation
data: 5 X 10 ’ C/+,/V%. excitation at CO,,= 4.5 Hz.
is equal
to: (a) 8.15 V: (h)
Z.J.
Kowalik and J. Stankowski
I The bifurcations in the rf-SQUID
the trajectory of motion, is the bifurcation effect. For increasing external driving forces this effect becomes more evident (fig. 2c) and next, second bifurcation takes place (fig. 2d). Because of the strong negative feedback no further evolution was observed. The effect described here can still be an example of the universal behaviour in the dynamical systems as well as the new method of estimation of the linear range of operation of the rf-SQUID system. We believe that such treatment can be useful for many other devices.
system
401
References J. Stat. Phys. 21 (1979) 25. [II M. Feigenbaum, PI For a review, see e.g., Physica D 7 (1983) 1-362. and P. Pieraliski, Phys. Rev. [31 Z.J. Kowalik, M. Franaszek A 37 (1988) 4016. [41 K. Fesser, A.R. Bishop and P. Kumar, Appl. Phys. Lett. 43 (1983) 123. [51 R.K. Ritala and M.M. Salomaa. Phys. Rev. B 29 (1984) 6143. [61 Z.J. Kowalik, Acta Phys. Pol. A 64 (1983) 357. [71 Z.J. Kowalik, Thesis, Pozna6 (1986). Seminary, J. Stankowski, PI Z.J. Kowalik, in: Cryogenics ed. (Poznali, 1982, in Polish).
THE BIFURCATIONS IN THE rf-SQUID SYSTEM Zbigniew
J. KOWALIK’
and Jan STANKOWSKI
Institute of Molecular Physics, Polish Academy Received 14 December 1987 Revised manuscript received 30 September
Measurements of the response points have been found
of Sciences,
0921-4526/89/$03.50 0 (North-Holland Physics
17119, PL-60-179
Poznari, Poland
1988
of the rf-SQUID
system
Many practical devices exhibit universal behaviour which was predicted theoretically by Feigenbaum for nonlinear dissipative structures [l]. The period doubling scenario in the route to chaos with increasing of the nonlinear parameter or the amplitude of an external driving force is characteristic and universal for a wide class of dynamical systems. This scenario describes the creation of the modes of motion which implicates new components in the Fourier spectrum with frequencies being the subharmonical of the fundamental frequency w,,. These new components appear successively with growth of the amplitude of the nonlinear term in the equation of motion and they have frequencies equal to 4 oO, $ oO, . . . and so on. For this reason that mechanism was called period doubling bifurcation. The limit of the bifurcation cascade is the chaotic behaviour. After this limit many phenomena like periodic windows, intermittencies, crises [2], selfreanimating chaotic attractors [3], and others can occur. It is known that chaotic motion (or deterministic chaos) occurs also in the Josephson devices [4,5]. Moreover, it was recently studied in the Phase-Locked-Loop (PLL) model of the Josephson junction [6]. We expected such behaviour also in the rf-SQUID system treated as a whole, ’ Present address: Institute of Applied Miinster, Corrensstr. 2-4, D-4400 Germany.
Smoluchowskiego
Physics, University of Miinster, Fed. Rep.
Elsevier Science Publishers Publishing Division)
on the external
driving
force have been made.
Two bifurcation
i.e., the rf-SQUID sensor coupled with a detection circuit in the flux-locked-loop operation mode as this system has properties similar to the PLL device [7]. Figure 1 presents the experimental setup. Here, the typical rf-SQUID system [7,8] is periodically disturbed by the low-frequency (If) oscillator. (Except for the oscilloscope and lfoscillator, all elements are substantial parts of the SQUID measurement system which was constructed in our laboratory.) The frequency of this signal was chosen as w,,/2n-4.5 Hz, which is close to the value of the characteristic frequency of the whole system related to the time constant of the integrator and cut-off frequency of the detector filter. The driving current is supplied to the SQUID sensor through the tank coil L, together with the feedback signal and audiofrequency bias. We observed the response of the system on the external driving force using the technique of the Lissajous-figures. The driving signal was put on the X-axis of the screen and the total response of the SQUID system on the Y-axis. The experimental results are presented in fig. 2. The regular motion is characterized by a simple limit cycle (fig. 2a). Here the influence of the nonlinearity appears only in the asymmetry of the phase portrait shape. Next, the trajectory in the phase space presented in fig. 2b splits with growing excitation amplitude. The phenomenon shown in this picture, consisting of doubling of B.V.
400
Fig. I. Observation of the nonlinear rcsponse of the rf-SQUID system. The SQUID (characteristic regime ot Integration (T = 100 ms). External w,,i2n = 31 MHz, w,,i2n = 9 kHz) works in the “middle” The sen SOT is of the Zimmermann type
Fig. 2. Phase portraits of the periodically 9.05 V; (c) 9.40 V: (d) 9.64 V.
disturbed
SQUID
system.
The amplitude
of excitation
data: 5 X 10 ’ C/+,/V%. excitation at CO,,= 4.5 Hz.
is equal
to: (a) 8.15 V: (h)
Z.J.
Kowalik and J. Stankowski
I The bifurcations in the rf-SQUID
the trajectory of motion, is the bifurcation effect. For increasing external driving forces this effect becomes more evident (fig. 2c) and next, second bifurcation takes place (fig. 2d). Because of the strong negative feedback no further evolution was observed. The effect described here can still be an example of the universal behaviour in the dynamical systems as well as the new method of estimation of the linear range of operation of the rf-SQUID system. We believe that such treatment can be useful for many other devices.
system
401
References J. Stat. Phys. 21 (1979) 25. [II M. Feigenbaum, PI For a review, see e.g., Physica D 7 (1983) 1-362. and P. Pieraliski, Phys. Rev. [31 Z.J. Kowalik, M. Franaszek A 37 (1988) 4016. [41 K. Fesser, A.R. Bishop and P. Kumar, Appl. Phys. Lett. 43 (1983) 123. [51 R.K. Ritala and M.M. Salomaa. Phys. Rev. B 29 (1984) 6143. [61 Z.J. Kowalik, Acta Phys. Pol. A 64 (1983) 357. [71 Z.J. Kowalik, Thesis, Pozna6 (1986). Seminary, J. Stankowski, PI Z.J. Kowalik, in: Cryogenics ed. (Poznali, 1982, in Polish).