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Tunable power law in the desynchronization events of coupled chaotic electronic circuits

We study the statistics of the amplitude of the synchronization error in chaotic electronic circuits coupled through linear feedback. Depending on the coupling strength, our system exhibits three qualitatively different regimes of synchronization: weak coupling yields independent oscillations; moderate to strong coupling produces a regime of intermittent synchronization known as attractor bubbling; and stronger coupling produces complete synchronization. In the regime of moderate coupling, the probability distribution for the sizes of desynchronization events follows a power law, with an exponent that can be adjusted by changing the coupling strength. Such power-law distributions are interesting, as they appear in many complex systems. However, most of the systems with such a behavior have a fixed value for the exponent of the power law, while here we present an example of a system where the exponent of the power law is easily tuned in real time.

João Pessoa, PB, Brazil. We study the statistics of the amplitude of the synchronization error in chaotic electronic circuits coupled through linear feedback. Depending on the coupling strength, our system exhibits three qualitatively different regimes of synchronization: weak coupling yields independent oscillations; moderate to strong coupling produces a regime of intermittent synchronization known as attractor bubbling; and stronger coupling produces complete synchronization. In the regime of moderate coupling, the probability distribution for the sizes of desynchronization events follows a power law, with an exponent that can be adjusted by changing the coupling strength. Such power-law distributions are interesting, as they appear in many complex systems.
However, most of the systems with such a behavior have a fixed value for the exponent of the power law, while here we present an example of a system where the exponent of the power law is easily tuned in real time.

I. INTRODUCTION
Synchronization phenomena are widely studied in many real and idealized systems, such as electronic circuits 3-5 , lasers [6][7][8] , and maps [9][10][11][12] . While one often studies the synchronization of nonlinear dynamical systems in periodic states 13 , the synchronization of chaotic systems is even more interesting, due to the counterintuitive effect of synchronization between two systems whose trajectories are exponentially sensitive to perturbations in phase-space 14 , and also due to potential applications, such as masked communications 1,3 . Besides complete synchronization, one can find other generalizations to the concept, such as phasesynchronization 15 and generalized synchronization 16,17 . When subject to a generic coupling, most systems will wander off the synchronized state because of noise or mismatched parameters. This is often the case of generic feedback-coupled nonlinear oscillators in which there are riddled basins of attraction 18-20 that give rise to a phenomenon called attractor bubbling 19, [21][22][23][24][25][26][27] . As a consequence of attractor bubbling, the difference between the state variables, observed in the coupled elements (which we refer to as an error signal) shows long intervals of low values interspersed with sudden and brief departures to large values, which we call bubbles, bursts, or desynchronization events.

2
The distribution of sizes of these bursts have characteristic parameters similar to the ones observed in extreme events that occur in complex systems 2, 28 . This similarity suggests that we can use coupled chaotic oscillators as a proxy for the study of extreme events in complex systems, motivated by the problem of catastrophic behavior of many natural and artificial systems 28-32 . Such complex system variables usually present non-normal statistical distributions with large values of event sizes associated to the asymptotic behavior of the distribution, which can be a power law or other heavy-tailed distribution [33][34][35][36][37] . The value of this slope can reveal information about the mechanism producing the bursts. Complex systems typically follow power laws P (x) = Cx −α with a characteristic exponent α. For instance, α = 1.5 in neurological systems 44 , or α = 2.0, in coupled oscillators desynchronization 2 , or α = 1.0 in earthquakes analysis 45 . Power-law distributions with exponents α > 3 have finite first and second moments (average and variance, respectively), while for 3 ≥ α > 1 the second moment (variance) does not exist, and for 2 ≥ α > 1 even the first moment (average) is infinite. Here we show that, beyond the analysis of the probabilities of events of different sizes it is possible to tune the power law exponent itself, which has as a consequence the change of the value of the maximum event expected to occur during a given observation time. It is known that variables of many complex systems, including certain chaotic systems and systems on the edge of chaos 28 , follow power law statistics. Understanding the physical mechanisms for the origin of a heavy-tailed distribution of event sizes in a specific complex system may give important information about this system. For instance: power laws are scale-free, implying that large-, medium-and small-sized events all share a common mechanisms of formation. Such a lack of distinction affects the predictability of large events, as they are not discernible from the smaller ones in the earlier stages of formation.
We analyze here a nonlinear dynamical system formed by a pair of coupled chaotic oscillators and we show that the statistical distribution of one of the variables describing the state of this system presents a heavy-tailed behavior, with a very interesting property: it can evolve through a large range of the power-law exponent, which can be tuned through a control parameter. Systems with a tunable-exponent power law do not occur very often in Nature, and finding one example of a simple and rather generic system with this property can give a hint on how the exponent could be controlled in other systems where it is important. The system studied here is comprised of two electronic oscillators, each governed by a second-order differential equation driven by an external periodic force 38 . We use the difference between the values of the same dynamical variable in the two oscillators to generate a unidirectional feedback, coupling the oscillators in a drive/response configuration.
We call error signal the difference between the variables of the oscillators, and we analyse its statistics for different values of the coupling strength. Our system exhibits three qualitatively different behaviors as we change the coupling strength: for strong coupling we observe perfect (or high-quality) synchronization; for weak or negligible coupling, the two subsystems are independent, while for intermediate ranges of coupling strength the distribution of burst sizes has a heavy tail, with large events becoming increasingly rare, as for example in power-law (Pareto) distributions found in Lévy Flights 39-41 . Analysis of Kirchhoff's laws reveals that the dynamical state of the circuit can be expressed in terms of the voltages V 1d , V 2d and V E (t), which obey a second-order differential equation with an external pumping when R 2 = R 3 and C 2 = C 3 38 . This second-order differential equation can be written as two first-order equations:

II. CIRCUIT DESCRIPTION
, and the characteristic time scale of the evolution of our system is 1/θ, given by R 2 C 2 = 220.2 µs. V 1d and V 2d are the voltages at the outputs of OP1 and OP2, respectively, V E = A sin 2πf t is the external pump voltage with amplitude A and frequency f , and I(V 2d ) is the current passing through the diodes D1 and D2 and going to the inverting input of OP1, approximately given by and plotted in Fig. 2.
Let us recall that, in order for a continuous, autonomous system to be chaotic its dynamics needs to i) be embedded in, at least, a three-dimensional (3D) phase space 42 , and ii) have a nonlinear term. Our system has a nonlinearity in the current of Eq. (3), and the phase of the external forcing can be recast as a dynamical variable that provides for the third dimension.
By changing the values of frequency and amplitude of the external drive one can tune the system to periodic or chaotic states, in both the experimental circuit and in the numerical model. In the experimental setup, the system exhibits windows of chaotic dynamical state, for example for frequencies between 760 Hz and 820 Hz and between 950 Hz and 1.0 kHz, both with amplitude 4.0 V. Figure 3 shows a typical chaotic trajectory in phase space, acquired from the circuit (Fig. 3(a)), and obtained by numerical integration of the differential equations (1) and (2) (Fig. 3(b)). As we use a simplified model to describe the circuit of In order to couple two identical circuits, the voltage V 1 d of the drive circuit is injected, through a feedback circuit, into the second chaotic system. In Fig. 4 we show the response circuit, highlighting the additional circuitry where the feedback signal is produced by a subtractor (operational-amplifier OP3) whose input voltages are V 1d and V 1r and the output voltage is V 1d −V 1r . This feedback signal is added to the dynamics of the response subsystem, providing the coupling between the drive and the response. The parameter that measures the coupling level between the circuits is ǫ = R 2 /R RE , where R RE is a variable resistor placed at the output of the subtractor.
The values of the components of the response circuit are chosen to be as close as possible to their counterparts in the drive circuit, within a tolerance of 0.5%. Therefore, the components OP2, are also the same models as in the drive circuit, and the operational amplifier OP3 is a LF411CN. A single signal generator provides the sinusoidal signal V E applied to both the drive and response systems.
FIG. 4. Schematic diagram of the response system. The dashed rectangle highlights the feedback circuit that couples the response system to the drive system.
The dynamic equations are the same for both circuits, except the coupling term, which 7 is present only in the response circuit: In the numerical integration of the coupled system we used amplitude A = 3.0 V and frequency f = 720 Hz in Eqs. (1), (2), (4) and (5), with a parameter mismatch ∼ 1%, compound of the tolerances of the components. We set the parameters so that the experimental and numerical systems were both in the same chaotic state when uncoupled (ǫ = 0).
In the experimental setup, the circuits were then coupled and we used a digital oscilloscope to acquire temporal series of the differences V 1d − V 1r and V 2d − V 2r . From these signals we obtained the distance |x ⊥ | between the drive and the response systems in the 3D phasespace. In order to facilitate the analog calculation of the distance, we used the L-1 norm to define this distance as

III. STATISTICS OF THE ERROR SIGNAL
We acquired time series of |x ⊥ | with 10 7 points at a sample rate of 100 kHz (sampling time  for strong coupling (ǫ > 0.8) there is high-quality synchronization, with the distance between the systems fluctuating around the origin. As we tune the coupling strength from a (completely) unsynchronized state to a highly-synchronized state, 0.5 ≤ ǫ < 0.8, the system exhibits attractor bubbling, characterized by the occurrence of brief escapes from the state of high-quality synchronization and by a heavy-tailed distribution for |x ⊥ | n .
In this regime of moderate coupling, the distributions are visually similar to power laws.  Table I. The error bars in this table only measure the uncertainty in the slope of the optimal linear fit (in log-scale), not the true error in the value of α, which we estimate to be on the order of 15%. We see that, for ǫ between 0.5 and 0.6 the system crosses the critical value of exponent α = 3, which for the pure power law implies the divergence of the second moment of the distribution, although in our system the power law is truncated, because there is a maximum value of |x ⊥ | n .
The variation of the power-law exponents with the coupling level is confirmed by the numerical model. However, due to the simplicity of our model, the parameters A and f (respectively, amplitude and frequency of the external pumping) need to be adjusted to obtain better agreement between the experimental and numerical distributions. As we can see from Fig. 6(a), 6(b) and Table I, by changing the value of ǫ in the regime of moderate coupling we are able to control the slope of the power-law distribution of the maxima series |x ⊥ | n both in the experimental circuit and in the simplified numerical model. In other words, we tune the average value of |x ⊥ | n , and the overall range of the distribution of this variable, turning the system more stable or unstable.

IV. CONCLUSION
We investigated experimentally a pair of coupled chaotic oscillators and observed the regimes of high-quality synchronization, attractor bubbling and independent chaotic oscillation. The oscillators display attractor bubbling for moderate coupling strengths. The statistics of the size of the desynchronization events shows a heavy-tailed distribution, similar to truncated power laws appearing in many complex systems. Most interesting, we are able to tune the power-law exponent via the coupling parameter and, as a consequence, to change the range and the probability of large events. As far as we know there is no other  Table I.