Emergent hybrid synchronization in coupled chaotic systems

PHYSICAL REVIEW E 91, 022920 (2015)

Emergent hybrid synchronization in coupled chaotic systems E. Padmanaban,1,* Stefano Boccaletti,2,3 and S. K. Dana1 1

CSIR-Indian Institute of Chemical Biology, Jadavpur, Kolkata 700032, India 2 Consiglio Nazionale delle Ricerche, Institute of Complex Systems, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Florence, Italy 3 Italian Embassy in Israel, 25 Hamered Street, 68125 Tel Aviv, Israel (Received 8 September 2014; revised manuscript received 2 December 2014; published 24 February 2015) We evidence an interesting kind of hybrid synchronization in coupled chaotic systems where complete synchronization is restricted to only a subset of variables of two systems while other subset of variables may be in a phase synchronized state or desynchronized. Such hybrid synchronization is a generic emergent feature of coupled systems when a controller based coupling, designed by the Lyapunov function stability, is first engineered to induce complete synchronization in the identical case, and then a large parameter mismatch is introduced. We distinguish between two different hybrid synchronization regimes that emerge with parameter perturbation. The first, called hard hybrid synchronization, occurs when the coupled systems display global phase synchronization, while the second, called soft hybrid synchronization, corresponds to a situation where, instead, the global synchronization feature no longer exists. We verify the existence of both classes of hybrid synchronization in numerical examples of the Rössler system, a Lorenz-like system, and also in electronic experiment. DOI: 10.1103/PhysRevE.91.022920

PACS number(s): 05.45.Xt, 05.45.Gg

I. INTRODUCTION

Synchronization of chaotic systems is nowadays an almost well understood topic; its categorization in literature [1,2] includes complete synchronization (CS) [3], phase synchronization (PS) [4–6], lag synchronization (LS) [7], and generalized synchronization (GS) [8,9]. Each of the above states was observed and characterized in various circumstances for linear diffusive (instantaneous or delayed) coupling, and a wealth of studies have been carried out with unidirectional or bidirectional configurations, including some special types of coupling such as inhibitory [10] or excitatory [11] synaptic and repulsive coupling [12]. One of the most important problems connected with the study of synchronization in coupled chaotic systems is the influence of a parameter mismatch. Major efforts of the investigations, in the past, were concentrated on playing with the coupling strength and the mismatch parameters to observe the onset of different synchronization regimes and their instabilities. The most common observation of synchronized behaviors in coupled systems is that all the state variables of the two coupled systems maintain a particular type of coherence, say, CS or PS. Alternatively, an approach of engineering synchronization [13–18] was attempted where one can implement a desired coherent or incoherent pattern in the dynamics of the coupled systems. The latter approach assumes the knowledge of the dynamical system, but the coupling is a priori unknown and properly designed to realize a desired synchronization state in an assembly of the given dynamical units. One way of implementing such a coupling design [17,18] is based on choosing a set of controllers, which establishes stability of the desired synchronous state by Lyapunov functions. Therefore, the relevant question is, once the desired synchrony is imprinted on the identical case, how such a state would be

*

[email protected]

1539-3755/2015/91(2)/022920(6)

affected if a system parameter would naturally be drifted or changed manually. In this paper, we address the question both numerically and experimentally, and consider two identical chaotic oscillators with a coupling design made explicitly to realize CS. We show that, if any parameter is then moved well beyond a prescribed tolerance level, a novel type of hybrid synchronization (HS) regime generically arises. The possibility of such HS states was originally predicted and termed as almost synchronization (AS) [2,19]. We opine that the title AS does not reflect the true character of the particular synchronization state and use the term HS as a closer definition of the state. This HS state corresponds to a mixed synchronization state, where only a subset of the state variables of the coupled systems maintains CS, whereas the other subset of variables evolves in a PS state or unsynchronized manner. More specifically we identify a hard hybrid synchronization (HHS) state, where the two systems display globally a PS state, as opposed to a soft hybrid synchronization (SHS) state, where, instead, no global synchronization features are set in the coupled systems. We furthermore show that HS states are robust and remain undisturbed for significant changes in the dynamics of the coupled systems, induced by large variations of a parameter mismatch. We distinguish this HS state from partial synchronization that was reported earlier [20] in two nonidentical systems where a subset of state variables was engineered to synchronize in a CS state using a controller based coupling design. We engineered a similar partial synchronization state previously [21,22] in two identical or mismatched oscillators using the Lyapunov function based controller design approach that can be easily extended to nonidentical systems. In both the cases, the partial synchronization was assumed a priori as a desired state and accordingly the coupling was designed to realize the targeted partially coherent state. There is a marked difference in our current observation of HS that we do not a priori target and design the controllers to realize it; rather the HS emerges in the coupled system with a parameter perturbation. Certainly, we use the Lyapunov function based stability approach to

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E. PADMANABAN, STEFANO BOCCALETTI, AND S. K. DANA


design the controllers for two identical oscillators and establish first a CS state in all the variables and then perturb a parameter, to a large extent, but do not change or readjust the coupling during this perturbation process. The novelty of our work lies in the emergent dynamics (a sequence of periodic and chaotic attractors) of the coupled systems with the parameter perturbation, and, as a consequence, the coupled system loses CS but, most interestingly, emerges into HS states, a HHS state and a SHS state which are unknown so far, to the best of our knowledge. Particularly, the HHS state reveals the coexistence of CS and PS or a global PS state but not a partial synchronization state, although the SHS reveals partial synchrony. The emergent HS state is theoretically explained by the presence of a partial Lyapunov stability that is satisfied even after a large parameter perturbation, as shown in detail in the specific examples below. The paper is organized as follows: In Sec. II, we briefly introduce the theory of coupling to the design of controllers and the HS is discussed by using mutually coupled Rössler oscillators, and then in Sec. III the Lorenz-like oscillator is investigated. For both cases, we focus on how a parameter mismatch can influence the quality of synchronization and, in the process, alter the dynamic of the coupled oscillators. The validity of analytical results is mainly checked through numerical simulation and experiments in Sec. IV using coupled Rössler oscillators. The results are summarized in Sec.V.

For identical systems, ω1,2 = ω, b1,2 = b, c1,2 = c, d1,2 = d, and one can design the controllers Ux,y in such a way that e → 0 with t → ∞. A specific choice of the controllers is, for instance, ⎞ ⎞ ⎛ 1 ⎛ − 2 (−e1 + ωe2 + e3 ) Ux1 ⎜Ux2 ⎟ ⎜ 1 (e1 + (1 + b)e2 ) ⎟ ⎟ ⎟ ⎜ 2 ⎜ ⎟ ⎜U ⎟ ⎜ 0 ⎟ ⎜ x3 ⎟ ⎜ (3) ⎟, ⎟=⎜ 1 ⎜ ⎜Uy1 ⎟ ⎜ 2 (−e1 + ωe2 + e3 ) ⎟ ⎟ ⎟ ⎜ 1 ⎜ ⎝Uy2 ⎠ ⎝ − (e1 + (1 + b)e2 ) ⎠ 2 Uy3 0 which satisfies the LFS condition, V˙ (e) = −e12 − e22 − de32 < 0 (if d > 0), and establishes a stable CS in the coupled system for which asymptotically y1 (t) = x1 (t), y2 (t) = x2 (t), and y3 (t) = x3 (t). Next, we introduce a mismatch in the coupled system by detuning the parameter d2 both positively and negatively from the identical condition d2 = d1 , while keeping all other parameters identical. The LFS criterion is then violated, but a partial CS stability [21] is still maintained (since a partial LFS condition V˙ (e) = −e12 − e22 < 0 is satisfied when x1 = y1 and x2 = y2 conditions hold) and it is not disturbed by the parameter perturbation. Therefore, we set the parameter values ω = 1.0,b = 0.22,c = 0.2, and d1 = 12.0, monitor only the (x3 ,y3 ) pair of variables with varying d2 , and estimate their similarity measure [1]: [xi (t) − yi (t)]2 , δi2 = xi2 (t)yi2 (t)

¨ II. COUPLED ROSSLER OSCILLATORS

where . stands for the time average over a sufficiently large time window, and for this particular example i = 3. Figure 1 reports δ3 as a function of d2 and reveals a sharp dipping into a global minimum δ3 = 0 at d2 = 12 when all parameters are identical. In the following, we elaborate that, away from the identical case, regimes of SHS and HHS are observed when δ3 = 0. For identification of the HHS and SHS regimes, we estimate the instantaneous phases of the two systems by using the x3 and y3 variables. The uncorrelated phases of the two variables rotate with different velocities (φ1 − φ2 ) ∼ t,

4

ln(δ3)

Let us start by considering two identical chaotic systems, x˙ = f (x) + Ux and y˙ = f (y) + Uy , where f (x) and f (y) represent identical flows of the two uncoupled systems, while the controllers Ux and Uy establish a mutual coupling. A stable CS state can be set by applying Lyapunov function stability (LFS) principles, in a way that the error function [defined by e(t) = y(t) − x(t)] of the coupled system satisfies limt→∞ e(t) = 0, so that asymptotically one has x(t) = y(t). To realize such a desired synchronized state, we first formulate a Lyapunov function, V (e) = 12 eT e, where T denotes the transpose of a matrix and V (e) is a positive definite function. Assuming that the parameters of the coupled system are known, the controllers Ux and Uy can then appropriately be chosen such that V˙ < 0. This latter condition, called the LFS criterion [17], actually warrants asymptotic stability of the error function and thereby establishes a global stability of the desired CS state. Starting from the coupling design establishing the CS state, we then proceed, by detuning one of the parameters of the coupled system, to test the effect of large parameter mismatches on the stability of CS. As an illustrating example, we take two mutually coupled Rössler systems: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x˙1 −ω1 x2 − x3 Ux1 ⎝x˙2 ⎠ = ⎝ x1 + b1 x2 ⎠ + ⎝Ux2 ⎠ , (1) x˙3 c1 + x1 x3 − d1 x3 Ux3 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ y˙1 −ω2 y2 − y3 Uy1 ⎝y˙2 ⎠ = ⎝ y1 + b2 y2 ⎠ + ⎝Uy2 ⎠ . (2) y˙3 c2 + y1 y3 − d2 y3 Uy3

(4)

−10 4

12

d2

24

FIG. 1. (Color online) Degradation of CS for a parameter detuning in two coupled Rössler oscillators. b = 0.22, c = 0.2, and d1 = 12.

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st

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FIG. 2. (Color online) Mean frequency difference (see text for definition) as a function of d2 and b1,2 = b.

until they arrive at a phase coherent state when the phase difference stops growing with time |φ1 − φ2 | < const and the frequency difference = 0 [4,23]. The region of PS in the (b − d2 ) parameter plane is presented in Fig. 2. The plateau region in blue (black) identifies a PS ( = 0) regime, which persists for a large positive or negative detuning of d2 for a range of b = 0.1 to 0.16 (b1,2 = b) values. This region is defined as the HHS regime where the (x1 , y1 ) and (x2 , y2 ) pairs are in CS, and the two oscillators are in a globally PS regime. This global PS scenario continues for b > 0.16 and d2 0.16 and d2 > 10.5 and beyond, the two oscillators escape from PS [see the green (light gray) and red (dark gray) regions of Fig. 2], and the system enter the SHS regime. The topology of the coupled Rössler attractors in the SHS regime now becomes phase noncoherent (a funnel type attractor) [24]. We emphasize that the other pairs of variables (x1,2 = y1,2 ) remain always in a CS state. Conventionally, in two coupled chaotic systems, a transition to PS is indicated by a switching of one originally vanishing Lyapunov exponent to a negative value, with the coupled system still remaining hyperchaotic [7], i.e., still displaying two positive and distinct Lyapunov exponents. On the other hand, in case of HS, the hyperchaotic regime is not found, as it can be seen from Fig. 3, where we report the two largest Lyapunov exponents [LLEs, in dotted blue (gray) and dotted black lines] and the mean frequency difference (in red or dark gray line) against d2 . It is, instead, seen that the LLE remains positive for varying d2 , while the second LLE remains zero, except in small periodic windows. At the same time, the remains zero almost everywhere for a negative detuning of d2 ( 10.5, the coupled systems abandon the HHS state ( = 0, red or dark gray line in the inset of Fig. 3) in chorus of a slight elevation in the SLE (dotted green line). In the overall process, the coupled system shows various emergent dynamics, periodic to chaotic in the HHS regime. Later in the

10

d

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FIG. 3. (Color online) The first and second largest Lyapunov exponents (dotted blue and dotted black line, respectively) and the mean frequency difference (red or dark gray line) of the coupled Rössler systems are plotted as a function of the parameter d2 . b = 0.24. Inset: Plot of the SLE (dotted green line) and the frequency difference (red or dark gray line) in the proximity of the transition from HHS to SHS.

text, we will support all these results with an electronic circuit experiment. III. COUPLED LORENZ-LIKE OSCILLATORS

In search of a generality of the phenomenon, we move now to search for HS in another system, and consider a coupled Lorenz-like system (LLS) [25] which is described by the following equations: ⎛ ⎞ ⎛ ⎞ ⎛ 1 ⎞ x˙1 (ae2 ) a1 (x2 − x1 ) 2 ⎝x˙2 ⎠ = ⎝−x1 x3 + C1 x2 ⎠ + ⎝ 1 (C + 1)e2 ⎠ , 2 x1 x2 − b1 x3 x˙3 0 (5) ⎞ ⎛ ⎞ ⎛ ⎞ − 12 (ae2 ) a2 (y2 − y1 ) y˙1 ⎝y˙2 ⎠ = ⎝−y1 y3 + C2 y2 ⎠ + ⎝− 1 (C + 1)e2 ⎠ . 2 y˙3 y1 y2 − b2 y3 0 ⎛

(6) A specific choice of the controllers is made first for identical systems (a1,2 = a, C1,2 = C, and b1,2 = b), with the assumption that e → 0 for t → ∞, i.e., satisfying the LFS condition V˙ (e) = −ae12 − e22 − be32 < 0 (if a,b > 0) when the coupled systems are in a CS state. The parameter C2 is then detuned both positively and negatively to estimate the similarity measures (4) of all three pairs of state variables for the selected values of a1,2 = 36, b1,2 = 3, and C1 = 13. They reveal that δ2,3 = 0 (x2,3 = y2,3 ), and only the (x1 ,y1 ) pair of variables always maintains CS (δ1 = 0). Away from the critical value C2 (=13), regimes of HHS and SHS clearly emerge (δ2,3 = 0). Further we scrutinize the nonsynchronized subsets of the coupled LLS by measuring the mean frequency differences

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the other two pairs remain in PS; as a whole, two systems are in a globally PS state [ = 0, dotted blue (dark gray) line]. For values C2 > 18 and beyond, the two systems escape from PS [dotted (blue dark) gray line in Fig. 4, = 0]. In this example, subsequent to the transition from HHS to SHS, the system displays a hyperchaotic regime, which is signified by a transition of the second LLE (black line) to positive values for C2 > 18 in Fig. 4. The third LLE is shown in magenta or light gray line.

2 ΔΩ

st

1 LLE nd 2 LLE rd 3 LLE

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ΔΩ

LE

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0

¨ IV. EXPERIMENT: COUPLED ROSSLER OSCILLATORS

−1

An experimental evidence of HS is now reported using a pair of electronic oscillators whose dynamics mimics the Rössler system as shown in Fig. 5. The op-amps are wired as summing integrators with various inputs representing the right hand side of the differential equations. The op-amps are all LM348 or equivalent and multipliers AD633 are used to design the oscillators OS1 and OS2 and the coupling circuit. The circuit is powered by ±12V and oscillators are scaled to 100 k. The voltages at the oscillator nodes are labeled by (X1,X2,X3) for OS1 and (Y 1,Y 2,Y 3) for OS2. In the coupling circuit, the voltages at the nodes are labeled as E1N, E2N, E3N, E1, E2, and E3 and are analogs of −e1 , − e2 , − e3 , e1 , e2 , and e3 , respectively. For bidirectional interactions, the continuity between the oscillators, OS1 and

−1

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25

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FIG. 4. (Color online) Mutually coupled LLS: three LLE and the mean frequency differences are plotted as a function of the parameter C2 .

[in dotted blue (dark gray) line] using the (x2 , y2 ) pair along with three LLEs as reported in Fig. 4 for different values of C2 . For detuning of the C2 < 18, the coupled system remains in the HHS regime where the (x1 ,y1 ) pair is in CS and OS1 2R

E1N

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Coupling Circuit Ra

AD 633 X1 W X2 Y1 Z Y2

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AD 633 X1 W X2 Y1 Z Y2

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− U1 +

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Ra − U16 +

Ra Ra

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Y3 E3N E2

Y2 E2N

FIG. 5. (Color online) Coupled Rössler circuit: All Op-amps are LM348 including integrators (U 1,U 3,U 4) and (U 5,U 7,U 8). Multipliers AD633 (A1,A2), resistances (R = 100 k, Ra = 10 k), and capacitances (100 nF) are used to design the OS1 and OS2. The coupling circuit is composed of summing amplifiers, inverters, and resistances. Power supply is ±12V. Components values are noted in the circuit (1% tolerance). 022920-4

EMERGENT HYBRID SYNCHRONIZATION IN COUPLED . . .


right. In the third row, plots of X1 vs Y1 in the left panel, and of X2 vs Y2 plot in the second panel from left, both confirm that x1,2 = y1,2 is maintained for all the detuned values of Rd . The X3 vs Y3 plot for Rd = 23k is shown in the third panel from left that indicates a PS regime. It basically represents the HHS regime where a global PS is seen. The output voltage plot of X3 vs Y3 (Rd = 12.5k ) is shown at the rightmost panel that belongs to the SHS regime where the global PS is not preserved. V. SUMMARY

FIG. 6. (Color online) Oscilloscope pictures (Rössler): 2D attractors of oscillator 1 measured at X1 vs X2 for different values of resistance Rd (in k) are 50, 40, 35.2, and 33 in the first row and 31.7, 28.3, 21.6, and 14.5 in the second row, respectively. Third row: Output voltage measured at X1 vs Y1 shown in the first panel and X2 vs Y2 shown in the second panel; both confirm CS. Output voltage measured at X3(0.2V/div) vs Y3(2V/div) shown in the third panel (HHS) and X3 vs Y3 shown in the fourth panel (SHS).

OS2, is maintained via incoming and outgoing terminals in the circuit. Measurements of the OS1 and the OS2 variables (analogs of x1 ,x2 ,x3 and y1 ,y2 ,y3 ) are made at the output of (U 1,U 3,U 4) and (U 5,U 7,U 8), respectively, using a fourchannel digital oscilloscope (Yokogawa DL9140, 1 GHz, 5 GS/s). The potentiometer Rd (analog of d2 ) is connected in the negative terminal of op-amp U 8 that is tuned to obtain different values of d2 . The oscilloscope pictures of phase portraits of emergent dynamics for detuning a parameter (Rd ) are presented in Fig. 6 and all belong to the HHS regime as mentioned above. The two-dimensional (2D) projections of OS1 as X1 versus (vs) X2 plots for different values of resistance (Rd ) are shown in the first and second rows: in the first row, period 1 at left, period 2 at the second panel, period 4 at the third panel, and chaos at right; in the second row, period 5 at left, period 3 at the second panel, period 3 at the third panel, and chaos at

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We demonstrated HS as a generic feature of coupled chaotic systems, using a coupling design approach whose CS stability is first established by the Lyapunov function stability and then by detuning a parameter that breaks the stability condition only partially. Furthermore, we demonstrated the robustness of HS in an electronic experiment displaying a high tolerance of a parameter shift (>100%). We evidenced that a parameter perturbation allows the coupled systems to emerge into two distinct kinds of HS regimes, a hard HS and a soft HS. An important point to note is that, in the hard HS, a globally PS regime is maintained, while it is lost in the soft HS state. As a general conclusion we can state that, once the controllers are designed based on the Lyapunov function stability to establish a stabilized synchronization manifold, y1,2,3 = x1,2,3 , if one annihilates the Lyapunov condition partially by detuning a parameter connected with a particular state variable, the respective pair will get affected, and in addition it will affect other pairs if the parameter is attached to a nonlinear term of the system. Due to its generality and robustness, the novel collective state can therefore be expected to emerge in a wealth of other coupled chaotic systems. In conclusion, we emphasize that this HS phenomenon is not engineered but an emergent behavior of the coupled system due to parameter perturbation (as drifting or by manual shifting) and clearly different from the engineered partial synchronization. ACKNOWLEDGMENTS

S.K.D. acknowledges support by the Emeritus Scientist Scheme of the Council of Scientific and Industrial Research (India) under Project No. 21(0928)/12/EMR-II.

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