Engineering generalized synchronization in chaotic oscillators

CHAOS 21, 013106 (2011)

Engineering generalized synchronization in chaotic oscillators P. K. Roy,1 C. Hens,2 I. Grosu,3 and S. K. Dana2,a) 1

Department of Physics, Presidency University, Kolkata 700073, India Central Instrumentation, Indian Institute of Chemical Biology, Council of Scientific and Industrial Research, Kolkata 700032, India 3 Faculty of Bioengineering, University of Medicine and Pharmacy, “Gr. T. Popa” Iasi, Romania 2

(Received 1 September 2010; accepted 17 December 2010; published online 15 March 2011) We report a method of engineering generalized synchronization (GS) in chaotic oscillators using an open-plus-closed-loop coupling strategy. The coupling is defined in terms of a transformation matrix that maps a chaotic driver onto a response oscillator where the elements of the matrix can be arbitrarily chosen, and thereby allows a precise control of the GS state. We elaborate the scheme with several examples of transformation matrices. The elements of the transformation matrix are chosen as constants, time varying function, state variables of the driver, and state variables of another chaotic oscillator. Numerical results of GS in mismatched Ro¨ssler oscillators as well as C 2011 American nonidentical oscillators such as Ro¨ssler and Chen oscillators are presented. V Institute of Physics. [doi:10.1063/1.3539802]

Synchrony is an amazing natural phenomenon ubiquitous in the living world and also verified in laboratory scale systems: chemical reaction, laser, and electronic circuits. Over the last two decades, the theoretical framework of synchronization of nonlinear oscillators, limit cycle and chaotic, is mainly established under unidirectional or mutual interactions and with complex topology. However, most of the theories start with an approximation of identical oscillators to reduce the burden of mathematical complexity to develop an understanding of the physical mechanism of the phenomenon. On the other hand, in largely mismatched or nonidentical oscillators, a generalized definition of synchronization is proposed to explain a functional relationship that may emerge between the coupled oscillators in drive– response mode. Such a generalized synchrony is not immediately recognizable by visual inspection from experimental time series and it is also not possible so far to determine the exact functional relationship of the coupled systems. An indirect method of auxiliary system approach is mainly used for verification of generalized synchronization, although it is unable to retrieve the exact correlation. We attempt here a reverse engineering approach to target a desired GS state as a functional relation between a driver and a response oscillator. The response may be a mismatched or a nonidentical oscillator. Additionally, we allow a flexible and precise control of the response GS state. We start with a given functional relation and implement the functionality, y (t) 5 f(x(t)), between the driver y(t) and the response, x(t), by designing an appropriate coupling using an open-plus-closed-loop coupling and ensure stability of the GS state.

a)

Author to whom correspondence should be addressed. Electronic mail: [email protected].

1054-1500/2011/21(1)/013106/7/$30.00

I. INTRODUCTION

Studies on synchronization of chaotic oscillators, using the conventional linear coupling under varying coupling strength, reveal that various degrees of entrainment or coherence can be achieved: complete synchronization (CS),1 phase synchronization (PS),2 lag synchronization (LS),3 generalized synchronization (GS),4 and time-scale synchronization.5 Among all these, CS is an ideal situation and occurs only if the interacting oscillators are identical. If the parameters of the coupled oscillators are mismatched but closely identical, more realistically, an almost or near CS state can be observed in experiments. For largely mismatched and nonidentical oscillators, the concept of GS was introduced4 which tried to explain emergence of a kind of functional relationship between the coupled oscillators in drive–response mode. GS is thus seen as an evolution of a functional relationship, y(t) ¼ f(x(t)), between the state vector of chaotic driver, x [ Rn, and the state vector of a chaotic response, y [ Rn. In other words, GS can be described as a state when the response becomes a map of the driver dynamics, y(t) ¼ Ax(t); A is a mapping operator (invertible or noninvertible) or a transformation matrix. However, by visual inspection of the measured time series, no inference can be drawn about the existence of such correlation as usually possible, to an extent, for other forms of synchrony. In case of GS, it is even more difficult to identify the mathematical structure of the transformation matrix or correlation between the driver and the response from experimental measurements. To circumvent this difficulty, an auxiliary system approach was proposed4 as an indirect way of verifying GS in chaotic systems. Other methods were also proposed6–8 for identification of GS but none of them is able to identify the exact transfer function. Very recently, experimental evidences of GS and its potential applications were reported in electronic circuits,4 in lasers,9 and in microwave oscillators.10 From the viewpoint of applications, we attempt to realize GS between two chaotic oscillators with a reverse

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engineering approach: given a desired functional relation or a transformation matrix to define a GS state, we address the question how to implement the desired GS state or the functional relation, y(t) ¼ f(x(t), between a chaotic driver and a chaotic response oscillator. We also incorporate an option for flexible and a precise control of the predefined GS state, which is most desirable for practical applications. For this, we use the open-plus-closed-loop (OPCL) coupling11,12 strategy. This scheme was implemented in chaotic oscillators using a multiplicative constant (a) to define a goal dynamics, gðtÞ ¼ a: xðtÞ), where a can be varied continuously for smooth transition from CS to AS state without disturbing the stability of synchronization. Here, we extend this OPCL coupling to the GS regime where the constant a: is now replaced by a matrix (n  n) whose elements may be constants, time varying functions or state variables of the driver or state variables of a different dynamical system and even a complex function taking combinations of state variables and constants. Effectively, the a-matrix is the transformation matrix that defines a targeted GS state. The important task, for implementing this scheme, is first to design a coupler based on a given transformation matrix, and then to connect the coupler between the oscillators for realizing a preassigned GS state. This scheme is possible to implement in experiment where one can tune at least few elements of the a-matrix. This incorporates a precise control of the final GS state. Thus, the method allows more flexibility to encounter existing challenges in secure communication using GS13,14 and also opens up windows of new applications such as in image processing.15 In the next section, we described briefly the basics of the OPCL coupling for master–slave configuration. In Sec. III, we illustrated the coupler design for various transformation matrices and then implemented the GS states using numerical simulations. Results are summarized in Sec. IV. II. OPCL COUPLING: GENERALIZED SYNCHRONIZATION

We first explain the general OPCL coupling approach of targeting a GS state using drive–response type unidirectional coupling in chaotic systems. We consider a dynamical system as a driver in the chaotic regime, _ ¼ FðxðtÞÞ; xðtÞ

xðtÞ 2 Rn :

(1)

yðtÞ 2 Rn :

(2)

The response dynamics is _ ¼ GðyðtÞÞ; yðtÞ

In the proposed coupling scheme, the response oscillator is to follow a desired goal state, g(t) ¼ ax(t), where a is a multiplicative factor which was originally11,12 assumed as a constant. For example, if a ¼1, the response or goal dynamics is in CS with the driver. Instead, we consider a as a transformation matrix (n  n) to realize a desired response GS state by choosing the elements (aij) of the matrix arbitrarily. For three-dimensional systems, let us define the goal state, g(t) ¼ ax(t), in general form,

0

1 0 g1 a11 @ g2 A ¼ @ a21 g3 a31

a12 a22 a32

10 1 0 P 1 x1 a13 P a1i xi a23 A@ x2 A ¼ @ P a2i xi A: (3) a3i xi a33 x3

Now, given a dynamical system, we make arbitrary choices of the elements of the a-matrix. Apart from constants, its elements may contain a function of time or state variable of the driver, and their combinations or even states of a different dynamical system. The response dynamics after coupling is _ ¼ GðyðtÞÞ þ DðyðtÞ; gðtÞÞ; yðtÞ

yðtÞ 2 Rn

(4)

where the second term on R.H.S. is the OPCL coupler, _  GðgðtÞÞ þ ðH  JðgðtÞÞÞðyðtÞ  gðtÞÞ: DðyðtÞ; gðtÞÞ ¼ gðtÞ (5) J(g(t)) is the Jacobian of the flow G(g(t)) and H is a constant matrix (n  n). The e(t)¼y(t)g(t) defines the error between the trajectories of the response and the goal state at any instant. We expand the G(y) ¼ G(g þ e) in a Taylor’s series, GðyÞ ¼ GðgÞ þ

@GðgÞ e þ ::: @g

(6)

and truncate the series at the first order term with an approximation of small error e. Using Eqs. (1) and (4)–(6), the error dynamics is found to obey e_ ¼ He. If the elements of the Hmatrix are now so chosen that its eigenvalues all have negative real parts, the error dynamics guarantees asymptotically stable synchronization (e ! 0 as t ! 1), i.e., a GS state. To implement the above strategy, we design the coupler D(y,g) in Eq. (5) to target the goal dynamics g(t). The unknown H-matrix in Eq. (5) is constructed from the knowledge of the Jacobian. We follow a general rule to construct the H-matrix: if the element of the Jacobian, Jij, is a constant (i.e., does not involve any state variable) we use it as the ijth element of the H-matrix. Otherwise, if the Jij consists of a state variable, we replace it by a constant pk (k ¼ 1, 2, …) in the H-matrix. The values of these parameters pk are then so chosen as to satisfy the Routh–Hurwitz (RH) stability criterion.12 To elaborate, let us consider a 3  3 matrix whose characteristic equation is k3 þ a1k2 þa2k þa3 ¼ 0, where ai (i ¼ 1, 2, 3, …) are the coefficients of the polynomial. The RH stability criterion follows the inequalities a1 > 0, a3 > 0, a1a2 > a3. It ensures that eigenvalues of H all have negative real parts. The H-matrix now becomes a Hurwitz matrix. Obviously, the choice of the elements of the a-matrix, a constant or a time dependent state, never affects the stability of synchronization. As a consequence, once synchronization is attained in the coupled system, by the choice of the Hurwitz parameters, one can change the elements of the a-matrix without disturbing the synchronization. This completes the design of the coupler, which is connected between the driver and response to realize the response GS state. We can continuously vary and tune any of the elements of the a-matrix (e.g., varying the amplitude of a periodic function or a constant) to allow a flexible control of the target GS state.

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In the next section, we explain the details how to design the coupler using several examples of a-matrices and thereby to implement the GS strategy using numerical simulations of different coupled systems. III. GENERALIZED SYNCHRONIZATION: EXAMPLES

Given the definition of a dynamical system, we first select a transformation matrix that decides a desired goal dynamics and then design the coupler to realize the targeted GS state. We present several examples of transformation matrices using various dynamical systems. We start with two mismatched Ro¨ssler oscillators. The driver x_ ¼ FðxÞ is 1 0 1 0 x1 x2  x3 x_ 1 @ x_ 2 A ¼ @ x1 x1 þ b1 x2 A (7) c1 þ x3 ðx1  d1 Þ x_ 3 while the response y_ ¼ GðyÞ is 1 0 1 0 y_1 x2 y2  y3 @ y_2 A ¼ @ x2 y1 þ b2 y2 A: y_3 c2 þ y3 ðy1  d2 Þ

(8)

Since synchronization is robust11,12 to parameter mismatch under the OPCL coupling, we consider large mismatch in all the parameters: x1 ¼ 1.0, b1 ¼ 0.18, c1 ¼ 0.2, d1 ¼ 10.0 for the driver and x2 ¼ 1.7, b2 ¼ 0.25, c2 ¼ 0.15, d2 ¼ 11.0 for the response. The Jacobian of the response is 0 1 1 0 x2 (9) b2 0 A: JðyÞ ¼ @ x2 0 y1  d2 y3 We construct the H-matrix from the Jacobian as described in the Sec. 2, 0 1 0 x2 1 H ¼ @ x2 (10) b2 0 A: 0 p2  d2 p1

when the expression of the coupler, 1 1 0 1:5x_ 1 3x_ 2 1:1x_ 3 x2 g2  g3 D ¼ @ 2x_ 1 þ 0:3x_ 2 A  @ x2 g1 þ b2 g2 A 1:6x_ 1 2x_ 2 þ x_ 3 c2 þ g3 ðg1  d2 Þ 0 100 1 0 11 0 0 0 y1 g1 þ@ 0 0 0 A@@ y2 A  @ g2 AA: p1  g1 0 p2  g2 y3 g3 (13) 0

Next, we add the coupler to the RHS of Eq. (8) to produce the response dynamics, which converges to the desired goal dynamics after the transients. Numerical results are presented in Fig. 1 to reveal the desired GS state. The time series of driver (x1) and response (y1) in Fig. 1(a) apparently shows no correlation and the x1 versus y1 plot in Fig. 1(b) appears as a Ro¨ssler-like structure. This structure is not to be confused with any filtered Ro¨ssler attractor.16 In fact, the x1 vs y1 plot is a projection of the synchronization manifold. Actually, the functional correlation between the driverPand response is embedded in the goal dynamics, g1 ¼ i¼1;3 a1i xi ¼ 1:5x1  3x2  1:1x3 which is plotted against y1 in Fig. 1(c). This plot shows 1:1 correlation to confirm the GS relation. Other response variables also follow similar GS relations with the driver as shown in Figs. 1(d)– 1(e). To verify a GS state, we follow a simple procedure: check 1:1 correlation between the response state and the transformed driver state. In rest of the examples, we use the Ro¨ssler system (8) as a response and hence the Hurwitz matrix remains unchanged. In a second example, we use systems (7) and (8) again, but include periodic functions in the a-matrix, 0

1 1:5 sinð0:3tÞ 0 0 a¼@ 0 0:3 cosð0:5tÞ 0 A: 0 0 1

(14)

For simulations, we choose the parameters, p1 ¼ 1.5 and p2 ¼ 9 to make the H-matrix to be a Hurwitz. A range of other values of the parameters is also available,12 which satisfies the Hurwitz property for stable synchronization. Once the stable synchronization is established, we propose an arbitrary choice of the elements of the transformation matrix, with all as constants first, 0 1 1:5 3 1:1 (11) a ¼ @ 2 0:3 0 A: 1:6 2 1 which targets a goal dynamics, 0

1 0 10 1 1:5 3 1:1 g1 x1 @ g2 A ¼ @ 2 0:3 0 A @ x2 A 1:6 2 1 g3 x3 0 1 1:5x1 3x2 1:1x3 ¼ @ 2x1 þ 0:3x2 A 1:6x1 2x2 þ x3

(12)

FIG. 1. (Color online) GS in mismatched Ro¨ssler oscillators defined by a constant matrix: (a) time series of driver x1 in solid (red) and response y1 dotted (blue) lines, respectively; (b) plot of x1 vs y1. Transformed driver variables plotted against response variables: (c) g1 ¼ (1.5x1  3x2  1.1x3) against y1; (d) g2 ¼ (2x1 þ 0.3x2) against y2; (e) g3 ¼ (1.6x1  2x2 þ x3) against y3 confirming GS relation (Ref. 12).

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0

1 0 10 1 1:5 sinð0:3tÞ 0 0 x1 g1 B C B CB C ¼ 0 0:3 cosð0:5tÞ 0 g x @ 2A @ A@ 2 A g3 x3 0 0 1 0 1 1:5 sinð0:3tÞx1 B C (15) ¼ @ 0:3 cosð0:5tÞx2 A: x3 and the coupler is 0

1 0 1 1:5 sinð0:3tÞx_ 1 0:45cosð0:3tÞx1 B C B C Dðx; gÞ ¼ @ 0:015sinð0:5tÞx2 A þ @ 0:3 cosð0:5tÞx_ 2 A x3 x_ 3 1 0 1 0 0 0 0 x2 g2  g3 C B C B 0 0 A  @ x2 g1 þ b2 g2 A þ @ 0 c2 þ g3 ðg1  d2 Þ p1  g1 0 p2  g2 00 1 0 11 g1 y1 BB C B CC  @@ y2 A  @ g2 AA: y3 g3 (16) We arbitrarily choose a few sinusoidal functions in Eq. (14); however, other choices are also possible without disturbing the stability of synchronization. Figure 2(a) shows the simulated time series of the driver x1 and its response y1. Both the time series and the x1 versus y1 plot in Fig. 2(b) does not indicate any correlation. However, the GS relation (15) is confirmed in Fig. 2(c) when g1(t) is plotted against y1(t). It is clear that, although the a-matrix contains time dependent functions, synchronization is still maintained, since its selection never affects the stability of synchronization as explained in Sec. II. In another example, we try the same combination of Ro¨ssler systems (7) and (8) but with a amatrix comprising of state variables of the driver, 0

0:01x1 a ¼ @ 0:05x2 0

0:1x2 0:01x1 0

10 1 1 0 g1 x1 0:01x1 0:1x2 0 @ g2 A ¼ @ 0:05x2 0:01x1 0 A@ x2 A g3 x3 0 0 1 0 1 0:01ðx1 Þ2 þ 0:1ðx2 Þ2 ¼ @ 0:05x2 x1 þ 0:01x1 x2 A x3 0

when the goal dynamics becomes

1 0 0 A: 1

(17)

Such matrix elements, obviously, elevates the degree of nonlinearity in the goal dynamics,

and, also in the coupler 0 1 0:02x1 x_ 1 þ 0:2x2 x_ 2 B C D ¼ @ 0:05ðx_ 2 x1 þ x2 x_ 1 Þ þ 0:01ðx_1 x2 þ x1 x_ 2 Þ A x_ 3 0 1 0 0 0 x2 g2  g3 B C B 0  @ x2 g1 þ b2 g2 A þ @ 0 c2 þ g3 ðg1  d2 Þ 00 1 0 11 g1 y1 BB C B CC  @@ y2 A  @ g2 AA: y3

p1  g1

0 0

(18)

1 C A

0 p2  g2

(19)

g3

Numerically simulated time series in Fig. 3(a) again demonstrates no apparent correlation. The x1 versus y1 plot in Fig. 3(b) shows an unfamiliar structure and different from the Figs. 1(b) and 2(b). This is due to higher degree of nonlinearity introduced in the GS relation by the a-matrix. However, as expected, the plot of g1 ¼ 0:01x1 2 þ 0:1x2 2 against the response state y1 in Fig. 3(c) shows 1:1 correlation and thereby confirms GS between the driver and the response. We have checked the other response variables also to follow the same GS relation. We extend the concept of GS further when the elements of the a-matrix are chosen as state variables of another dynamical system, say, a Lorenz oscillator (20), while the driver and response remains mismatched Ro¨ssler systems (7) and (8). The dynamics of the Lorenz system is 0 1 0 1 crðz2  z1 Þ z_1 @ z_2 A ¼ LðzÞ ¼ @ cðz1 ðr  z3 Þ  z2 Þ A: (20) cðz2 z1  bz3 Þ z_3 The parameters of the Lorenz system are typically selected in the chaotic regime, r ¼ 10, r ¼ 28, b ¼ 8=3. The time scale of the original Lorenz system is rescaled by a factor of c ¼ 0.05 for convenience without loss of generality. The elements of the a-matrix are selected using the state variables of the Lorenz system,

FIG. 2. (Color online) GS in mismatched Ro¨ssler oscillators with periodic function in a-matrix: (a) time series of driver x1 in solid (blue) and response y1 dotted (red) lines, respectively; (b) x1 vs y1 plot; (c) transformed driver variable g1 ¼ 1.5 sin(0.3t) x1 is plotted against response variable y1.

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FIG. 3. (Color online) GS in mismatched Ro¨ssler oscillators with transformation matrix consisting driver variables. (a) Time series of driver x1 in solid (blue) and response y1 in dotted (red) lines; (b) x1 vs y1 plot; (c) transformed driver g1 ¼ 0:01x1 2 þ 0:1x2 2 and response variables y1 show one-to-one GS relation, only one variable is shown while other also comply.

0

0:05z1 a ¼ @ 0:01z2 0

0:01z2 0:05z1 0

1 0 0 A; 1

(21)

so that the coupler becomes 1 0 1 0 0:05z_1 x1 0:01z_2 x2 0 0:05z1 x_ 1 þ 0:01z2 x_2 C B C B D ¼ @ 0:01z_2 x1 0:05z_1 x2 0 A þ @ 0:01z2 x_ 1 þ 0:05z1 x_ 2 A x3 0 0 0 0 1 0 1 0 0 0 x2 g2  g3 B C B C 0 0 A  @ x2 g1 þ b2 g2 A þ @ 0 c2 þ g3 ðg1  d2 Þ p1  g1 0 p2  g2 00 1 0 11 g1 y1 BB C B CC (22)  @@ y2 A  @ g2 AA; y3 g3 when the goal dynamics g(t) is set at, 10 1 0 1 0 x1 g1 0:05z1 0:01z2 0 @ g2 A ¼ @ 0:01z2 0:05z1 0 A@ x2 A 0 0 1 g3 x3 0 1 0:05z1 x1 þ 0:01z2 x2 ¼ @ 0:01z2 x1 þ 0:05z1 x2 A: x3

only to Ro¨ssler and Lorenz systems but valid for other combinations of systems for realizing any desired GS state. Finally, we present a more complex scenario of GS: a Chen oscillator17 (24) drives a Ro¨ssler oscillator (8), 0

1 0 1 bðx2  x1 Þ x_ 1 @ x_ 2 A ¼ FðxÞ ¼ @ ðb2  b1 Þx1  x1 x3 þ b2 x2 A; x1 x2  b 3 x3 x_ 3

(24)

with the parameters b1¼ 34, b2¼ 23, and b3¼ 3. Moreover, we choose the elements of the a-matrix from state variables of another chaotic system, say, a Sprott system whose dynamics is, 0 1 1 0:2z2 z_1 A: @ z_2 A ¼ SðzÞ ¼ @ z1 þ z3 2 ðz1 þ z2  z3 Þ z_3 0

(25)

The transformation matrix is now selected as 0

(23)

We find that the OPCL coupling scheme works equally efficiently to realize the GS relationship (23) between the driver and response as detailed in Fig. 4. Both the time series of the driver, response (x1, y1) in Fig. 4(a) and the synchronization manifold x1 versus y1 in Fig. 4(b) again shows no correlation from visual inspection. Similar to the previous cases g1(t) versus y1(t) plot in Fig. 4(c) clearly confirms the targeted GS correlation. Note that, this scheme is not restricted

0:05z1 a ¼ @ 0:01z3 0:1z2

0:02z2 0:05z1 1

1 0:3 0:05z3 A; 0:01z3

(26)

when the goal dynamics becomes 0 1 0 1 0:05z1 0:02z2 g1 0:3 B C B C @ g2 A ¼@ 0:01z3 0:05z1 0:05z3 A g3 0:1z2 1 0:01z3 0 1 0 1 0:05z1 x1 þ 0:02z2 x2 þ 0:3x3 x1 B C B C  @ x2 A ¼ @ 0:01z3 x1  0:05z1 x2 þ 0:05z3 x3 A; x3 0:1z2 x1 þ x2 þ 0:01z3 x3 (27)

FIG. 4. (Color online) GS in mismatched Ro¨ssler oscillators with transformation matrix consisting state variables of other dynamical system. (a) Time series of driver and response (x1, y1) in solid (blue) and dotted (red) black lines respectively; (b) x1 vs y1 plot shows no correlation; (c) response y1 against g1¼0.05x1z1 þ 0.01x2z2 plot.

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FIG. 5. (Color online) GS in Chen and Ro¨ssler oscillators with transformation matrix consisting state variables of a Sprott system. (a) Widely varied time series of driver and response (x1, y1) in solid (blue) and dotted (red) black lines, respectively; (b) x1 vs y1 plot shows no correlation; (c) transformed driver, g1¼0.05x1z1 þ 0.01x2z2 þ 0.3x3, shows 1:1 correlation with response y1.

and the coupler is obtained as 0 10 1 0:05z_1 0:02z_2 0 x1 B CB C D ¼ @ 0:01z_3 0:05z_1 0:05z_3 A@ x2 A 0:1z_2 0 0:05z1 B þ @ 0:01z3 0 B @

0:01z_3

0

0:02z2 0:05z1

0:1z2

1

x2 g2  g3 x2 g1 þ b2 g2

x3 10

1 0:3 x_ 1 CB C 0:05z3 A@ x_ 2 A 0:01z3 x_ 3 1 0 0 0 C B 0 Aþ@ 0

c2 þ g3 ðg1  d2 Þ 00 1 0 11 g1 y1 BB C B CC  @@ y2 A  @ g2 AA: y3 g3

p1  g1

0

0 0

1 C A

p2  g2 (28)

Even with such a diverse choice of subsystems, numerical results in Fig. 5 clearly show that the desired GS relation is successfully realizable using the proposed OPCL scheme. The driver (x1) and the response (y1) time series in Fig. 5(a) show no visual correlation while the x1 versus y1 plot appears as if they follow a type of antiphase synchronization (APS).18 In reality, they develop a GS relation as confirmed by the 1:1 correlation between g1(t) and y1(t) in Fig. 5(c). The OPCL coupling scheme is thus able to establish a robust GS relation in mismatched oscillators; it is successful in realizing a designated functional relation or transformation even when the driver and response oscillators are completely different and the transformation matrix is defined using a third type of dynamical system. The method allows great flexibility in engineering a GS state. Even for the simplest case of a constant a-matrix (n  n), we have interesting options to choose aij as positive or negative values (1