Ring intermittency near the boundary of the synchronous time scales of chaotic oscillators
In this paper we study both experimentally and numerically the intermittent behavior taking place near the boundary of the synchronous time scales of chaotic oscillators being in the regime of time scale synchronization. We have shown that the observed type of the intermittent behavior should be classified as the ring intermittency.
Chaotic synchronization is one of the most important directions of nonlinear dynamics which attracts great attention due to the large fundamental significance Pikovsky et al. (2001); Boccaletti et al. (2002a) and the wide range of practical applications, e.g., in the microwave systems Rosa et al. (2000). It is also helpful for information transmission Argyris et al. (2005), for diagnostics of dynamics of some biological systems Glass (2001); Hramov et al. (2006a), etc. There are several main types of chaotic synchronization, such as phase synchronization (PS) Pikovsky et al. (2001), lag synchronization (LS) Rosenblum et al. (1997), generalized synchronization (GS) Paoli et al. (1989); Rulkov et al. (1995), complete synchronization (CS) Pecora and Carroll (1990), time scale synchronization (TSS) Hramov and Koronovskii (2004); Hramov and Koronovskii (2005a). Among these types of chaotic synchronization the time scale synchronization regime plays an important role, since all other types of chaotic synchronous behavior are known to be the particular cases of TSS Hramov and Koronovskii (2004); Hramov and Koronovskii (2005a); Hramov et al. (2005b).
Along with the intensive study of the synchronous behavior of the coupled chaotic oscillators the careful attention of scientists is paid to the dynamics observed in the vicinity of the onset of the synchronization regime. Considering the pre-transitional behavior allows to find the mechanisms responsible for the formation/destruction of the synchronous regime Pazó et al. (2002); Hramov et al. (2007b) and to reveal the essential features of this type of dynamics Rosenblum et al. (1997); Boccaletti and Valladares (2000); Hramov and Koronovskii (2005b).
It is well known that near the boundaries of chaotic synchronization regimes the intermittent behavior can be observed, when the de-synchronization mechanism involves persistent intermittent time intervals during which the synchronized oscillations are interrupted by the non-synchronous behavior. These pre-transitional intermittencies have been described in details for the onset of different types of synchronous dynamics, such as lag synchronization Rosenblum et al. (1997); Boccaletti and Valladares (2000), generalized synchronization Hramov and Koronovskii (2005b), phase synchronization Pikovsky et al. (1997); Boccaletti et al. (2002b); Hramov et al. (2006b) (except for time scale synchronization), and their main statistical properties have been shown to be common to other important physical processes. Moreover, the intermittency is not limited to the physical objects, it is observed widely in the nature, e.g., in the living systems Perez Velazquez and et al. (1999).
At the same time, as it has been mentioned above, the pre-transitional system behavior has not been studied hitherto for the time scale synchronization regime.
In this paper we, for the first time, report on the intermittent behavior observed near the boundary of the range of the synchronous time scales of chaotic oscillators which are in the time scale synchronization regime Hramov and Koronovskii (2004); Hramov and Koronovskii (2005a); Hramov et al. (2005b). Having studied this type of behavior both experimentally and numerically we came to the conclusion that ring intermittency Hramov et al. (2006b) takes place in the case under study.
The time scale synchronization regime means the presence of the synchronous dynamics in certain range of the time scales , introduced with the help of the continuous wavelet transform Torresani (1995)
with Morlet mother wavelet function , . Each of time scales can be characterized by the phase , where is the complex wavelet surface given by (1). For the two coupled chaotic systems time scale synchronization takes place, if there is the range of the synchronous time scales where the phase locking condition
is satisfied and the part of the wavelet spectrum energy fallen on this range is not equal to zero
The intermittent behavior near the boundary of the range of the synchronous time scales of chaotic oscillators being in the time scale synchronization regime has been studied experimentally (Fig. 1). In the experiment we have used a simple electronic oscillator where all parameters may be controlled precisely. As a basis element of the scheme we have used the generator with the linear feedback and nonlinear converter (NC) Rulkov (1996). The coupling strength between generators has been governed by resistor (Fig. 1). The main frequencies of the autonomous chaotic oscillations have been kHz and kHz for the drive and response systems, respectively. The behavior of the coupled oscillators has been analyzed by means of the Agilent E4402B spectrum analyzer and L-Card L-783 analog–digital converter (ADC) PCI-card with 12-bit resolution.
For the coupling resistor value Ohm the oscillators under study demonstrate the time scale synchronization regime (which may be considered for the given values of the control parameters also as phase synchronization), with the range of the synchronous time scales being , s, s. In other words, for the synchronous dynamics is observed, since the phase locking condition (2) is satisfied. For the time scales being outside this area the dynamics of the phase difference features time intervals of the synchronized motion (laminar phases) persistently and intermittently interrupted by sudden phase slips (turbulent phases) during which the value of jumps up by . In other words, for time scales lying both below and above the intermittent behavior is observed, with the observation time scale being a criticality parameter. Note, the time scale parameter is not a parameter of the original dynamical system (but a parameter of observation), although usually intermittency is observed by changing a control parameter of the dynamical system under study. For and the mean length of the laminar phase goes to infinity whereas the turbulent phases become very rare events. Alternatively, away from the boundaries of the synchronous time scales, for and the phase slips take place with great regularity that means the presence of the asynchronous dynamics on the observation time scale. The schematic representation of the relationship between the observed regimes and time scales is shown in Fig. 2. Note also, that since the dynamics of the original system does not depend on the observation time scale , from the point of view of the phase synchronization theory, the phase synchronization regime takes place for Ohm.
Having studied the characteristics of intermittency such as the laminar phase distribution, the dependence of the mean length of the laminar phases and the probability of the turbulent phase detection we have come to conclusion that the observed type of the intermittent behavior taking place near the boundary of the range of the synchronous time scales of chaotic oscillators being in the time scale synchronization regime should be classified as a ring intermittency Hramov et al. (2006b). Indeed, in Hramov et al. (2006b) it has been shown that the ring intermittency is characterized by the exponential distribution of the laminar phase lengths
whereas the dependence of the mean length of the laminar phases on the criticality parameter obeys the law
where is a mean length of the laminar phase for the time scale bounding the region of ring intermittency, is the probability of detecting the turbulent phase on the time interval of the observation with the length on the time scale . Typically, the dependence of the probability on the criticality parameter is close to linear, and, therefore, for
The time scale corresponds to the lower boundary of the linear form of the dependence and may be determined from the condition . The theoretical relation (5) is applicable only in the range .
To separate in the experimental time series the laminar phases from the turbulent ones we have used the approach described in Zhuravlev et al. (2010). The distribution of the laminar phase lengths obtained experimentally for three different time scales s (line 1, symbols “”), s (line 2, symbols “”) and s (line 3, symbols “”) are shown in Fig. 3,a. One can see that the experimentally obtained distributions of the laminar phase lengths agree very well with the theoretical predictions for ring intermittency (3) given in Hramov et al. (2006b). The dependence of the mean length of the laminar phase on the time scale (playing a role of the criticality parameter) is also in the strict accordance with the theoretical law (4), with the probability of detecting the turbulent phase on the observation time interval with the length ms obeying the linear law (Fig. 3, b and c, respectively).
So, we came to the conclusion that the intermittent behavior observed near the boundary of the range of the synchronous time scales of the considered chaotic oscillators being in the regime of time scale synchronization should be classified as ring intermittency. The fact that the mechanism resulting in the phase slips on the certain time scales in the system under study is exactly the same as in the case of ring intermittency is the irrefutable evidence of the correctness of the decision made above. Indeed, the origin of the intermittent behavior for the ring intermittency regime is connected with the events when the phase trajectory on the plane rotating according to the drive system state starts enveloping the origin (see Hramov et al. (2006b) for detail), and the phase slips are observed all the times that the phase trajectory envelops the origin of that plane. If we consider and as the variables determining the system state, then on the rotating plane , where, in accordance with Hramov et al. (2006b), , , the ring intermittency mechanism is revealed evidently. Indeed, in the region of the intermittent behavior the trajectory envelops the zero of the coordinate system (see Fig. 4), with the boundary of the synchronous time scales corresponding to the situation when the trajectory of the second system starts enveloping the origin of the rotating plane. When the observation time scale is shifted from to the trajectory on the rotating plane envelops the origin more and more often, and for this event is observed with great regularity (Fig. 4, c) in accordance with the ring intermittency theory Hramov et al. (2006b).
To prove the generality of our findings we have also studied numerically the intermittent behavior near the boundary of the range of the synchronous time scales for two coupled chaotic Rössler oscillators
where  are the cartesian coordinates of the drive (the response) oscillator, dots stand for temporal derivatives, and is a parameter ruling the coupling strength. The other control parameters of Eq. (6) have been set to , , . The –parameter has been selected to be ; the analogous parameter for the drive system has been fixed to . For such a choice of the parameter values the boundary of the time scale synchronization regime occurs around , with the boundaries of the range of the synchronous time scales being , .
The distributions of the laminar phase lengths detected on the asynchronous scales for two coupled Rössler systems (6) being in the time scale synchronization regime are shown in Fig. 5, a. Again, as well as for the experimental data (compare with Fig. 3) the excellent agreement between the calculated distributions and theoretical exponential law (3) is observed. As far as the dependence of the mean length of the laminar phases on the criticality parameter (i.e., the observation time scale ) is concerned (see Fig. 5, b), the obtained curve also corresponds to the theoretical relation reported in Hramov et al. (2006b) for the ring intermittency regime. Moreover, the probability of detecting the turbulent phase on the observation time interval with the length is found to be linear (Fig. 5, c) that also is an evidence of the ring intermittency presence. Thus, the ring intermittency regime is detected on the asynchronous time scales for two coupled Rössler systems being in the regime of time scale synchronization as well as in the case of the experimental study of two coupled chaotic generators. Therefore, we can make a decision that the ring intermittency is a typical feature of the behavior observed on the asynchronous time scales for synchronized coupled chaotic systems. At the same time, if the coupling strength between oscillators is too weak for the interacting systems to be synchronized, the ring intermittency on the asynchronous time scales is not realized.
In conclusion, we have reported for the first time on the ring intermittency observed near the boundary of the range of the synchronous time scales of chaotic oscillators being in the time scale synchronization regime. It may be observed in a certain range of the observation time scales lying outside the area of time scales where the synchronous behavior is detected. The experimentally and numerically obtained data are in the perfect agreement with the theoretical equations reported in Hramov et al. (2006b). We expect that the very same phenomenon can be observed in many other relevant circumstances, as e.g. laser systems Boccaletti et al. (2002b), or in the case of the interaction between the main rhythmic processes in the human cardiovascular system Hramov et al. (2006a), etc.
We thank the Referees of our manuscript for valuable comments and remarks that allowed us to improve our paper. This work has been supported by Federal special purpose programme “Scientific and educational personnel of innovation Russia (2009–2013)” and the President Program (NSh-3407.2010.2).
- The analogous relation may be also deduced easily for , in this case
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