- Email: [email protected]
The strange attractor of giant optical fluctuations of 2D electrons in the quantum Hall effect regime
Physica E 44 (2012) 1653–1656
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
The strange attractor of giant optical fluctuations of 2D electrons in the quantum Hall effect regime A.L. Parakhonsky n, M.V. Lebedev, A.A. Dremin, I.V. Kukushkin Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Russia
a r t i c l e i n f o
abstract
Article history: Received 21 March 2012 Received in revised form 6 April 2012 Accepted 10 April 2012 Available online 7 May 2012
Strange attractor of the giant optical fluctuations of 2D system was reconstructed in phase space. Grassberger–Procaccia algorithm was explored to compute correlation sum of the attractor. The embedding and correlation dimensions are estimated to be 4 and 3.6, respectively. Thus, the revealed attractor is characterized by a system of 4 nonlinear differential equations. & 2012 Elsevier B.V. All rights reserved.
1. Introduction In our previous works on the giant optical fluctuations (GOF) of 2D electron system in the quantum Hall effect regime, we have noticed that the character of a spectrum of such fluctuations shows similarity with that considered in the theory of open dissipative dynamic systems [1,2]. Indeed, in our experiments, a 2D electronic gas is an open dissipative system being far from equilibrium. This system continuously gains energy due to the laser excitation and continuously consumes energy through the recombination of 2D electrons with photoexcited holes. With a certain value of the control parameter the fluctuation spectrum becomes continuous, without clearly defined features. This state of the system is a precursor for the appearance of incommensurate frequencies in the fluctuation spectrum and the onset of instability in the system (‘‘Fibonacci number’’ regime) [2]. Such behavior of a system corresponds to the regime that became known as deterministic chaos and is described by the Ruelle– Takens–Newhouse model [3]. This model states that under small perturbation of a dissipative dynamic system, a strange attractor can arise in it. It is such attracting set of unstable trajectories in the phase space that determines the number of interrelated nonlinear differential equations describing the system. Using the methods developed in this theory, it would be reasonable to define a characteristic image and the dimension of a possible strange attractor of the fluctuating 2D system. Therefore, it enables one to define the number of nonlinear differential equations describing the process under study. In the present work, we reveal the results allowing us to justify the occurrence
n
Corresponding author. E-mail address: [email protected] (A.L. Parakhonsky).
1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2012.04.014
of a strange attractor in our fluctuating 2D system. The revealed attractor is characterized by a system of four nonlinear differential equations.
2. Experiment In the work, we used the samples grown with a molecularbeam epitaxial technique on a GaAs substrate according to the ˚ following scheme: a 3000-A-thick GaAs buffer layer, an undoped ˚ super lattice with a total thickness GaAs/Al0.3Ga0.7As (30–100 A) ˚ a 250-A-wide ˚ of 13,000 A, GaAs quantum well, a 400-A˚ ˚ Al0.3Ga0.7As spacer, and a 650-A-thick Al0.3Ga0.7As: Si doped (at a level of 1018 cm 3) layer. The concentration and mobility of electrons were nS ¼4 1011 cm 2 and m ¼ 2 106 cm2 =ðV sÞ, respectively. These are the standard structures where the giant optical fluctuations are observed. The sample under study was placed inside a superconducting solenoid, immersed in a liquidhelium cryostat (at temperature T¼1.6–1.7 K). The solenoid produced a magnetic field up to 12 T. In order to analyze the giant optical fluctuations and their properties we used a special optical scheme, previously described in Ref. [4]. The time dependences of PL intensities measured for the maximum of radiation line, corresponding to the zero Landau level at magnetic field in the vicinity of filling factor n ¼ 2, were studied. A semiconductor charge-coupled detector (CCD) was used to record the PL signal. Sufficiently long (up to t ¼7200 s) sequences of the 2D electron PL time dependences I(t) were recorded with steps of 1 s. In order to analyze the fluctuation level, we estimated the average 2Delectron radiation intensity over the measurement time /IS, the variance of the intensity D ¼ /DI2 S, and the ratio of the variance to the mean value of the intensity D=/IS. The GOF
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A.L. Parakhonsky et al. / Physica E 44 (2012) 1653–1656
regime ðD=/IS b 1Þ at n ¼ 2 corresponded to magnetic field B ¼7.78 T.
3. Pseudo-attractor of 2D fluctuating system The phase portrait of a possible strange attractor of dissipative dynamic systems can be received using the methods described in Refs. [2,3,5]. Takens has shown [5] that a time series of measurements of a single observable X(t) of the dynamic system trajectory can be used to reconstruct qualitative features of the strange attractor in phase space. In our case, such a component is the time dependence of 2D electron PL intensity I(t). The measured time series are used with some time delay t. The sequence of m-phase space vectors of one component X(t) is given by 8 ð1Þ X ðtÞ ¼ IðtÞ > > > > ð2Þ > > < X ðtÞ ¼ Iðt þ tÞ X ð3Þ ðtÞ ¼ Iðt þ2tÞ ð1Þ > > >^ > > > : ðmÞ X ðtÞ ¼ Iðt þ ðm1ÞtÞ: Here, m is the parameter designated as the embedding dimension (see below). We used system (1) for the reconstruction of a possible strange attractor of the fluctuating 2D system at various values of magnetic field. We revealed that at the minimal integer value m¼3, there is such mode of noise in the vicinity of n ¼ 2 where the value of vectors (1) are eventually grouped in the vicinity of each of three axes of phase space (Fig. 1). The type of experimental time dependence I(t), where the strange attractor was reconstructed, is shown on the inset of Fig. 1. The similar phase portrait at m ¼3 was observed in Ref. [6]. In this work, methods of the theory of dynamic systems were applied to the analysis of geology-geophysical experimental data. This similarity suggests nonlinear dynamic mechanisms being common to a wide range of dynamic systems. Fig. 2 demonstrates the result of a similar reconstruction under the close conditions; however, it has no such a characteristic phase portrait as shown in Fig. 1. A set of trajectories in phase space forms a sphere in this case. Such a phase portrait can be usually obtained in the absence of a strange attractor of the system [3]. It is necessary to notice that a noise character differs for these two cases. In the first instance, fluctuations have discrete regions where the noise amplitude is insignificant. While in the latter the giant optical fluctuations occur in a continuous mode, practically,
Fig. 2. Phase space reconstructed for the dependence I(t) measured under the same conditions as that presented in Fig. 1 measured after a time. The characteristic phase portrait is absent.
Fig. 3. Structureless cloud of the trajectories in the lag space I(t) obtained under different conditions: B¼ 8 T, T¼ 1.63–1.71 K.
throughout all time series I(t). Besides, the noise amplitude in the first instance is 1.5–2 times larger than in the former. For the time sequences measured at the other magnetic fields, a strange attractor was not observed as well (Fig. 3). If one considers the grouping of trajectories along the axes of three-dimensional phase space, then one can conclude that for such a phase portrait, the necessary requirement is the presence of discrete parts in the noise time dependence. In the other words, the intensity of the studied signal must hold small amplitude long enough (magnitude as lag time t).
4. Correlation dimension of GOF strange attractor
Fig. 1. Pseudo-attractor reconstructed in m-phase space with m¼ 3 for a 2D electron PL time dependence I(t) (see the inset) measured at B¼ 7.78 T ðn ¼ 2Þ, T¼ 1.63–1.71 K.
It is well known that correlation dimension Dc can serve as the complexity measure of a strange attractor of dynamic systems. This parameter is fractional. For the determination of this value, it is necessary to know the above mentioned embedding dimension m (the smallest integer dimension of the space containing the whole attractor [3]; m4 2). One can determine the embedding dimension by the signal time-dependent measurements. This procedure is known as Grassberger–Procaccia algorithm [7]. The time series studied is given by system (1): XðtÞ ¼ IðtÞ,Iðt þ tÞ, Iðt þ 2tÞ, . . . ,Iðt þ ðm1ÞtÞ. Autocorrelation function of the signal
A.L. Parakhonsky et al. / Physica E 44 (2012) 1653–1656
Fig. 4. Autocorrelation function of the PL signal presented on the inset of Fig. 1.
1655
Fig. 5. The log–log scale dependence of correlation sum C(r) from embedding dimension m. The inset shows the dependence of correlation dimension Dc from m for the pseudo-attractor presented in Fig. 1.
I(t) is calculated to estimate the time shift value t:
GðtÞ ¼ /IðtÞ Iðt þ tÞS
ð2Þ
t is taken to be the time value whereby the autocorrelation function of a time sequence goes to zero or has the first minimum [8]. This parameter approximately equals to 165 s in our experiment (Fig. 4). Further, one needs to execute the following procedure. One should take some point Ii belonging to the attractor and calculate the number of points of the attractor distant from Ii less than or equal to the distance r. Then one should repeat this procedure for the next step point, etc. Thus, calculation of the correlation dimension Dc is reduced to that for the correlation sum [7]: CðrÞ ¼ lim
n 1 X
n-1 n2
Hðr9Ii Ij 9Þ
ð3Þ
i,j ¼ 1
where H(x) is the Heaviside step function for all pairs of i and j, n is the number of points Ii,j , and 9Ii Ij 9 is a distance between neighboring points in m-dimensional phase space. C(r) increases with r and has a maximum where r is comparable to size of the attractor ðCðrÞ-1Þ. For small r, the dependence holds: CðrÞpr Dc
ð4Þ
where Dc is the desired correlation dimension of the attractor. Having taken the logarithm, one receives a linear part of this dependence in the log–log scale. The asymptotic slope of which is equal to correlation dimension Dc: Dc ðmÞ ¼ lim r-0
lg CðrÞ lg r
ð5Þ
Such dependence, we are interested in, is shown in Fig. 5. Grassberger–Procaccia procedure is carried out several times for m¼2,3,y,mmax consecutively. The value mmax, where the slope does not change, corresponds to the true value of embedding dimension of the attractor. Thus, when the system goes to deterministic chaos, the correlation dimension for the time series converges to its true value (a situation presented in Fig. 1). At the same time, for random time series, points of the reconstructed pseudo-attractor form a structureless mixture in the phase space regardless of its dimension (Figs. 2 and 3). Consecutive calculation results of the correlation sum (3) for m¼2,3,y,7 are collected in the inset of Fig. 5. One can see that correlation dimension Dc reaches the saturation and does not vary from some value of mmax (namely, from m¼4). At that, the value of Dc approximately equals to 3.6.
5. Discussion and conclusions Thus, in the present work it is revealed that under the giant optical fluctuations of 2D electron system in the quantum Hall Effect regime ðn ¼ 2Þ, the strange attractor arises. Grassberger–Procaccia algorithm was explored to compute the correlation sum and fractal dimension of the pseudo-attractor. The calculations show that the correlation dimension approximately equals to 3.6. The embedding dimension was estimated to be 4 in our case, i.e., the number of independent variables of phase space is four. Thus, one can conclude that reconstructed strange attractor of the giant optical fluctuations is characterized by a system of four nonlinear differential equations. Hence, the process of giant optical fluctuations is not random but is governed by the limited number of control parameters. We can repeat this conclusion like in Ref. [6]. On the other hand, if the time series is random, the points of phase space form a structureless mixture in the phase space. Such behavior of the fluctuations has also been revealed by us at the filling factor n ¼ 2. However, a character of the time dependence I(t) qualitatively and quantitatively differs from that where the strange attractor arose. The fluctuations in this case proceed in a continuous-wave mode practically that corresponds to the deterministic chaos regime. While for the case of strange attractor, the time dependence of PL intensity includes discrete parts of the noise. For n a 2, a structureless mixture in the phase space also takes place, i.e., values of the time dependence studied are also random. Similarity of the phase portraits reconstructed in the given work and in Ref. [6] at m¼3 suggests the general regularities for the nonlinear small size (values m and Dc are not too large) dynamics for a wide range of systems. We previously noticed that the system of interactive two-dimensional electrons becomes highly uniform under the giant optical fluctuations conditions [1]. Such nonlinear small-size dynamics, which is defined by the coherence of elements of 2D electron system, can serve as the argument in favor of this statement. The described analysis points to a certain determinism in behavior of 2D electron system under the giant optical fluctuations conditions. However, such determinism takes place only at the beginning of instability in the system.
Acknowledgments We would like to express thanks to Professor G. Loukova for critical reading of the manuscript and many useful suggestions.
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A.L. Parakhonsky et al. / Physica E 44 (2012) 1653–1656
The work was supported by the Russian Foundation for Basic Research 10-02-00506. References [1] O.V. Volkov, I.V. Kukushkin, M.V. Lebedev, G.B. Lesovik, K. von Klitzing, K. Eberl, JETP Letters 71 (2000) 383. [2] A.L. Parakhonsky, M.V. Lebedev, I.V. Kukushkin, Y. Smet, K. von Klitzing, Physics of the Solid State 49 (5) (2007) 925.
[3] H.G. Schuster, W. Just, Deterministic Chaos: An Introduction, Physik Verlag, Weinheim, 1984 248 p. [4] A.L. Parakhonsky, M.V. Lebedev, A.A. Dremin, Physica E 43 (2011) 1449. [5] F. Takens, Detecting strange attractors in turbulence. in: Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1981, 898 p. [6] V.S. Zakharov, The Search of Determinism in Experimental Geosciences Data: The Analysis of Correlation Dimension for Time Sequences, Modern Geologic Processes (In Russian), 2002, 184 p. [7] P. Grassberger, I. Procaccia, Physical Review Letter 50 (5) (1983) 346. [8] J. Theiler, Lincoln Laboratory Journal 3 (1990) 63.
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
The strange attractor of giant optical fluctuations of 2D electrons in the quantum Hall effect regime A.L. Parakhonsky n, M.V. Lebedev, A.A. Dremin, I.V. Kukushkin Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Russia
a r t i c l e i n f o
abstract
Article history: Received 21 March 2012 Received in revised form 6 April 2012 Accepted 10 April 2012 Available online 7 May 2012
Strange attractor of the giant optical fluctuations of 2D system was reconstructed in phase space. Grassberger–Procaccia algorithm was explored to compute correlation sum of the attractor. The embedding and correlation dimensions are estimated to be 4 and 3.6, respectively. Thus, the revealed attractor is characterized by a system of 4 nonlinear differential equations. & 2012 Elsevier B.V. All rights reserved.
1. Introduction In our previous works on the giant optical fluctuations (GOF) of 2D electron system in the quantum Hall effect regime, we have noticed that the character of a spectrum of such fluctuations shows similarity with that considered in the theory of open dissipative dynamic systems [1,2]. Indeed, in our experiments, a 2D electronic gas is an open dissipative system being far from equilibrium. This system continuously gains energy due to the laser excitation and continuously consumes energy through the recombination of 2D electrons with photoexcited holes. With a certain value of the control parameter the fluctuation spectrum becomes continuous, without clearly defined features. This state of the system is a precursor for the appearance of incommensurate frequencies in the fluctuation spectrum and the onset of instability in the system (‘‘Fibonacci number’’ regime) [2]. Such behavior of a system corresponds to the regime that became known as deterministic chaos and is described by the Ruelle– Takens–Newhouse model [3]. This model states that under small perturbation of a dissipative dynamic system, a strange attractor can arise in it. It is such attracting set of unstable trajectories in the phase space that determines the number of interrelated nonlinear differential equations describing the system. Using the methods developed in this theory, it would be reasonable to define a characteristic image and the dimension of a possible strange attractor of the fluctuating 2D system. Therefore, it enables one to define the number of nonlinear differential equations describing the process under study. In the present work, we reveal the results allowing us to justify the occurrence
n
Corresponding author. E-mail address: [email protected] (A.L. Parakhonsky).
1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2012.04.014
of a strange attractor in our fluctuating 2D system. The revealed attractor is characterized by a system of four nonlinear differential equations.
2. Experiment In the work, we used the samples grown with a molecularbeam epitaxial technique on a GaAs substrate according to the ˚ following scheme: a 3000-A-thick GaAs buffer layer, an undoped ˚ super lattice with a total thickness GaAs/Al0.3Ga0.7As (30–100 A) ˚ a 250-A-wide ˚ of 13,000 A, GaAs quantum well, a 400-A˚ ˚ Al0.3Ga0.7As spacer, and a 650-A-thick Al0.3Ga0.7As: Si doped (at a level of 1018 cm 3) layer. The concentration and mobility of electrons were nS ¼4 1011 cm 2 and m ¼ 2 106 cm2 =ðV sÞ, respectively. These are the standard structures where the giant optical fluctuations are observed. The sample under study was placed inside a superconducting solenoid, immersed in a liquidhelium cryostat (at temperature T¼1.6–1.7 K). The solenoid produced a magnetic field up to 12 T. In order to analyze the giant optical fluctuations and their properties we used a special optical scheme, previously described in Ref. [4]. The time dependences of PL intensities measured for the maximum of radiation line, corresponding to the zero Landau level at magnetic field in the vicinity of filling factor n ¼ 2, were studied. A semiconductor charge-coupled detector (CCD) was used to record the PL signal. Sufficiently long (up to t ¼7200 s) sequences of the 2D electron PL time dependences I(t) were recorded with steps of 1 s. In order to analyze the fluctuation level, we estimated the average 2Delectron radiation intensity over the measurement time /IS, the variance of the intensity D ¼ /DI2 S, and the ratio of the variance to the mean value of the intensity D=/IS. The GOF
1654
A.L. Parakhonsky et al. / Physica E 44 (2012) 1653–1656
regime ðD=/IS b 1Þ at n ¼ 2 corresponded to magnetic field B ¼7.78 T.
3. Pseudo-attractor of 2D fluctuating system The phase portrait of a possible strange attractor of dissipative dynamic systems can be received using the methods described in Refs. [2,3,5]. Takens has shown [5] that a time series of measurements of a single observable X(t) of the dynamic system trajectory can be used to reconstruct qualitative features of the strange attractor in phase space. In our case, such a component is the time dependence of 2D electron PL intensity I(t). The measured time series are used with some time delay t. The sequence of m-phase space vectors of one component X(t) is given by 8 ð1Þ X ðtÞ ¼ IðtÞ > > > > ð2Þ > > < X ðtÞ ¼ Iðt þ tÞ X ð3Þ ðtÞ ¼ Iðt þ2tÞ ð1Þ > > >^ > > > : ðmÞ X ðtÞ ¼ Iðt þ ðm1ÞtÞ: Here, m is the parameter designated as the embedding dimension (see below). We used system (1) for the reconstruction of a possible strange attractor of the fluctuating 2D system at various values of magnetic field. We revealed that at the minimal integer value m¼3, there is such mode of noise in the vicinity of n ¼ 2 where the value of vectors (1) are eventually grouped in the vicinity of each of three axes of phase space (Fig. 1). The type of experimental time dependence I(t), where the strange attractor was reconstructed, is shown on the inset of Fig. 1. The similar phase portrait at m ¼3 was observed in Ref. [6]. In this work, methods of the theory of dynamic systems were applied to the analysis of geology-geophysical experimental data. This similarity suggests nonlinear dynamic mechanisms being common to a wide range of dynamic systems. Fig. 2 demonstrates the result of a similar reconstruction under the close conditions; however, it has no such a characteristic phase portrait as shown in Fig. 1. A set of trajectories in phase space forms a sphere in this case. Such a phase portrait can be usually obtained in the absence of a strange attractor of the system [3]. It is necessary to notice that a noise character differs for these two cases. In the first instance, fluctuations have discrete regions where the noise amplitude is insignificant. While in the latter the giant optical fluctuations occur in a continuous mode, practically,
Fig. 2. Phase space reconstructed for the dependence I(t) measured under the same conditions as that presented in Fig. 1 measured after a time. The characteristic phase portrait is absent.
Fig. 3. Structureless cloud of the trajectories in the lag space I(t) obtained under different conditions: B¼ 8 T, T¼ 1.63–1.71 K.
throughout all time series I(t). Besides, the noise amplitude in the first instance is 1.5–2 times larger than in the former. For the time sequences measured at the other magnetic fields, a strange attractor was not observed as well (Fig. 3). If one considers the grouping of trajectories along the axes of three-dimensional phase space, then one can conclude that for such a phase portrait, the necessary requirement is the presence of discrete parts in the noise time dependence. In the other words, the intensity of the studied signal must hold small amplitude long enough (magnitude as lag time t).
4. Correlation dimension of GOF strange attractor
Fig. 1. Pseudo-attractor reconstructed in m-phase space with m¼ 3 for a 2D electron PL time dependence I(t) (see the inset) measured at B¼ 7.78 T ðn ¼ 2Þ, T¼ 1.63–1.71 K.
It is well known that correlation dimension Dc can serve as the complexity measure of a strange attractor of dynamic systems. This parameter is fractional. For the determination of this value, it is necessary to know the above mentioned embedding dimension m (the smallest integer dimension of the space containing the whole attractor [3]; m4 2). One can determine the embedding dimension by the signal time-dependent measurements. This procedure is known as Grassberger–Procaccia algorithm [7]. The time series studied is given by system (1): XðtÞ ¼ IðtÞ,Iðt þ tÞ, Iðt þ 2tÞ, . . . ,Iðt þ ðm1ÞtÞ. Autocorrelation function of the signal
A.L. Parakhonsky et al. / Physica E 44 (2012) 1653–1656
Fig. 4. Autocorrelation function of the PL signal presented on the inset of Fig. 1.
1655
Fig. 5. The log–log scale dependence of correlation sum C(r) from embedding dimension m. The inset shows the dependence of correlation dimension Dc from m for the pseudo-attractor presented in Fig. 1.
I(t) is calculated to estimate the time shift value t:
GðtÞ ¼ /IðtÞ Iðt þ tÞS
ð2Þ
t is taken to be the time value whereby the autocorrelation function of a time sequence goes to zero or has the first minimum [8]. This parameter approximately equals to 165 s in our experiment (Fig. 4). Further, one needs to execute the following procedure. One should take some point Ii belonging to the attractor and calculate the number of points of the attractor distant from Ii less than or equal to the distance r. Then one should repeat this procedure for the next step point, etc. Thus, calculation of the correlation dimension Dc is reduced to that for the correlation sum [7]: CðrÞ ¼ lim
n 1 X
n-1 n2
Hðr9Ii Ij 9Þ
ð3Þ
i,j ¼ 1
where H(x) is the Heaviside step function for all pairs of i and j, n is the number of points Ii,j , and 9Ii Ij 9 is a distance between neighboring points in m-dimensional phase space. C(r) increases with r and has a maximum where r is comparable to size of the attractor ðCðrÞ-1Þ. For small r, the dependence holds: CðrÞpr Dc
ð4Þ
where Dc is the desired correlation dimension of the attractor. Having taken the logarithm, one receives a linear part of this dependence in the log–log scale. The asymptotic slope of which is equal to correlation dimension Dc: Dc ðmÞ ¼ lim r-0
lg CðrÞ lg r
ð5Þ
Such dependence, we are interested in, is shown in Fig. 5. Grassberger–Procaccia procedure is carried out several times for m¼2,3,y,mmax consecutively. The value mmax, where the slope does not change, corresponds to the true value of embedding dimension of the attractor. Thus, when the system goes to deterministic chaos, the correlation dimension for the time series converges to its true value (a situation presented in Fig. 1). At the same time, for random time series, points of the reconstructed pseudo-attractor form a structureless mixture in the phase space regardless of its dimension (Figs. 2 and 3). Consecutive calculation results of the correlation sum (3) for m¼2,3,y,7 are collected in the inset of Fig. 5. One can see that correlation dimension Dc reaches the saturation and does not vary from some value of mmax (namely, from m¼4). At that, the value of Dc approximately equals to 3.6.
5. Discussion and conclusions Thus, in the present work it is revealed that under the giant optical fluctuations of 2D electron system in the quantum Hall Effect regime ðn ¼ 2Þ, the strange attractor arises. Grassberger–Procaccia algorithm was explored to compute the correlation sum and fractal dimension of the pseudo-attractor. The calculations show that the correlation dimension approximately equals to 3.6. The embedding dimension was estimated to be 4 in our case, i.e., the number of independent variables of phase space is four. Thus, one can conclude that reconstructed strange attractor of the giant optical fluctuations is characterized by a system of four nonlinear differential equations. Hence, the process of giant optical fluctuations is not random but is governed by the limited number of control parameters. We can repeat this conclusion like in Ref. [6]. On the other hand, if the time series is random, the points of phase space form a structureless mixture in the phase space. Such behavior of the fluctuations has also been revealed by us at the filling factor n ¼ 2. However, a character of the time dependence I(t) qualitatively and quantitatively differs from that where the strange attractor arose. The fluctuations in this case proceed in a continuous-wave mode practically that corresponds to the deterministic chaos regime. While for the case of strange attractor, the time dependence of PL intensity includes discrete parts of the noise. For n a 2, a structureless mixture in the phase space also takes place, i.e., values of the time dependence studied are also random. Similarity of the phase portraits reconstructed in the given work and in Ref. [6] at m¼3 suggests the general regularities for the nonlinear small size (values m and Dc are not too large) dynamics for a wide range of systems. We previously noticed that the system of interactive two-dimensional electrons becomes highly uniform under the giant optical fluctuations conditions [1]. Such nonlinear small-size dynamics, which is defined by the coherence of elements of 2D electron system, can serve as the argument in favor of this statement. The described analysis points to a certain determinism in behavior of 2D electron system under the giant optical fluctuations conditions. However, such determinism takes place only at the beginning of instability in the system.
Acknowledgments We would like to express thanks to Professor G. Loukova for critical reading of the manuscript and many useful suggestions.
1656
A.L. Parakhonsky et al. / Physica E 44 (2012) 1653–1656
The work was supported by the Russian Foundation for Basic Research 10-02-00506. References [1] O.V. Volkov, I.V. Kukushkin, M.V. Lebedev, G.B. Lesovik, K. von Klitzing, K. Eberl, JETP Letters 71 (2000) 383. [2] A.L. Parakhonsky, M.V. Lebedev, I.V. Kukushkin, Y. Smet, K. von Klitzing, Physics of the Solid State 49 (5) (2007) 925.
[3] H.G. Schuster, W. Just, Deterministic Chaos: An Introduction, Physik Verlag, Weinheim, 1984 248 p. [4] A.L. Parakhonsky, M.V. Lebedev, A.A. Dremin, Physica E 43 (2011) 1449. [5] F. Takens, Detecting strange attractors in turbulence. in: Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1981, 898 p. [6] V.S. Zakharov, The Search of Determinism in Experimental Geosciences Data: The Analysis of Correlation Dimension for Time Sequences, Modern Geologic Processes (In Russian), 2002, 184 p. [7] P. Grassberger, I. Procaccia, Physical Review Letter 50 (5) (1983) 346. [8] J. Theiler, Lincoln Laboratory Journal 3 (1990) 63.