We numerically analyze the problem of how to drive a synchronized state in a complex system to other state with diferent complexity, keeping synchronization. The complex system used is obtained by synchronizing two identical chaotic Takens-Bogdanov sub-systems specially coupled to recover in the global system the symmetries of each oscillator. The global state is adjusted to have an initial synchronized hyperchaotic state (with two positive Lyapunov exponents). This work is an attempt, using small amplitude external signals, to drive the global system to other complex state keeping the synchronized state. The method used to overcome the problems that we had to select a useful frequency value for the driving signal will be discussed, together with a possible experiment in a thermo-convective flow for validating the results obtained

Mancini-Maza, H. L.  (Hector Luis)

Vidal, G. (Gerard)

Dadun, University of Navarra

Int. J. Complex Systems in Sciencevol.1(2) (2011), pp. 183–190Complexity control in a synchronized complex systemHector Mancini1,† and Gerard Vidal11 Physics Institute, University of NavarraAbstract. We numerically analyze the problem of how to drive a synchronizedstate in a complex system to other state with different complexity, keeping syn-chronization. The complex system used is obtained by synchronizing two identicalchaotic Takens-Bogdanov sub-systems specially coupled to recover in the globalsystem the symmetries of each oscillator. The global state is adjusted to have aninitial synchronized hyperchaotic state (with two positive Lyapunov exponents).This work is an attempt, using small amplitude external signals, to drive the globalsystem to other complex state keeping the synchronized state. The method usedto overcome the problems that we had to select a useful frequency value for thedriving signal will be discussed, together with a possible experiment in a thermo-convective flow for validating the results obtained.Keywords: Synchronization, Hyperchaotic systems, Pre-turbulent convection.† Corresponding author: hmancini@fisica.unav.esReceived: October 17, 2011Published: October 24, 2011184 Complexity control in a synchronized complex system1. IntroductionReduction in complexity of synchronized coupled systems has been frequentlyobserved in numerical experiments controlling the coupling since the firstworks on confined systems [1] to large populations discussed in the frameof complex networks [2]. When a low dimensional system (3D) is analized,synchronization frequently supress chaos. In systems with higher dimensionaldynamics, more recent works coupling hyperchaotic ODE have been reportedto obtain synchronized states between identical systems without chaos sup-pression [4].This work is focused on the driving with small amplitude signals two syn-chronized hyperchaotic systems, without destroy the synchronization betweenthem. We use two Takens Bogdanov systems (with symmetry D4) coupledadequately to recover for the ensamble the symmetry of each equation sys-tem. Recently has been shown that a small amplitude harmonic [5] can beused to change the dynamical state in this system, if the frequency of thedriving signal is properly chose. The first problem is that this frequency cannot be obtained from the Fourier spectrum. In the cited work, a histogramconstructed ad-hoc by observing the recurrent times in the output signal wasused as a reference. As we remark forward, the histogram is not the result ofthe return times from a Poincare section, but it is simply a display of the morefrequently visited times. The new state has been verified by obtaining the newFourier spectrum and the attractor in the phase space. As the system has aRiddley bassin, a control of the output against a small change in the initialconditions was necessary to verify the independence of the global dynamicsfrom initial conditions.Finnally, we present among the new results, the autocorrelation in theoutput signal both for the “free-running” (without injection signal) and thecontrolled state. The larger autocorrelation time that can be observed in thecontrolled state, is a good indicator that, even if the system remains chaotic,becomes more coherent.2. The dynamical systemTwo identical oscillators TB1 and TB2 have been coupled to obtain the chaoticsynchronized dynamical system that we will drive. Each one of them is a fourdimensional Takens-Bogdanov’s system as it is shown in Eq. (1).Hector Mancini et al 185Figure 1: Interconnection scheme. The harmonic signal fE is always injectedto the variable x of the oscillator Ax˙ = y (1)y˙ = µx+ x(a(x2 + z2) + bz2)z˙ = w (2)w˙ = µz + z(a(x2 + z2) + bx2)The symmetry properties of the dynamical equations are described by thesymmetry group D4 composed by reflection τ and rotation ρ (Eq. (3)):τ : (x, y, z, w)→ (z, w, x, y) (3)ρ : (x, y, z, w)→ (−x,−y, z, w)These symmetries play an important role in the way that we have toconnect A and B, as it is shown in Fig. 1.Symmetrical coupling of two identical Takens-Bogdanov through one vari-able x has been presented in reference [4]. Synchronization regime has beenanalized using different coupling schemes (symmetric or asymmetric) and con-sidering the coupling as a direct function of the error between both systems(acting as a feedback loop). Phase synchronization (PS) has been obtainedusing selected values for the parameters that must be fine-tuned to fit theLyapunov exponent windows [6]. The synchronized regimes obtained in thiscase are not very stable and depend strongly on the coupling coefficient value.Here we present a different approach. To obtain a robust synchronizationmanifold we recover for the coupled system the symmetry of each one oscil-lator. This is achieved by coupling the feedback on two different variables186 Complexity control in a synchronized complex system(x and z). Internal symmetry of the equations couples variables by pairs (xand y to z and w). Introducing the coupling bidirectionally between A and Bthrough x and z, we are constructing a closed “ring structure” that providesthe robustness and Complete Synchronization (CS) is obtained. This was im-possible to obtain in the case of a symmetrical coupling scheme based on onlyone variable [3]. The equations for the coupled system are now transformed inEq. (4), where variables are named by the subscripts (A, B) corresponding toeach original system. The coupled system is also hyperchaotic (having morethan one positive Lyapunov exponent) and chaos is not completely suppressedby synchronization.x˙A = yA + εx(xB − xA) + 0.01xMAX sin(2pifEt)y˙A = µxA + xA(a(x2A + z2A) + bz2A)z˙A = wAw˙A = µzA + zA(a(x2A + z2A) + bx2A)x˙B = yB (4)y˙B = µxB + xB(a(x2B + z2B) + bz2B)z˙B = wB + εz(zA − zB)w˙B = µzB + zB(a(x2B + z2B) + bx2B)The coupling terms εx(xB − xA) and εz(zA − zB) can be interpreted asthe feedback signals between both systems. These factors are equal to zerowhen complete synchronization is achieved and both systems reproduce thesame trajectory on the synchronous manifold separately whitout any feedbackbetween them. This kind of coupling extends the inner symmetries of eachequation [4]. system to the coupled one which has a higher dimension. Theresults can be seen in Fig. 2.Variables (x, y, z, w) in A are completely synchronized to (x, y, z, w) inB. This mean that xA is complete synchronized to xB and so on. Temporalsignals displaying the synchronized state will be shown in the next paragraphtogether with effects of a driving signal fE = 0.01xMAX sin(2pifEt).Fig. 2 (a) displays the Fourier spectrum for one variable (variable xA) anda low frequencies detail appears in Fig. 2 (b). This figure shows a characteristicspectrum of a chaotic signal. Synchronization can be appreciated in Fig. 2(c), where the synchronization error function against time goes to zero after ashort transient.Hector Mancini et al 1870 1 2 3 4 510−5100105Frequency (Hz)0 0.5 1 1.510−5100105Frequency (Hz)(a) (b)200 400 600 800 1000Time5 10 15 20Characteristic Times(c) (d)Figure 2: Figure (a) Fourier spectrum for variable x1. (b) Zoom on lowfrequencies. (c) Error function for synchronization in logarithmic scale. Thefunction going to zero against time illustrate the necessary condition for CS.(d) Histogram of recurrent times (in seconds) that allow us to obtain thesystem characteristic times.3. Driving the coupled systemThis information, (that can be easily obtained from an experiment), is notsufficient if we need to choose a frequency for driving the system. As inmany other cases (like in the Ro¨ssler attractor for the “Fu¨nnel” parameters),here also is not possible to construct a Poincare´ section, and under theserestrictions we cannot define an ”analytical phase” [1]. To overcome thisproblem we constructed an histogram considering the period between twoneighbours maximum (or minimum [5]), obtained from a very long data fileof the output signal. The amplitud in the histogram represents at each timevalue, the frequency (number of times) that this period value appears in theoutput signal, being the highest peak the period that most frequently appears.Resolution in time can be controled by choosing the width of the time interval188 Complexity control in a synchronized complex systemto count, and in amplitud, it is one count over the total number of counts. Atypical histogram constructed from the output signal of the coupled system isshown in Fig. 2 (d). From this histogram, with periods distributed aroundtwo main peaks, we can obtain the most recurrent values of the frequencies inthe system.4. Results and ConclusionsThe histogram has been used to localize the region of frequencies that shouldbe applied. To determine which of the frecuencies in the interval is the bestto act on the system, we reconstruct histograms injecting signals with severalfrecuencies between the two peaks. Clearly the central frequency gives a statewith a different dynamic (figure 3). To check it accurately we obtained thenew Forier spectrum, the new phase diagram and the autocorrelation for thevariable x. The results for autocorrelation are presented in figure 4. Thedriven state display a longer correlation (zero crossing) for the drived system.It is a clear indication that we obtained a more ordered system with a lowercomplexity and a higher coherence.0,1370,1270,1170,107 0,147 0,157fEFigure 3: Sequence of histograms for the system driven with a harmonic signalwith different frequencies fE injected to the variable x of the oscillator AHector Mancini et al 189Figure 4: Comparison of autocorrelation in the variable x obtained from thefree running (red) and the driven state (blue)AcknowledgementsWe aknowledge the finantial support fromMEC Project FIS2008-01126, Spain,and we also thank to Prof. Wenceslao Gonza´lez-Vin˜as for many helpful dis-cussions.References[1] M.G. Rosenblum, A.S. Pikovsky and J. Kurths, “From Phase toLag Synchronization in Coupled Chaotic Oscillators,” Phys. Rev. Lett.78, 4193-4196, 1997.[2] S. Boccaletti, “The synchronized dynamics of complex systems”Monograph Series on Nonlinear Sciences and Complexity, Vol 6, Else-vier, 2008.[3] G. Vidal and H. Mancini, Int. J. Bifurcation and Chaos, 20 (3),885-896, 2010.[4] G. Vidal and H. Mancini, Int. J. Bifurcation and Chaos, 19 (2),719-726, 2009.[5] H. Mancini and G. Vidal, Dynamics of two coupled chaotic systemdriven by external signals“Eur. Phys. J. D 6257-66 (2011).190 Complexity control in a synchronized complex system[6] J. Bragard, G. Vidal, H. Mancini, C. Mendoza and S. Boc-caletti, ”Chaos suppression through asymmetric coupling”,Chaos17(4), 2007.

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