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Self-sustained oscillations and chaotic transitions in current-carrying thin HTSC-films cooled by boiling nitrogen
26 September 1994 PHYSICS LETTERS A
Physics Letters A 193 (1994) 144-147
ELSEVIER
Self-sustained oscillations and chaotic transitions in current-carrying thin HTSC-films cooled by boiling nitrogen V.N. Skokov, V.P. Koverda, N.M. Semenova Institute of Thermophysics, Russian Academy of Sciences, UralDivision, Pervomaiskaya Street 91, GSP-828,Ekaterinburg 620219, Russian Federation
Received 1 April 1994; revised manuscript received 18 July 1994; acceptedfor publication 20 July 1994 Communicatedby A.R. Bishop
Abstract
An increase of the relaxation oscillations in the nonlinear system of a thin film of a high-temperature superconductor (HTSC) connected to an external electric circuit with an inductance, boiling nitrogen, has been shown experimentally. With increasing dc bias voltage one could observe an upset of the periodic oscillations and a transition to chaotic ones. Also, relaxation oscillations and chaos transitions have been studied experimentally in the case when adc bias voltage was added to a harmonic component.
A current-carrying superconductor is an example of a physical object in which, owing to nonlinear thermal properties, may occur various processes of self-organization and nonequilibrium phase transitions [ 1,2 ]. As a result of the Joule self-heating of normal or resistive zones a state of thermal bistability may arise in a superconductor. The voltage-current ( V - I ) characteristic of a bistable superconductor has an S-shaped form. Connecting a bistable superconductor to an external electric circuit with an inductance in the system, self-sustained relaxation oscillations may arise [ 2-4 ]. Such oscillations were often observed in the experimental study of "classical" low-temperature superconductors (see, for instance, Refs. [5-7] and Ref. [2] and references therein) and relatively recently in the experimental study of thin film microbridges of high-temperature superconductors (HTSC), cooled by liquid helium [ 8 ]. One of the conditions for the onset of self-sustained oscillations is the following: the current-relaxation time in the circuit must be larger than the time of thermal relaxation. At liquid nitrogen tempera-
tures (operating temperatures of HTSC) thermal relaxation times, due to the heat exchange by heat conduction, are much larger than the corresponding times at liquid helium temperatures. Therefore, self-sustained oscillations in a HTSC, cooled by heat conduction, can arise only with a connection to the circuit of a large inductance. However, if a superconductor is immersed in liquid nitrogen directly, the characteristic times of thermal relaxation, due to the heat removal in boiling nitrogen, are determined by the frequency of separation of vapor bubbles and decrease considerably to values of 10 - 3 10-2s. In this case self-sustained oscillations in a system may arise at relatively small circuit inductances. The onset of relaxation oscillations in an electric circuit must also influence the process of generation of vapor bubbles. Such an influence, in particular a transition from periodic relaxation oscillations to chaotic ones, is found in the present study. Also, the results of an investigation of the effect of the harmonic component of a bias voltage on the dynamic behavior of the system is presented.
0375-9601/94/$07.00 © 1994 ElsevierScience B.V. All rights reserved SSD10375-9601 (94) 00620-2
145
EN. Skokov et al. / Physics Letters A 193(1994) 144-147
Experiments were carried out on thin-film bridges YBa2Cu3OT_x obtained by the method of dc magnetron sputtering [ 9 ] and immersed directly into liquid nitrogen. The thickness of bridges was ~ 0.3 lxm, the width from 0.7 to 1.0 mm, the length varied from 1.0 to 7.0 mm. The temperatures of superconducting transition were T¢=86-88 K. The bridges were inserted into the circuit with a dc voltage supply successively with an inductance and a load resistance. Measurements were made with inductance L l = 14 m H and load resistance R I = 2.75 ohm, and L2 = 180 m H and R2= 3.62 ohm. The voltage on the potential contacts of the superconductor and the current in the circuit (voltage drop on the load resistor) were recorded on an XY-recorder, and oscillations on oscillographs connected with a PC. Fig. I gives a typical 11-1characteristic taken without an inductance. At point A in Fig. 1 in an HTSCbridge there arises a thermal instability. The section AB corresponds to the growth of the thermal domain ("hot spot" ), liquid nitrogen near the sample surface is superheated and heat exchange occurs at the expense of one-phase convection. Point B in Fig. 1 corresponds to the moment of nitrogen boiling-up at a place with increased heat release (nitrogen boiling was LJ,V
observed visually). As a result of the boiling-up the convective heat transfer from the heating surface improves, and the local nitrogen superheating decreases. This leads to a stepwise transition of the system from point B to point C along the load line of the electric circuit. When nitrogen boiled, chaotic oscillations with a frequency corresponding to the frequency of bubble separation were observed on the II-1 characteristic [ 9 ]. With insertion of an inductance into the circuit periodic oscillations arose. Fig. 2 gives oscillograms of the voltage drop on the potential contacts and the current in the circuit connected with the inductance L = 180 mH. A necessary condition of the onset of self-sustained oscillations is the presence of a section with a large negative differential conductance on the I1"-1 characteristic. Such sections arise in the case when d Q / d T > d W / d T , where Q ( T ) is the Joule heat release power and W(T) is the heat removal power. If a superconductor was immersed in boiling nitrogen directly, the condition of onset of thermal instability was satisfied only for films with a relatively large critical current density ( >/105 A/cm2), a sharp temperature dependence of the resistance and homogeneous along a length. In the presence of a strong inhomogeneity in a film a stable thermal domain was
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~
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0.0 0.0
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L
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . , ......... , ......... I,A
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. . . . . . . . .
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I, A Fig. 1. V-/characteristic of thin Y-Ba-Cu-O films immerseddirectly in liquid nitrogen. The dotted line shows the phase trajectory in the V-I plane under self-sustainedoscillations.
0.2 ......... , ......... i ......... , ......... , ......... o.oo 0.05 o.10 o.15 0.20 0.25 t,s
Fig. 2. Oscillogramsof the voltagedrop on the potential contacts ( 1) and transport current in the circuit (2) with connection of the inductance L-- 180 mH.
146
V.N. Skokov et al. / Physics Letters A 193 (1994) 144-147
formed and the V-I characteristics contained sections only with a positive inclination. It should be emphasized in particular that, when relaxation oscillations arise in a system, the nitrogen boiling ceases to be chaotic, and the frequency of generation and separation of vapor bubbles coincides with the frequency of the periodic oscillations. In Fig. 1 a dotted line shows the phase trajectory during one period of self-sustained oscillations at the minimum bias voltage at which self-excited oscillations are still possible. With increasing bias voltage the frequency of the oscillations increased. At a certain value of the bias voltage one observed an upset of the periodic oscillations and a transition to chaotic ones. The chaotic oscillations connected with nitrogen boiling at the film surface were the same as without an inductance in the circuit and had a considerably smaller amplitude than periodic oscillations. Fig. 3 gives the sequence U
qv~
(6)~ I'v 0.1A
0.00
0.05
0.10
0.15
t,s Fig. 3. Oscillogramsof the voltagedrop on the potential contacts and the phase trajectories in the 1I-1 plane at different de bias voltage Uo: (1) 1.79 V, (2) 1.87 V, (3) 1.93 V, (4) 1.94 V, (5) 1.95 V. Load resistanceR~= 2.75 ohm. Inductance L = 14 mH.
of oscillograms of the voltage drop on the potential contacts with increasing bias voltage. The same figure gives the phase trajectories of the system in the V - I plane. With increasing inductance in the circuit the frequency of periodic self-sustained oscillations decreased, and the amplitude increased. A much more diverse pattern was observed in the case when the constant dc bias voltage was supplemented with a harmonic component. For this purpose use was made of an additional harmonic oscillator, which was connected in parallel with an HTSC bridge. In a certain frequency range of the harmonic component, synchronization of the relaxation selfsustained oscillations and additional forcing ones was observed. The resulting frequency was defined by the forcing oscillations. With frequency detuning, i.e. decreasing or increasing the harmonic-component frequency below or above the threshold value, one observed a transition from simple periodic to more complicated oscillations. Fig. 4 gives oscillograms of the voltage drop on the potential contacts at different values of the frequency and constant amplitude of the forcing harmonic oscillations. The same figure gives the phase trajectories in the II-1 plane. The observed patterns resemble transitions from oscillations of the limit-cycle type to deterministic chaos [ 10, t 1 ]. As has already been mentioned above, the possibility of an onset of self-sustained oscillations in a nonlinear conductor with an S-shaped V-/characteristic connected to a circuit with an inductance is well known. Such self-sustained oscillations arise as a result of the H o p f bifurcation, i.e. transformation of a stable focus into an unstable one and are described by the generalized van der Pol equation [ 4,10,11 ]. An essential moment of the present study is the detection of a correlation between the nonlinear processes in the circuit and the thermal and hydrodynamic processes in the cooler. The self-sustained oscillations in the circuit control the process of bubble generation in the circuit. In the system arises synchronization of electric and thermo-hydrodynamic oscillations. During an upset of periodic oscillations one observes intermittency and a transition to chaos. Connecting the circuit with a source of external harmonic oscillations, there appears an additional dynamic variable in the system. Here we are dealing with an analogy with a driven van der Pol oscillator [ 11,12 ]. This causes a variety of patterns of transi-
V.N. Skokov et aL / Physics Letters A 193 (1994) 144-147
U
1 V 0.02
i i i
iii
ii
0.04
i tl
ii
0.0:5
I,
lllll
0.06
I II
quency a n d the a m p l i t u d e o f a driving force. In order to measure such a dependence it is necessary to establish quantitative criteria o f differences between periodic, quasi-periodic and chaotic motions. The results o f such investigations will be presented in our next publication. The observed patterns m a y be interpreted as nonequilibrium phase transitions in a complex nonlinear system: a superconductor with a current connected to an external circuit with an inductance, boiling nitrogen. Such n o n e q u i l i b r i u m phase transitions are acc o m p a n i e d by traditional " e q u i l i b r i u m " supercond u c t o r - n o r m a l conductor a n d l i q u i d - v a p o r phase transitions.
I / / l l l
0.10
1
0.1 A
147
0.15
The authors are grateful to Professor V.P. Skripov for his attention and encouragement to this work and Professor A.M. Iskoldsky a n d Dr. N.B. Volkov for helpful discussions.
(z)
References
,,,,,,,,ll,,,,,,,,,l,,,,,,,,,i
0.00
0.10
.
0.20
0.30
t,S Fig. 4. Oscillograms of the voltage drop and the phase trajectories in the V-I plane with an additional harmonic oscillator at different values of the frequency of forcing harmonic oscillations; ( 1) natural relaxation oscillation 0~ = 64 Hz); ( 2 ) - (7) forced oscillation with driving frequencyf (2) 10 Hz, (3) 15 Hz, (4) 100 Hz, (5) 180 Hz, (6) 220 Hz, (7) 900 Hz. tion from oscillations o f the limit-cycle to more complex motions. F o r a more complete investigation o f the response o f the system to the external h a r m o n i c force, it is necessary to measure the phase d i a g r a m o f the system in the f r e q u e n c y - a m p l i t u d e plane, i.e., to measure the dependence o f the response o f the system on the fre-
[ 1] V.A.Altov, V.B. Zenkevich, N.G. Kremlev and V.V. Sytchev, Stabilization of superconducting magnetic systems (Plenum, New York, 1984). [2] A.V1.Gurevich and R.G. Mints, Rev. Mod. Phys. 59 (1987) 941. [3 ] S.Ya. Berkovich, Radiotechn. Elektron. 10 (1965) 736. [4] Yu.M. Lvovsky, Zh. Techn. Fiz. 58 (1988) 2219. [ 5 ] W.J. Skocpol, M.R. Beasleyand M. Tinkham, J. Appl. Phys. 45 (1974) 4054. [6] R.P. Huebener, J. Appl. Phys. 46 (1975) 4982. [7 ] I.I. Era, S.A.Peskovatskii and A.V. Poladich, Fiz. Tver. Tela 15 (1973) 2228. [ 8 ] T. Kisu, K. Enpuku, K. Yoshida, M. Takeo and K. Yarnafuji, Japan. J. Appl. Phys~ 29 (1990) L. 2056. [ 9 ] V.N. Skokov and V.P. Koverda, Cryogenics33 (1993) 1072. [ 10] H.G. Schuster, Deterministic chaos (Physik-Verlag, Weinheim, 1984). [ 11 ] E. Sch611, Nonequilibrium phase transitions in semiconductors (Springer, Berlin, 1987). [ 12] J. Guckenheimer, Physica D 1 (1980) 227.
Physics Letters A 193 (1994) 144-147
ELSEVIER
Self-sustained oscillations and chaotic transitions in current-carrying thin HTSC-films cooled by boiling nitrogen V.N. Skokov, V.P. Koverda, N.M. Semenova Institute of Thermophysics, Russian Academy of Sciences, UralDivision, Pervomaiskaya Street 91, GSP-828,Ekaterinburg 620219, Russian Federation
Received 1 April 1994; revised manuscript received 18 July 1994; acceptedfor publication 20 July 1994 Communicatedby A.R. Bishop
Abstract
An increase of the relaxation oscillations in the nonlinear system of a thin film of a high-temperature superconductor (HTSC) connected to an external electric circuit with an inductance, boiling nitrogen, has been shown experimentally. With increasing dc bias voltage one could observe an upset of the periodic oscillations and a transition to chaotic ones. Also, relaxation oscillations and chaos transitions have been studied experimentally in the case when adc bias voltage was added to a harmonic component.
A current-carrying superconductor is an example of a physical object in which, owing to nonlinear thermal properties, may occur various processes of self-organization and nonequilibrium phase transitions [ 1,2 ]. As a result of the Joule self-heating of normal or resistive zones a state of thermal bistability may arise in a superconductor. The voltage-current ( V - I ) characteristic of a bistable superconductor has an S-shaped form. Connecting a bistable superconductor to an external electric circuit with an inductance in the system, self-sustained relaxation oscillations may arise [ 2-4 ]. Such oscillations were often observed in the experimental study of "classical" low-temperature superconductors (see, for instance, Refs. [5-7] and Ref. [2] and references therein) and relatively recently in the experimental study of thin film microbridges of high-temperature superconductors (HTSC), cooled by liquid helium [ 8 ]. One of the conditions for the onset of self-sustained oscillations is the following: the current-relaxation time in the circuit must be larger than the time of thermal relaxation. At liquid nitrogen tempera-
tures (operating temperatures of HTSC) thermal relaxation times, due to the heat exchange by heat conduction, are much larger than the corresponding times at liquid helium temperatures. Therefore, self-sustained oscillations in a HTSC, cooled by heat conduction, can arise only with a connection to the circuit of a large inductance. However, if a superconductor is immersed in liquid nitrogen directly, the characteristic times of thermal relaxation, due to the heat removal in boiling nitrogen, are determined by the frequency of separation of vapor bubbles and decrease considerably to values of 10 - 3 10-2s. In this case self-sustained oscillations in a system may arise at relatively small circuit inductances. The onset of relaxation oscillations in an electric circuit must also influence the process of generation of vapor bubbles. Such an influence, in particular a transition from periodic relaxation oscillations to chaotic ones, is found in the present study. Also, the results of an investigation of the effect of the harmonic component of a bias voltage on the dynamic behavior of the system is presented.
0375-9601/94/$07.00 © 1994 ElsevierScience B.V. All rights reserved SSD10375-9601 (94) 00620-2
145
EN. Skokov et al. / Physics Letters A 193(1994) 144-147
Experiments were carried out on thin-film bridges YBa2Cu3OT_x obtained by the method of dc magnetron sputtering [ 9 ] and immersed directly into liquid nitrogen. The thickness of bridges was ~ 0.3 lxm, the width from 0.7 to 1.0 mm, the length varied from 1.0 to 7.0 mm. The temperatures of superconducting transition were T¢=86-88 K. The bridges were inserted into the circuit with a dc voltage supply successively with an inductance and a load resistance. Measurements were made with inductance L l = 14 m H and load resistance R I = 2.75 ohm, and L2 = 180 m H and R2= 3.62 ohm. The voltage on the potential contacts of the superconductor and the current in the circuit (voltage drop on the load resistor) were recorded on an XY-recorder, and oscillations on oscillographs connected with a PC. Fig. I gives a typical 11-1characteristic taken without an inductance. At point A in Fig. 1 in an HTSCbridge there arises a thermal instability. The section AB corresponds to the growth of the thermal domain ("hot spot" ), liquid nitrogen near the sample surface is superheated and heat exchange occurs at the expense of one-phase convection. Point B in Fig. 1 corresponds to the moment of nitrogen boiling-up at a place with increased heat release (nitrogen boiling was LJ,V
observed visually). As a result of the boiling-up the convective heat transfer from the heating surface improves, and the local nitrogen superheating decreases. This leads to a stepwise transition of the system from point B to point C along the load line of the electric circuit. When nitrogen boiled, chaotic oscillations with a frequency corresponding to the frequency of bubble separation were observed on the II-1 characteristic [ 9 ]. With insertion of an inductance into the circuit periodic oscillations arose. Fig. 2 gives oscillograms of the voltage drop on the potential contacts and the current in the circuit connected with the inductance L = 180 mH. A necessary condition of the onset of self-sustained oscillations is the presence of a section with a large negative differential conductance on the I1"-1 characteristic. Such sections arise in the case when d Q / d T > d W / d T , where Q ( T ) is the Joule heat release power and W(T) is the heat removal power. If a superconductor was immersed in boiling nitrogen directly, the condition of onset of thermal instability was satisfied only for films with a relatively large critical current density ( >/105 A/cm2), a sharp temperature dependence of the resistance and homogeneous along a length. In the presence of a strong inhomogeneity in a film a stable thermal domain was
5.00 U,V
4.0
~
)
4 00
2.0 3.00
0.0 0.0
2.00
L
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . , ......... , ......... I,A
(2) 1.00
0.00
0.4
"~
. . . . . . . . .
0.00
, . . . . . . . . .
0.20
, . . . . . . . . . .
0.40
0.60
I, A Fig. 1. V-/characteristic of thin Y-Ba-Cu-O films immerseddirectly in liquid nitrogen. The dotted line shows the phase trajectory in the V-I plane under self-sustainedoscillations.
0.2 ......... , ......... i ......... , ......... , ......... o.oo 0.05 o.10 o.15 0.20 0.25 t,s
Fig. 2. Oscillogramsof the voltagedrop on the potential contacts ( 1) and transport current in the circuit (2) with connection of the inductance L-- 180 mH.
146
V.N. Skokov et al. / Physics Letters A 193 (1994) 144-147
formed and the V-I characteristics contained sections only with a positive inclination. It should be emphasized in particular that, when relaxation oscillations arise in a system, the nitrogen boiling ceases to be chaotic, and the frequency of generation and separation of vapor bubbles coincides with the frequency of the periodic oscillations. In Fig. 1 a dotted line shows the phase trajectory during one period of self-sustained oscillations at the minimum bias voltage at which self-excited oscillations are still possible. With increasing bias voltage the frequency of the oscillations increased. At a certain value of the bias voltage one observed an upset of the periodic oscillations and a transition to chaotic ones. The chaotic oscillations connected with nitrogen boiling at the film surface were the same as without an inductance in the circuit and had a considerably smaller amplitude than periodic oscillations. Fig. 3 gives the sequence U
qv~
(6)~ I'v 0.1A
0.00
0.05
0.10
0.15
t,s Fig. 3. Oscillogramsof the voltagedrop on the potential contacts and the phase trajectories in the 1I-1 plane at different de bias voltage Uo: (1) 1.79 V, (2) 1.87 V, (3) 1.93 V, (4) 1.94 V, (5) 1.95 V. Load resistanceR~= 2.75 ohm. Inductance L = 14 mH.
of oscillograms of the voltage drop on the potential contacts with increasing bias voltage. The same figure gives the phase trajectories of the system in the V - I plane. With increasing inductance in the circuit the frequency of periodic self-sustained oscillations decreased, and the amplitude increased. A much more diverse pattern was observed in the case when the constant dc bias voltage was supplemented with a harmonic component. For this purpose use was made of an additional harmonic oscillator, which was connected in parallel with an HTSC bridge. In a certain frequency range of the harmonic component, synchronization of the relaxation selfsustained oscillations and additional forcing ones was observed. The resulting frequency was defined by the forcing oscillations. With frequency detuning, i.e. decreasing or increasing the harmonic-component frequency below or above the threshold value, one observed a transition from simple periodic to more complicated oscillations. Fig. 4 gives oscillograms of the voltage drop on the potential contacts at different values of the frequency and constant amplitude of the forcing harmonic oscillations. The same figure gives the phase trajectories in the II-1 plane. The observed patterns resemble transitions from oscillations of the limit-cycle type to deterministic chaos [ 10, t 1 ]. As has already been mentioned above, the possibility of an onset of self-sustained oscillations in a nonlinear conductor with an S-shaped V-/characteristic connected to a circuit with an inductance is well known. Such self-sustained oscillations arise as a result of the H o p f bifurcation, i.e. transformation of a stable focus into an unstable one and are described by the generalized van der Pol equation [ 4,10,11 ]. An essential moment of the present study is the detection of a correlation between the nonlinear processes in the circuit and the thermal and hydrodynamic processes in the cooler. The self-sustained oscillations in the circuit control the process of bubble generation in the circuit. In the system arises synchronization of electric and thermo-hydrodynamic oscillations. During an upset of periodic oscillations one observes intermittency and a transition to chaos. Connecting the circuit with a source of external harmonic oscillations, there appears an additional dynamic variable in the system. Here we are dealing with an analogy with a driven van der Pol oscillator [ 11,12 ]. This causes a variety of patterns of transi-
V.N. Skokov et aL / Physics Letters A 193 (1994) 144-147
U
1 V 0.02
i i i
iii
ii
0.04
i tl
ii
0.0:5
I,
lllll
0.06
I II
quency a n d the a m p l i t u d e o f a driving force. In order to measure such a dependence it is necessary to establish quantitative criteria o f differences between periodic, quasi-periodic and chaotic motions. The results o f such investigations will be presented in our next publication. The observed patterns m a y be interpreted as nonequilibrium phase transitions in a complex nonlinear system: a superconductor with a current connected to an external circuit with an inductance, boiling nitrogen. Such n o n e q u i l i b r i u m phase transitions are acc o m p a n i e d by traditional " e q u i l i b r i u m " supercond u c t o r - n o r m a l conductor a n d l i q u i d - v a p o r phase transitions.
I / / l l l
0.10
1
0.1 A
147
0.15
The authors are grateful to Professor V.P. Skripov for his attention and encouragement to this work and Professor A.M. Iskoldsky a n d Dr. N.B. Volkov for helpful discussions.
(z)
References
,,,,,,,,ll,,,,,,,,,l,,,,,,,,,i
0.00
0.10
.
0.20
0.30
t,S Fig. 4. Oscillograms of the voltage drop and the phase trajectories in the V-I plane with an additional harmonic oscillator at different values of the frequency of forcing harmonic oscillations; ( 1) natural relaxation oscillation 0~ = 64 Hz); ( 2 ) - (7) forced oscillation with driving frequencyf (2) 10 Hz, (3) 15 Hz, (4) 100 Hz, (5) 180 Hz, (6) 220 Hz, (7) 900 Hz. tion from oscillations o f the limit-cycle to more complex motions. F o r a more complete investigation o f the response o f the system to the external h a r m o n i c force, it is necessary to measure the phase d i a g r a m o f the system in the f r e q u e n c y - a m p l i t u d e plane, i.e., to measure the dependence o f the response o f the system on the fre-
[ 1] V.A.Altov, V.B. Zenkevich, N.G. Kremlev and V.V. Sytchev, Stabilization of superconducting magnetic systems (Plenum, New York, 1984). [2] A.V1.Gurevich and R.G. Mints, Rev. Mod. Phys. 59 (1987) 941. [3 ] S.Ya. Berkovich, Radiotechn. Elektron. 10 (1965) 736. [4] Yu.M. Lvovsky, Zh. Techn. Fiz. 58 (1988) 2219. [ 5 ] W.J. Skocpol, M.R. Beasleyand M. Tinkham, J. Appl. Phys. 45 (1974) 4054. [6] R.P. Huebener, J. Appl. Phys. 46 (1975) 4982. [7 ] I.I. Era, S.A.Peskovatskii and A.V. Poladich, Fiz. Tver. Tela 15 (1973) 2228. [ 8 ] T. Kisu, K. Enpuku, K. Yoshida, M. Takeo and K. Yarnafuji, Japan. J. Appl. Phys~ 29 (1990) L. 2056. [ 9 ] V.N. Skokov and V.P. Koverda, Cryogenics33 (1993) 1072. [ 10] H.G. Schuster, Deterministic chaos (Physik-Verlag, Weinheim, 1984). [ 11 ] E. Sch611, Nonequilibrium phase transitions in semiconductors (Springer, Berlin, 1987). [ 12] J. Guckenheimer, Physica D 1 (1980) 227.