Synchronization transitions in coupled timedelay electronic circuits with a threshold nonlinearity
Abstract
Experimental observations of typical kinds of synchronization transitions are reported in unidirectionally coupled timedelay electronic circuits with a threshold nonlinearity and two time delays, namely feedback delay and coupling delay . We have observed transitions from anticipatory to lag via complete synchronization and their inverse counterparts with excitatory and inhibitory couplings, respectively, as a function of the coupling delay . The anticipating and lag times depend on the difference between the feedback and the coupling delays. A single stability condition for all the different types of synchronization is found to be valid as the stability condition is independent of both the delays. Further, the existence of different kinds of synchronizations observed experimentally is corroborated by numerical simulations, and from the changes in the Lyapunov exponents of the coupled timedelay systems.
pacs:
05.45.Xt,05.45.PqI Introduction
Timedelay is a veritable blackbox which can give rise to several interesting and novel phenomena such as multistable states Kim et al. (1977), amplitude death Reddy et al. (2000), chimera states Sethia et al. (2008), phase flip bifurcation Prasad et al. (2006), NeimarkSacker type bifurcations Atay et al. (2004), etc., which cannot be observed in the absence of delay in the underlying systems. Further, it has also been shown that delay coupling in complex networks enhances the synchronizability of networks and interestingly it leads to the emergence of a wide range of new collective behavior Atay et al. (2004); Masoller et al. (2005). On the other hand, it has also been shown that connection delays can actually be conducive to synchronization so that it is possible for delayed systems to synchronize, whereas the undelayed systems do not Atay et al. (2004). Enhancement of neural synchrony, that is, the existence of a stable synchronized state even for a very low coupling strength for a significant timedelay in the coupling has also been demonstrated Dhamala et al. (2004). Timedelay feedback has been used to generate highdimensional, highcapacity waveforms at high bandwidths to sucessfully transfer digital information at gigabit rates by chaotically fluctuating laser light travelling over 120 kilometers of a commercial fibreoptic link around Athens, Greece Argyris et al. (2005). Timedelay feedback control has also been used to control pattern formation in neuroscience to prevent the pathological activity in cortical tissues Dahlem et al. (2008); Schneider et al. (2009).
Synchronization in dynamical systems with timedelay feedback and in intrinsic timedelay systems with/without timedelay coupling has been receiving central importance during the past decade both theoretically and experimentally Atay et al. (2004); Masoller et al. (2005); Dhamala et al. (2004); Argyris et al. (2005); Dahlem et al. (2008); Schneider et al. (2009); Sivaprakasam et al. (2001); Wedekind and Parlitz (2002); Peil et al. (2002); Liu et al. (2002); Uchida et al. (2003, 2003); Boccaletti et al. (2001, 2006, 2001); Shahverdiev et at. (2001, 2002); Voss (2002); Sano et al. (2007); Kim et al. (2006); Wagemakers et al. (2008); Senthilkumar et al. (2006); Senthilkumar and Lakshmanan (2005); Senthilkumar et al (2009). However, experimental investigations/confirmations of theoretical results of synchronization transitions in coupled timedelay systems remain lagging in the available literature. Nevertheless, experimental investigations on differerent kinds of synchronization transitions in semiconductor laser systems with a delay feedback have been carried out recently Atay et al. (2004); Masoller et al. (2005); Dhamala et al. (2004); Argyris et al. (2005); Dahlem et al. (2008); Schneider et al. (2009); Sivaprakasam et al. (2001); Wedekind and Parlitz (2002); Peil et al. (2002); Liu et al. (2002); Uchida et al. (2003, 2003); Boccaletti et al. (2001, 2006, 2001); Shahverdiev et at. (2001, 2002). However, experimental investigations in intrinsic timedelay systems, whose dynamics cannot be realized in the absence of timedelay such as the paradigmatic MackeyGlass or Ikeda systems, using electronic circuits remain poorly explored and very few experimental results have been reported so far Voss (2002); Sano et al. (2007); Kim et al. (2006); Wagemakers et al. (2008).
In particular, real time anticipatory synchronization of chaotic states using timedelayed electronic circuits with singlehumped smooth nonlinearity was demonstrated by Voss Voss (2002). Dual synchronization of chaos in two pairs of unidirectionally coupled MackeyGlass electronic circuits with timedelayed feedback was demonstrated in Sano et al. (2007). These authors have also investigated the regions for achieving dual synchronization of chaos when the delay time is mismatched between the drive and response circuits. The effect of frequency bandwidth limitations in communication channels on the synchronization of two unidirectionally coupled MackeyGlass analog circuits was demonstrated in Kim et al. (2006). Recently, experimental demonstration of simultaneous bidirectional communication between two chaotic systems by means of isochronal synchronization was carried out using MackeyGlass electronic circuits with timedelay feedback Wagemakers et al. (2008).
Further, experimental observation of both anticipated and retarded synchronization has been demonstrated using unidirectionally coupled semiconductor lasers with delayed optoelectronic feedback Shahverdiev et at. (2002). It has been shown that depending on the difference between the transmission time and the feedback delay time the lasers fall into either anticipated or retarded synchronization regimes, where the driven receiver laser leads or lags behind the driving transmitter laser, confirming the theoretical works of Voss and Masoller Voss (2000); Masoller (2001, 2001). Recently, we have demonstrated theoretically the transition from anticipatory to lag via complete synchronization as a function of the coupling delay with suitable stability condition in a system of unidirectionally coupled timedelay systems Senthilkumar and Lakshmanan (2005). Further, it was also shown that anticipatory/lag synchronizations can be characterized using appropriate similarity functions and the transitions from a desynchronized state to an approximate anticipatory/lag synchronized state is characterized by a transition from onoff intermittency to periodicity in the laminar phase distribution settling the skepticism on characterizing anticipatory/lag synchronization using the similarity function as discussed by Zhan etal Zhan et al. (2002).
In the present manuscript, we will demonstrate experimentally all the aforesaid synchronization transitions, along with their inverse counterparts with inhibitory coupling, in a unidirectionally coupled timedelay electronic circuit with a threshold nonlinearity supported by an appropriate theoretical analysis.
Specifically, in this manuscript we demonstrate the transition from anticipatory to complete and then from complete to lag synchronizations as a function of the coupling delay, for a fixed set of other system parameters, in a unidirectionally coupled piecewise linear (designed using a threshold controller) timedelay electronic circuit. Further we will also show the existence of their inverse counterparts, that is the transition from inverse anticipatory to inverse lag synchronizations via inverse complete synchronization, with inhibitory coupling. The importance of inhibitory coupling and its intrinsic role in neural synchrony are discussed in Masoller (2001); Zhan et al. (2010); Senthilkumar et al (2009). Furthermore, we will also show that neither inverse complete synchronizations can be realized with an excitatory coupling nor direct/conventional synchronizations can be realized with an inhibitory coupling as a result of the nature of the nonlinear function and the parametric relation obtained from the stability analysis using the KrasvoskiiLyapunov stability theory. Numerical simulations are presented in confirmation with the experimental results and the transitions in the spectrum of Lyapunov exponents of the coupled timedelay systems also confirm the observed synchronization transitions.
The plan of the paper is as follows. In Sec. II, we present the details of the delay dynamical system under consideration and the experimental implementation of the system using an appropriate analog electronic circuit. Unidirectionally coupled timedelay system and its circuit details are discussed in Sec. III. In Sec. IV, we analyze the different synchronization manifolds and identify the conditions for the stability of the synchronized states of unidirectionally coupled timedelay systems. In Sec. V, we demonstrate experimentally the existence of anticipatory, complete, and lag synchronizations with excitatory coupling, and their inverse counterparts with inhibitory coupling are discussed in Sec. VI, along with their numerical confirmation. Finally in Sec. VII, we summarize our results.
Ii The Scalar Delayed Chaotic System With Threshold Nonlinearity
We consider the following firstorder time delay differential equation (DDE) describing the delay feedback oscillator,
(1) 
where and are positive parameters, is a dynamical variable, is a nonlinear activation function and is the time delay. The function is taken to be a symmetric piecewise linear function defined by Srinivasan et al. (2007)
(2a)  
Here  
(2b) 
where is a controllable threshold value, and and are positive parameters. In our analysis, we chose , , , and . It may be noted that for , the function has the negative slope and it lies in all the four quadrants of the plane (Fig. 1(a)). The figure reveals the piecewise linear nature of the function. Experimental implementation (see below) of the function is shown in Fig.1(b) in the form of voltage characteristic of the nonlinear device unit of Figs. 2 and 3.
This function employs a threshold controller for flexibility. It efficiently implements a piecewise linear function. The control of this piecewise linear function facilitates controlling the shape of the attractors. Even for a small delay value this circuit system exhibits hyperchaos and can produce multiscroll chaotic attractors by just introducing more number of threshold values, for example a square wave. In particular, this method is effective and simple to implement since we only need to monitor a single state variable and reset it if it exceeds the threshold and so has potential engineering applications for various chaosbased information systems.
ii.1 Experimental setup
The system described by Eq. (1) with the nonlinear function is constructed using analog electronic devices. The circuit (Fig. 2) has a ring structure and comprises of a diode based nonlinear device unit (Fig. 3) with amplifying stages (), a time delay unit (Fig. 4) with a buffer () and an amplifying stage (). The dynamics of the circuit in Fig. 2 is represented by a DDE of the form
(3) 
where is the voltage across the capacitor , is the voltage across the delay unit (DELAY), is the delay time and is the static characteristic of the .
In order to analyze the above circuit, we transform it onto the dimensionless oscillator (1) on the basis of the following relations by defining the dimensionless variables and dimensionless parameters as
(4) 
A nonzero is chosen such that . In addition, the other parameters and variables are described by the relations , , , . These relations reveal that the circuit equation (3) is identical to Eq. (1) with . Without loss of generality, is treated as itself in our further analysis.
The experimental circuit parameters are : , , , , (trimmerpot), , (trimmerpot), , , (trimmerpot), (trimmerpot), , , , . From (5), we can see that ms, ms, so that the timedelay for the chosen values of the circuit parameters. The delay time can be simply varied by using the variable resistance . In our circuit, are employed as operational amplifiers. The constant voltage sources , and , and the voltage supply for all active devices are fixed at Volts. The threshold value of the three segments involved in Eq. (2) can be altered by adjusting the values of voltages and .
For the above choice of the circuit parameters, the values of the dimensionless parameters turns out to be , , and the delay time .
ii.2 Results
To start with, Eq. (1) has been numerically integrated with the chosen nonlinear function for the parameter values , , , , , and , with the initial condition in the range . A oneband chaotic attractor is shown in Fig. 5a(i) for , while for a doubleband hyperchaotic attractor is obtained (Fig. 5a(ii)). The corresponding experimental results are shown in Figs. 5b(i) and 5b(ii) for the values of the parameter (in this case ) and (now ), respectively. The experimental results are in good agreement with the numerical ones and also in their corresponding parameter values.
The system described by Eqs. (1) and (2) exhibit multiple positive Lyapunov exponents for large values of the delay time, a typical feature of timedelay systems. The seven maximal Lyapunov exponents for the above parameter values as a function of the timedelay in the range are shown in Fig. 6, which are evaluated using the procedure of Farmer (1982). Now it is evident from the maximal Lyapunov exponents that the single band chaotic attractors shown in Figs. 5a(i) and b(i) for the value of delay time and the resistance , respectively, has one positive Lyapunov exponent, while the double band chaotic attractor shown in Figs. 5a(ii) and b(ii) for the value of the delay time and the resistance , respectively, has two positive Lyapunov exponents corroborating its hyperchaotic nature. We will demonstrate in the following sections the existence of different kinds of synchronization transitions in the hyperchaotic regime in coupled systems.
Iii Coupled time delay systems with threshold nonlinearity
Now let us consider the following set of unidirectionally coupled firstorder delay differential equations,
(6a)  
(6b) 
where are positive constants, contributes to the parameter mismatch resulting in coupled nonidentical systems, is the coupling strength, is the feedback delay and is the coupling delay. The nonlinear function is of the same form as in Eq. (2).
Now to analog simulate the coupled timedelay systems (Eqs. (6b)) and to demonstrate experimentally the existence of different types of synchronizations, a unidirectionally coupled timedelay electronic circuit is constructed as shown in the block diagram of Fig. 7. One of the electronic oscillator circuits is used as the drive system, while the other structurally identical circuit is used as the response system with some parameter mismatches. The drive voltage after the delay line in the drive system is fed back to the nonlinear part of the drive system and a fixed filter with time delay to generate chaotic/hyperchaotic oscillations. Similarly, the response circuit with a nonlinear part , a delay line and a fixed filter is capable of generating chaotic/hyperchaotic oscillations. The signal after the nonlinear function of drive is used as the transmission signal, which is unidirectionally transmitted through the lowpass filter , delay line and nonlinear part to the response circuit. All the parameters need to be matched between the drive and the response circuits, whereas the parameters of the nonlinear activation functions of the drive, the response and the coupling are to be fixed according to the parametric relation obtained from the stability analysis (given below in Sec. IV).
The state equations of the coupled electronic circuit (Fig. 7) can be written as
(7a)  
where the variables and correspond to the output variables of each circuit. By defining the new normalized variables as , , , , one can check that the circuit equation (7) is identical to Eq. (6b) with , , , and . and
Before demonstrating the experimental results and the corresponding numerical confirmation of various synchronizations in the coupled timedelay systems (6b) and (7), we deduce a sufficient stability condition, using the KrasovskiiLyapunov theory, valid for different synchronization manifolds. After choosing the appropriate parameter values satisfying the obtained stability condition, we will demonstrate the existence of anticipatory, complete and lag synchronizations as a function of the coupling delay for excitatory coupling and their inverse counterparts for inhibitory coupling in the same system both experimentally and numerically. It is to be noted that neither inverse synchronizations can be realized with excitatory coupling nor direct/conventional synchronizations can be realized with inhibitory coupling as a result of the nature of the nonlinear function and the parametric relation between and obtained from the stability analysis.
Iv Synchronization manifold and its stability condition
Consider the direct synchronization manifold of the coupled timedelay equation (6b) with excitatory coupling (correspondingly the inverse complete synchronization manifold becomes with the inhibitory coupling in Eq. (6bb)), where , which corresponds to the following distinct cases:

Anticipatory synchronization (AS) occurs when with , where the state of the response system anticipates the state of the drive system synchronously with the anticipating time . In contrast, in the case of the inverse anticipatory synchronization (IAS), the state of the response system anticipates exactly the inverse state of the drive system, that is, .

Complete synchronization (CS) results when with , where the state of the response system evolves in a synchronized manner with the state of the drive system, while in the case of inverse complete synchronization (ICS), the state of the response system evolves exactly identical but inverse to the state of the drive system, that is, .

Lag synchronization (LS) occurs when with , where the state of the response system lags the state of the drive system in synchronization with the lag time . However, in the case of inverse lag synchronization (ILS), the state of the response system lags exactly inverse to the state of the drive system, that is, .
Now, the time evolution of the difference system with the state variable can be written for small values of by using the evolution Eqs. (6b) as
(9)  
The above inhomogeneous equation can be rewritten as a homogeneous equation of the form
(10) 
for the specific choice of the parameters
(11) 
so that the stability condition can be deduced analytically. The synchronization manifold corresponding to Eq. (10) is locally attracting if the origin of the above error equation is stable. Following the KrasovskiiLyapunov functional approach Krasovskii (1963); Senthilkumar and Lakshmanan (2005), we define a positive definite Lyapunov functional of the form
(12) 
where is an arbitrary positive parameter, .
The above Lyapunov function, , approaches zero as . Hence, the required solution to the error equation, Eq. (10), is stable only when the derivative of the Lyapunov functional along the trajectory of Eq. (10) is negative. This requirement results in the condition for stability as
(13) 
Again as a function of for a given has an absolute minimum at with . Since , from the inequality (13), it turns out that a sufficient condition for asymptotic stability is
(14) 
Now from the form of the piecewise linear function given by Eq. (2), we have,
(15) 
Consequently the stability condition (14) becomes along with the parametric relation . Since the deduced stability condition is independent of the delay times and , the same general stability condition is valid for anticipatory, complete and lag synchronizations with excitatory coupling and to their inverse counterparts with inhibitory coupling.
We remark here that if one substitutes in Eq. (6bb), then the excitatory coupling becomes an inhibitory coupling and the inhibitory coupling becomes an excitatory coupling due to the nature of the nonlinear function, . Furthermore, one obtains the parametric relation along with the same stability condition (14) for both the cases of excitatory coupling with an inverse synchronization manifold and inhibitory coupling with a direct synchronization manifold. Therefore, to obtain both direct and inverse synchronizations either from excitatory or from inhibitory coupling both the parametric relations, that is and given by (11), have to be satisfied for fixed values of the nonlinear parameters or and for positive values of the coupling strength . The only way to satisfy both the parametric relations and the stability condition, , is to choose negative values for the coupling strength and this changes the nature of the coupling. Hence, one cannot obtain inverse (anticipatory, complete and lag) synchronizations with excitatory coupling or direct (anticipatory, complete and lag) synchronizations with inhibitory coupling for the chosen form of the unidirectional nonlinear coupling due to the nature of the parametric relation (11) and the stability condition (14).
V Direct synchronizations with excitatory coupling
In this section, we will demonstrate the existence of anticipatory, complete and lag synchronizations as a function of the coupling delay , both experimentally and numerically, for the choice of the parameters satisfying the stability condition (14) for the case of excitatory coupling.
v.1 Anticipatory Synchronization
For , the synchronization manifold becomes an anticipatory synchronization manifold as described above. We have fixed the value of the feedback delay and the coupling delay , while the other parameters are fixed as , , , , and . The value of the nonlinear parameters are fixed as and such that both the stability condition (14) and the parametric relation (11) are satisfied. All the above parameter values are fixed to be the same except for the coupling delay for the remaining part of the study. The experimental and the numerical time series plots of both the drive and the response systems are shown in Figs. 8a and 9a, respectively, for demonstrating the existence of anticipatory synchronization. Both the experimental and the numerical phase space plots corresponding to the anticipatory synchronization manifold of the drive and the response systems are shown in Figs. 10(a)i and 10(a)ii, respectively.
The seven largest Lyapunov exponents of the coupled timedelay systems are shown in Fig. 11a as a function of the nonlinear parameter for the anticipatory synchronization manifold. For the values of delay times and the uncoupled systems exhibit only two positive Lyapunov exponents as may be seen from Fig. 6. The two positive Lyapunov exponents of the drive system remain positive, while one of the positive Lyapunov exponents of the response system becomes negative at and the second positive Lyapunov exponent becomes negative at confirming the existence of exact anticipatory synchronization for . It is to be noted that the Lyapunov exponents of the coupled systems indicate the existence of exact anticipatory synchronization also in the range in which the stability condition (14) is not satisfied confirming that it is only a sufficiency condition but not a necessary one.
v.2 Complete Synchronization
For , the synchronization manifold becomes a complete synchronization manifold . Now, we have fixed the value of the coupling delay as for fixed values of the other parameters as mentioned in the previous section. The experimental and the numerical time series plots of both the drive and the response systems are shown in Figs. 8b and 9b, respectively, demonstrating the existence of complete synchronization between the coupled timedelay systems. The phase space plots of both the systems corresponding to the complete synchronization manifold are shown in Figs. 10b. The seven largest Lyapunov exponents (Fig. 11b) of the coupled timedelay systems corresponding to the complete synchronization manifold indicate that both the positive Lyapunov exponents of the response system become negative for , while the two Lyapunov exponents of the drive system remain positive, confirming the existence of complete synchronization between the drive and response systems. Note that the coupled systems remain in a hyperchaotic state, that is, this transition to complete synchronization is a transition from one hyperchaotic regime to another one.
v.3 Lag Synchronization
The synchronization manifold becomes a lag synchronization manifold for . Both the time series and the phase space plots of the coupled timedelay systems obtained using our experimental realization are shown in Figs. 8c and 10(c)i, respectively, and those obtained using numerical simulations are shown in Figs. 9c and 10(c)ii, respectively, indicating the existence of a lag synchronization. Again, the seven largest Lyapunov exponents of the coupled timedelay systems shown in Fig. 11c for the lag synchronization manifold confirm the existence of it for .
Vi inverse synchronizations with inhibitory coupling
Now we consider the inhibitory coupling, , in Eq. (6bb) instead of the excitatory coupling to demonstrate the transition from inverse anticipatory to inverse lag synchronization via an inverse complete synchronization as a function of the coupling delay for the same values of parameters as in the Sec. V.
vi.1 Inverse anticipatory synchronization
As discussed above, the inverse synchronization manifold becomes an inverse anticipatory synchronization manifold for . For the same values of all the parameters as in Sec. V.1, the coupled timedelay system (6b) exhibits an inverse anticipatory synchronization in the presence of inhibitory coupling as shown in Figs. 12a and 13a. The experimental and numerical phase plots of the coupled timedelay system corresponding to the inverse anticipatory synchronization manifold are shown in Figs. 14(a)i and 14(a)ii, respectively. The seven largest Lyapunov exponents of the coupled systems corresponding to the inverse anticipatory synchronization manifold are shown in Fig. 15a as a function of the nonlinear parameter . The two largest positive Lyapunov exponents of the drive system remain unaltered in their values, while that of the response system become negative for confirming the existence of inverse anticipatory synchronization between the coupled timedelay systems with inhibitory coupling.
vi.2 Inverse complete synchronization
An inverse complete synchronization manifold is obtained for . The time series plot of both the drive and the response variables obtained from experimental realization are depicted in Fig. 12b and those obtained from numerical simulation are shown in Fig. 13b illustrating the existence of inverse complete synchronization. The experimental and numerical phase space plots of the coupled timedelay systems corresponding to the inverse complete synchronization manifold are depicted in Figs. 14(b)i and 14(b)ii, respectively. The seven largest Lyapunov exponents of the coupled timedelay systems (Fig. 15) confirm the existence of inverse complete synchronization indicated by a change in the signs of both the positive Lyapunov exponents of the response system for , while that of the drive system remain unchanged.
vi.3 Inverse lag synchronization
Again, for , the synchronization manifold becomes an inverse lag synchronization manifold. The experimental and the numerical time series plots, indicating the existence of inverse lag synchronization, of both the drive and response systems are shown in Figs. 12c and 13c, respectively. The corresponding phase space (of inverse lag synchronization) plots are also depicted in Figs. 14(c)i and 14(c)ii, respectively. The seven largest Lyapunov exponents of the coupled timedelay systems corresponding to inverse lag synchronization manifold are shown in Fig. 15c again as a function of . The two positive Lyapunov exponents of the drive system remain positive, while that of the response system become negative for confirming the existence of inverse lag synchronization between the coupled timedelay systems.
Vii Summary and Conclusion
In this paper, we have presented experimental observations of typical kinds of synchronization transitions in a system of unidirectionally coupled piecewiselinear timedelay electronic circuit designed using a threshold controller. In particular, we have shown the transition from anticipatory synchronization to lag synchronization through complete synchronization and their inverse counterparts with excitatory and inhibitory couplings, respectively, as a function of the coupling delay and for a fixed set of other parameters. A common stability condition valid for all these synchronized states is deduced and it is independent of both the feedback and the coupling delays. Futher, experimental observations are confirmed by numerical simulations and also from transitions in the Lyapunov exponents of the coupled timedelay systems. We also note that the nature of the piecewise linear function in the proposed circuit can be easily changed by using multiple threshold values and that multiscroll hyperchaotic attractors can also be produced even for a small value of delay time for further study and applications.
Acknowledgements.
The work of K.S. and M.L. has been supported by the Department of Science and Technology (DST), Government of India sponsored IRHPA research project, and DST Ramanna program of M.L. D.V.S. has been supported by the Alexander von Humboldt Foundation. J.K. acknowledges the support from EU under project No. 240763 PHOCUS(FP7ICT2009C).References
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