Engineering synchronization of chaotic oscillators

Engineering synchronization of chaotic oscillators using controller based coupling design E. Padmanaban, Chittaranjan Hens, and Syamal K. Dana Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 21, 013110 (2011); doi: 10.1063/1.3548066 View online: http://dx.doi.org/10.1063/1.3548066 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/21/1?ver=pdfcov Published by the AIP Publishing

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CHAOS 21, 013110 (2011)

Engineering synchronization of chaotic oscillators using controller based coupling design E. Padmanaban,a) Chittaranjan Hens, and Syamal K. Danab) Central Instrumentation, Indian Institute of Chemical Biology, (Council of Scientific and Industrial Research), Jadavpur, Kolkata 700032, India

(Received 18 June 2010; accepted 5 January 2011; published online 24 March 2011) We propose a general formulation of coupling for engineering synchronization in chaotic oscillators for unidirectional as well as bidirectional mode. In the synchronization regimes, it is possible to amplify or to attenuate a chaotic attractor with respect to other chaotic attractors. Numerical examples are presented for a Lorenz system, Ro¨ssler oscillator, and a Sprott system. We physically realized the controller based coupling design in electronic circuits to verify the theory. We extended the theory to a network of coupled oscillators and provided a numerical example with C 2011 American Institute of Physics. [doi:10.1063/1.3548066] four Sprott oscillators. V

During the last two decades, there has been a great interest in understanding the mechanism of synchronization in chaotic systems and their stability criterion. Mainly, the unidirectional and bidirectional linear couplings were explored in two or many oscillators. The studies were focused on collective behaviors of nonlinear oscillators and on understanding several coherent patterns observed in natural systems: physical, chemical, and biological. An alternative approach is attempted recently for engineering synchronization and coherent patterns and even for inducing desynchronization by design of coupling. The methods focus on defining an appropriate coupling for targeting a desired synchronization state such as complete synchronization (CS), antisynchronization (AS), amplitude death (AD), anticipation synchronization, and generalized synchronization (GS), which can be induced in chaotic oscillators, given a model of the system. This reverse engineering approach is important for applications. We propose a general formulation of coupling by designing a nonlinear open loop controller (NOLC) to target synchronization states such as CS and AS with amplification or attenuation in chaotic oscillators. The stability of synchronization is ensured by using Lyapunov function stability (LFS) theorem for both unidirectional and bidirectional mode of coupling. We support the theory with experimental verification in electronics circuits. We extended the theory to a network of coupled oscillators.

I. INTRODUCTION

Chaos synchronization is extensively studied using unidirectional coupling, commonly known as master–slave coupling,1 particularly, from the viewpoint of secure communication.2 On the other hand, a bidirectional or mutual coupling is usually found in many natural systems,3 for example, in gap junction4 of neurons. The gap junctions allow chemical ions to diffuse across a cell membrane and thereby develop bidirectional a)

Electronic mail: [email protected]. Electronic mail: [email protected].

b)

1054-1500/2011/21(1)/013110/10/$30.00

interactions. Using both the modes of coupling, a variety of possible coherent states are observed3 in chaotic oscillators such as complete synchronization (CS),1 lag synchronization5 and anticipation synchronization,6 phase synchronization (PS),3,7 antiphase synchronization (APS),8 generalized synchronization (GS),9 and time-scale synchronization.10 Specific coupling definition such as inhibitory11 or excitatory and even repulsive coupling12 was also used, particularly, in context of studies related to the collective behaviors of neurons. However, one of the main targets of the investigations so far was on playing with the coupling strength and with the mismatch parameters to observe the onset of the synchronization regimes and their stability in two or more oscillators. An alternative approach of engineering synchronization13–19 in nonlinear oscillators is attempted recently, which is important from the practical viewpoint. It is recently reported17,18 that, given a definition of a model system, one can always engineer CS, AS, and amplitude death (AD) in chaotic oscillators using an open-plusclosed-loop (OPCL) coupling. Additionally, the coupling is able to induce amplification (or attenuation) of a driver attractor onto response oscillator. Such scaling of an attractor is an alternative way of looking at projective synchronization,20 when a driver attractor is enlarged/reduced in size at the response exactly at a predefined scale. The scaling factor, in practical systems, can be varied smoothly to implement CS and AS via AD with a precise and continuous control21 from one to the other form of synchronization. However, the OPCL method is limited by local stability. Moreover, the OPCL method in the AS regime under mutual coupling is restricted to inversion symmetric systems only. As an alternative, we explore a controller based coupling design22 here using the LFS technique that ensures global stability and rules out the restriction of inversion symmetry. The method so far remained only of theoretical interest and the studies were confined to master–slave coupling mode. In this paper, we show that the LFS based NOLC design is not difficult to realize in electronic circuits. Furthermore, we extend the theory to bidirectional mode with scaling of one chaotic attractor relative to another one by using an asymmetric coupling. A linear asymmetric coupling is

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reported23 earlier for synchronization in chaotic oscillators. But there, the purpose was to enhance the critical coupling strength for CS and to suppress chaos. On the other hand, we propose a design of coupling for engineering various forms of synchronization and scaling of attractors. To our best knowledge, the mutual scaling of attractors with synchronization is not reported so far. An immediate question arises if one can use the standard linear amplification method after the synchronization process for scaling an attractor, at least, for unidirectional coupling mode. In reality, all amplifier devices are limited by their bandwidth, which will not allow the whole spectrum of a chaotic signal to be equally amplified. This practical limitation will certainly induce distortion in the output of an amplifier. For mutually coupled oscillators, the question of scaling after the synchronization process does not arise. The scaling of a chaotic attractor in drive-response mode has potential applications in encryption and decryption of message in secure communication.24 We test the coupling design in real experiment using electronic circuits. In experiments, we use two closely identical chaotic oscillators, results of which are found in good agreement with the numerical results. We generalize the theory for a network of coupled oscillators and provide a numerical example of four coupled oscillators. The paper is organized as follows: we explained the theoretical basis of the NOLC design in Sec. II. In Sec. III, the theory for unidirectional coupling is elaborated how to realize CS and AS using numerical examples of Lorenz systems and Sprott system. We demonstrated practical feasibility of the unidirectional mode using electronic Sprott circuit. In Sec. IV, we extended the theory to bidirectional coupling and elaborated with numerical examples of a Lorenz system, Ro¨ssler oscillator, and a Sprott system, and presented experimental verification. Finally, we described the general theory for a network of coupled oscillators in Sec. V with mathematical details in the Appendix. The results are summarized in Sec. VI.

VðeÞ ¼ 12 eT e:

(4)

T denotes transpose of a matrix, where V(e) is a positive definite function. Assuming that the parameters of the coupled system are known, we make an appropriate choice of the _ < 0. And this ensures ascontrollers Ux and Uy so that VðeÞ ymptotic global stability of synchronization and thereby realizes any desired synchronization state. For unidirectional coupling, one can assume either (Ux ¼ 0, b ¼ 1) or (Uy ¼ 0, a ¼ 1), while for bidirectional or mutual coupling, (Ux, Uy) and (a, b) all have nonzero values. Signs of a and b decide whether it is CS or AS and the ratio of a and b decides the scaling factor: amplification or attenuation of one oscillator relative to another oscillator. It is to be mentioned that, for identical systems, the mutual coupling is symmetric when a ¼ b, and we can only achieve CS but no amplification or attenuation. For a = b, the coupling is asymmetric when we realize relative amplification or attenuation of attractors. Two different cases arise in the asymmetric situation: case I: a ¼ 1, jbj > 1 when one attractor is attenuated from its original size and case II: jaj > 1, b ¼ 1 when one attractor is amplified in size. In the following sections, we describe the exact structure of the Lyapunov function, error dynamics, and coupling design based on the NOLC controllers using several numerical examples.

III. UNIDIRECTIONAL COUPLING

To elaborate the unidirectional coupling scheme, which is always asymmetric, we first choose a dynamical system such as a Lorenz system, Master Lorenz oscillator: x_ 1 ¼ r1 ðx2 x1 Þ; x_ 2 ¼ r1 x1 x2 x1 x3 ; x_ 3 ¼ b1 x3 þ x1 x2 ;

(5a)

Slave Lorenz oscillator: II. THEORY OF COUPLING DESIGN

Consider two identical chaotic systems x_ ¼ f ðxÞ þ Ux ; y_ ¼ f ðyÞ þ Uy ;

(1) (2)

where f(x) [ Rn and f(y) [ Rn are the flow of the systems in uncoupled state. The mutual coupling of the systems is defined by the NOLC, Ux and Uy. The error function is defined by e(t) ¼ by(t) ax(t), where a and b are constants. A stable synchronization state is realizable if one applies the LFS technique when the error function of the coupled system follows the limit lim keðtÞk ! 0;

t!1

(3)

so that ax ¼ by. To design the coupling based on the NOLCs, we define a Lyapunov function

y_ 1 ¼ r2 ðy2 y1 Þ þ Uy1 ; y_ 2 ¼ r2 y1 y2 y1 y3 þ Uy2 ; y_ 3 ¼ b2 y3 þ y1 y2 þ Uy3 ;

(5b)

where Uy ¼ [Uy1 Uy2 Uy3]T is the NOLC that defines the coupling to be designed. Here, we choose, Ux ¼ 0 and b ¼ 1. For identical systems, we assume r1 ¼ r2 ¼ r; r1 ¼ r2 ¼ r; and b1 ¼ b2 ¼ b. The error function is now defined by e ¼ (y ax): e1 ¼ y1 ax1, e2 ¼ y2 ax2, and e3 ¼ y3 ax3, where e ¼ [e1 e2 e3]T, x ¼ [x1 x2 x3]T, and y ¼ [y1 y2 y3]T. We derive the error dynamics by taking a difference of Eqs. (5a) and (5b) e_ 1 ¼ rðe2 e1 Þ þ Uy1 ; e_ 2 ¼ re1 e2 y1 y3 þ ax1 x3 þ Uy2 ; e_ 3 ¼ be3 þ y1 y2 ax1 x2 þ U y3 :

(6)

For stable synchronization, e ! 0 with t ! 1, when y1 ¼ ax1, y2 ¼ ax2, and y3 ¼ ax3. By substituting the


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conditions in Eq. (6) and taking the time derivative of Lyapunov function (4) _ VðeÞ ¼ e1 rðe2 e1 Þ þ Uy1 þ e2 re1 e2 aða 1Þx1 x3 þ Uy2 þ e3 be3 þ aða 1Þx1 x2 þ U y3 ;

(7)

we make a choice of the NOLC or, in other words, propose a design of the controllers Uy1 ¼ ðre2 Þ; Uy2 ¼ ðre1 Þ að1 aÞx3 x1 ; Uy3 ¼ að1 aÞx2 x1 ;

(8)

such that

}

_ VðeÞ ¼ e1 e_ 1 þ e2 e_2 þ e3 e_ 3 2

2

2

¼ re1 e2 be3 ;

(9)

_ when VðeÞ < 0 if r > 0 and b > 0. Equation (9) ensures asymptotic global stability of synchronization. Note that the choice of the controllers is not unique, and one can always _ make other choices so as to satisfy, VðeÞ < 0. Based on the proposed design (8) of the NOLC that defines the coupling, we can target a desired synchronization with a choice of the a-value. Numerical results are shown in Fig. 1 for two coupled Lorenz oscillators (5a) and (5b). The three-dimensional (3D) trajectories of the driver and response are shown in Fig. 1(a) for a ¼ 3. The response attractor is enlarged by three times and inverted. The x1 vs y1 plot in Fig. 1(b) confirms that they are in AS state. All the time series (response, yi, in dotted lines and driver, xi, in solid lines) in Figs. 1(c)– 1(e) are in AS state. Thus, similar to the OPCL method,17,18 this method is also able to induce complete AS in axial symmetric system like the Lorenz system while the conventional linear coupling can only achieve partial AS. But, in contrast to the OPCL method, we are able to establish global stability of synchronization. The error plot ei ¼ (yi axi) (i ¼ 1, 2, 3) in Fig. 1(g) shows an exponential convergence to zero that confirms the amplification of the driver exactly by a factor of 3 as expected for a ¼ 3. Such amplification is also realizable in the CS regime by simply taking a ¼ 3, as shown in Fig. 1(f). Attenuation of the driver is also realized in both AS and CS regimes by taking 0 < jaj < 1, but figures are not presented. For physical implementation in electronic circuit, we consider a Sprott system and present the numerical results first. Master Sprott oscillator: x_ 1 ¼ a1 x2 ; x_ 2 ¼ x1 þ x3 ;

(10)

Slave Sprott oscillator:

2

y_3 ¼ y1 þ y2 y3 þ Uy3 :

For identical system a1 ¼ a2 ¼ a and after coupling if we make the following choice of the controller, Uy ¼ [Uy1 Uy2 Uy3]T, Uy1 ¼ e1 þ ae2 ; Uy2 ¼ e1 e2 e3 ;

(12) 2

Uy3 ¼ e1 þ að1 aÞx2 ; it satisfies

x_ 3 ¼ x1 þ x2 2 x3 ;

y_1 ¼ a2 y2 þ Uy1 ; y_2 ¼ y1 þ y3 þ Uy2 ;

FIG. 1. (Color online) Coupled Lorenz attractor (r ¼ 28, r ¼ 10, b ¼ 8/3), a ¼ 3. (a) 3D trajectories of driver and response (amplified and inverted) attractors and (b) plot of x1 vs y1 in AS state. (c)–(e) Time series of (x1, y1), (x2, y2), and (x3, y3), respectively. (f) Plot of x1 vs y1 confirms CS for a ¼ 3. (g) Plot of errors ei ¼ (yi þ 3xi).

(11)

_ VðeÞ ¼ e1 2 e2 2 e3 2 and hence limt!1 keðtÞk ! 0 is established. Numerical results of the coupled Sprott systems (10) and (11) are shown in Fig. 2. The two-dimensional (2D) projections in Fig. 2(a) show that the response is enlarged and in the reverse direction to the driver attractor. The x1 vs y1 plot is shown in Fig. 2(b) to confirm AS (a ¼ 2). The time series of the response variable (y1) in dotted lines and of the driver


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}

FIG. 2. (Color online) Coupled Sprott system (a ¼ 0.225). (a) 2D plot of driver and its amplified and inverted response, (b) plot of x1 vs y1 confirms AS, (c) time series of x1 in solid lines and amplified y1 in dotted lines, and (d) plot of error ei ¼ (yi þ 2xi) confirms an amplification by a factor of 2 (a ¼ 2).

variable (x1) in solid lines in Fig. 2(c) also confirm AS while the response is enlarged. The plot of error ei ¼ (yi þ 2xi) in Fig. 2(d) converges to zero after the initial transients. It confirms that the response is an amplified version of the driver by a factor of 2. Next, we describe physical realization of the Sprott systems (10) and (11) with coupling circuit in Fig. 3. The Op-amp U1–U4 (U5–U8) and the multiplier A1 (A2) with resistances R1–R8 (R9–R16) and capacitances C1–C3 (C4–C6) represent the driver oscillator OS1 (response oscillator OS2). The coupling is designed using Op-amp U9–U14 with resistances R17–R38 to implement the controller in Eq. (12). Measurements of the driver and response variables (analog of x1,2 and y1,2) are made at the output of (U1, U3) and (U5, U7), respectively, using a fourchannel digital oscilloscope (Yokogawa DL9140, 1 GHz, 5 GS/s). Experimental oscilloscope pictures are shown in Fig. 4 and they are in good agreements with the numerical results in Fig. 2. The response attractor in the upper row middle is enlarged and in reverse direction to the driver attractor in the left (a ¼ 2). The output voltages, U1 vs U5, in upper row right confirms AS. The time series of measured outputs at U3 and U7 at lower row also confirms AS and amplification of the driver at response.

IV. BIDIRECTIONAL COUPLING

We extend the theory of NOLC design to bidirectional coupling and show that a scaling as a form of relative enlargement or reduction of size of the attractors is also possible for mutually interacting oscillators. Either of the oscillators can be amplified in comparison to another with an appropriate selection of the constants (a, b) in the error function. As mentioned above, the controllers (Ux, Uy) and the constants (a, b) in Eqs. (1) and (2) now all have nonzero values. If the coupling is symmetric for a ¼ b in identical systems, one can only

}

FIG. 3. Coupled Sprott circuit. Driver circuit OS1 at the top, response circuit OS2 at the middle, and the coupling circuit at the bottom. Linear integrators U1–U4 (U5–U8), multipliers AD633 A1(A2), resistances R1–R8 (R9–16), and capacitances C1–C3 (C4–C6) are used to design the driver OS1 and response OS2. The coupling circuit is composed of summing amplifiers with resistances R17–R38. Components values (1% tolerance) are noted in the circuit.

observe 1:1 synchronization. Here, we present only the results of the asymmetric case (a = b) that clearly explains the general principle of the NOLC based coupling design. We again choose the Lorenz system for elaboration.


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In the final step, we make a following choice of the NOLC: 1 re2 ; 2a 1 Ux2 ¼ re1 ; 2a Ux3 ¼ 0;

(17a)

1 re2 ; 2b 1 a a Uy2 ¼ re1 1 x3 x1 ; 2b b b a a 1 Uy3 ¼ x2 x1 : b b

(17b)

Ux 1 ¼

} Uy1 ¼

FIG. 4. (Color online) Oscilloscope pictures: 2D projection of the attractors at the upper panel, driver at the left, amplified and inverted response at the middle (axes in same scale), and output voltage of U1 (horizontal axis, 500 mV/div.) vs U5 (vertical axis 1 V/div.) at the right. Lower panel: time series measured at U3 (upper row) and U7 (lower row).

Oscillator-1: x_ 1 ¼ r1 ðx2 x1 Þ þ Ux1 ; x_ 2 ¼ r1 x1 x2 x1 x3 þ Ux2 ; x_ 3 ¼ b1 x3 þ x1 x2 þ Ux3 ;

(13)

Oscillator-2: y_ 1 ¼ r2 ðy2 y1 Þ þ Uy1 ; y_ 2 ¼ r2 y1 y2 y1 y3 þ Uy2 ; y_ 3 ¼ b2 y3 þ y1 y2 þ Uy3 ;

(14)

where Ux ¼ [Ux1 Ux2 Ux3]T and Uy ¼ [Uy1 Uy2 Uy3]T are the controllers. Note the difference between Eqs. (5a) and (13), while Eq. (5b) is same as Eq. (14). A set of controllers is now added to the first oscillator (5a) to set up the mutual interactions in Eq. (13). The error function is defined by e ¼ by ax For identical systems, r1 ¼ r2 ¼ r; r1 ¼ r2 ¼ r, and b1 ¼ b2 ¼ b, the error dynamics then becomes e_ 1 ¼ rðe2 e1 Þ þ bUy1 aUx1 ; e_ 2 ¼ re1 e2 by1 y3 þ ax1 x3 þ bUy2 aUx2 ; e_ 3 ¼ be3 þ by1 y2 ax1 x2 þ bU y3 aUx3 ;

(15)

where e ¼ [e1 e2 e3]T and e1 ¼ by1 ax1, e2 ¼ by2 ax2, and e3 ¼ by3 ax3. As usual when e ! 0 with t ! 1, y1 ¼ (a/b)x1, y2 ¼ (a/b)x2, and y3 ¼ (a/b)x3. By substituting these conditions in Eq. (15) and taking the time derivative of Lyapunov function, VðeÞ ¼ 12 eT e, _ VðeÞ ¼ e1 rðe2 e1 Þ þ bUy1 aUx1 a 1 x1 x3 þ bUy2 aUx2 þ e2 re1 e2 a b a 1 x1 x2 þ bU y3 aUx3 : (16) þ e3 be3 þ a b

By substituting Eq. (17) into Eq. (16), we obtain the stability _ condition for synchronization VðeÞ ¼ re1 2 e2 2 be3 2 when r > 0 and b > 0. In the asymmetric case I: (a = b), we decide to maintain the original attractor size of the oscillator-1 and to attenuate the attractor size of the oscillator-2 by choosing a ¼ 1 and jbj > 1. In case II: jaj > 1 and b ¼ 1, the attractor size of oscillator-1 still remains unchanged while the oscillator-2 is amplified. Note that the oscillator-1 has linear controller terms in Eq. (17), while the oscillator-2 has nonlinear controllers. Effectively, oscillator-1 behaves like a driver and oscillator-2 as a slave. Alternatively, one can easily amplify or attenuate the oscillator-1 keeping the size of oscillator-2 unchanged by making a choice of the NOLC, such that oscillator-1 has the nonlinear controller and the oscillator-2 has the linear controller. Numerical results of two mutually coupled Lorenz systems (13) and (14) are presented in Fig. 5 for case I: a ¼ 1 and b ¼ 2. The 3D trajectories of the oscillators are shown in Fig. 5(a). It is seen that one is larger and the other is inverted and smaller. The x1 vs y1 plot in Fig. 5(b) confirms AS between the oscillators. The time series plots in Figs. 5(c)–5(e) show all three pairs of similar variables in AS state while each of xi in dotted lines for oscillator-1 is larger than each of yi in solid lines for oscillator-2. The error plot ei ¼ (byi axi), i ¼ 1, 2, 3 in Fig. 5(g) shows an exponential decay to zero. The attractor of oscillator-2 is inverted and reduced by a factor of 2. We also realize CS by simply taking a ¼ 2 and b ¼ 1, as shown in Fig. 5(f). We illustrate case II using an example of Ro¨ssler system, where one attractor is enlarged while the other remains unchanged. Two mutually coupled Ro¨ssler systems plus the controllers are given by Ro¨ssler oscillator-1: x_ 1 ¼ x1 x2 x3 þ Ux1 ; x_ 2 ¼ x1 þ b1 x2 þ Ux2 ; x_ 3 ¼ C1 þ x3 x1 d1 x3 þ Ux3 ;

(18)

Ro¨ssler oscillator-2: y_1 ¼ x2 y2 y3 þ Uy1 ; y_2 ¼ y1 þ b2 y2 þ Uy2 ; y3 ¼ C2 þ y3 y1 d2 y3 þ Uy3 :

(19)


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}

Numerical results of mutually coupled Ro¨ssler systems in Eqs. (18) and (19) are shown in Fig. 6. The 3D attractors in Fig. 6(a) for a ¼ 2 are shown where the size of oscillator-1 attractor remain unchanged while the oscillator-2 attractor is larger but inverted confirming their AS state. It is also confirmed from the plot of x1 vs y1 in Fig. 6(b). The time series y1 in dotted lines in Fig. 6(c) is also larger than x1 (solid lines) in amplitude but opposite in phase. The plot of error ei ¼ (yi þ 2xi), i ¼ 1, 2, 3, in Fig. 6(d) converges to zero with time that confirms an amplification of the oscillator-2 by a factor of 2. Next, we implement the design of the NOLC based bidirectional coupling in electronic circuit using two Sprott systems. We present the numerical results for verification first, Sprott oscillator-1: x_ 1 ¼ a1 x2 þ Ux1 ; x_ 2 ¼ x1 þ x3 þ Ux2 ;

(21)

2

x_ 3 ¼ x1 þ x2 x3 þ Ux3 ; Sprott oscillator-2: y_ 1 ¼ a1 y2 þ Uy1 ; y_ 2 ¼ y1 þ y3 þ Uy2 ;

(22)

2

y_ 3 ¼ y1 þ y2 y3 þ Uy3 :

FIG. 5. (Color online) Mutually coupled Lorenz oscillators (r ¼ 28, r ¼ 10, b ¼ 8/3). (a) 3D plots of oscillator-1 (original size) and oscillator-2 (attenuated and inverted). (b) Plot of x1 vs y1 confirm AS (b ¼ 2) in and (c)–(e) corresponding time series of (x1, y1), (x2, y2), and (x3, y3), respectively. (f) Plot of x1 vs y1 confirm CS (a ¼ 2). (g) Plot of error ei function with time.

For identical systems, a1 ¼ a2 ¼ a, and we make a choice of the controllers Ux and Uy as given in the following equations: 1 ðe1 þ ae2 Þ; Ux1 ¼ 2a 1 (23a) ðe1 e2 e3 Þ; Ux2 ¼ 2a 1 Ux3 ¼ e1 ; 2a

For identical systems, x1 ¼ x2 ¼ x, b1 ¼ b2 ¼ b, C1 ¼ C2 ¼ C, d1 ¼ d2 ¼ d. We choose the controllers Ux and Uy as defined in Eqs. (20a) and (20b) for oscillator-1 and oscillator-2 1 ðe1 þ xe2 þ e3 Þ; 2a 1 Ux2 ¼ ðe1 þ ð1 þ bÞe2 Þ; 2a Ux3 ¼ 0;

(20a)

1 Uy1 ¼ ðe1 þ xe2 þ e3 Þ; 2b 1 Uy2 ¼ ðe1 þ ð1 þ bÞe2 Þ; 2b a a a Uy3 ¼ C 1 þ 1 x3 x1 ; b b b

(20b)

Ux1 ¼

when they satisfy the LFS criterion, _ VðeÞ ¼ e21 e22 de23 ;

if d > 0

and establish stability of synchronization.

}

FIG. 6. (Color online) Mutually coupled Ro¨ssler attractor (x ¼ 1, b ¼ 0.15, C ¼ 0.2, d ¼ 10). (a) 3D plot of oscillator-1 (original size) and oscillator-2 (amplified and inverted). (b) Plot of x1 vs y1 confirm AS (a ¼ 2, b ¼ 1) and (c) the time series of (x1, y1). (d) Plot of errors ei.


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1 ðe1 þ ae2 Þ; 2b 1 ðe1 e2 e3 Þ; Uy 2 ¼ 2b 1 a a 1 U y 3 ¼ e1 þ x2 2 ; 2b b b

Chaos 21, 013110 (2011)

Uy 1 ¼

(23b)

where ei (i ¼ 1, 2, 3) is the error function that may be easily derived taking a difference of Eqs. (21) and (22). The controllers as defined above then satisfy _ VðeÞ ¼ e1 2 e2 2 e3 2 :

(24)

Numerical results of mutually coupled Sprott systems are shown in Fig. 7. The 2D projections of the attractors in Fig. 7(a) shows them in AS but one of them is enlarged since (a ¼ 2, b ¼ 1). Plot of x1 vs y1 in Fig. 7(b) and time series in Fig. 7(c) reconfirms the AS state. The error function, ei ¼ (yi þ 2xi), i ¼ 1, 2, 3, in Fig. 7(d) shows convergence to zero thereby confirming amplification by a scale of 2. Figure 8 shows two Sprott circuits, OSC1 and OSC2: Op-amp U1–U4 (U5–U8) and multiplier A1 (A2) with resistances R1–R8 (R9–R16) and capacitances C1–C3 (C4–C6) represent the OSC1 (OSC2). The coupling circuit is same as shown in Fig. 3, which uses Op-amp U9–U14 and resistance R17–R31. But for bidirectional interactions, the continuity between the oscillators, OSC1 and OSC2, and their mutual coupling are maintained via different in-coming and out-going terminals in the circuit and input resistances R32–R44. Measurements of output voltages of OSC1 and OSC2 (analogs of x1,2 and y1,2) are made at the outputs of (U1, U3) and (U5, U7), respectively, using the same digital oscilloscope as above. Experimental results in Fig. 9 are clearly seen in good agreements with the numerical results in Fig. 7. By appropriate choice of (a, b), we also realized CS and attenuation. Thus we are able to derive a method of mutual scaling in coupled oscillators with experimental support. To our best knowledge, this issue was not explored earlier.

}

FIG. 8. Mutually coupled Sprott circuit. Integrators U1–U4 (U5–U8), multiplier A1 (A2) with resistances R1–R8 (R9–16), and capacitances C1–C3 (C4–C6) represent OSC1 (OSC2). The coupling circuit is same as in Fig. 3. Components values (all of 1% tolerance) are noted in the circuit.

V. COUPLED NETWORK OF OSCILLATORS

We extend the NOLC based bidirectional coupling to a network of N-oscillators. In the case of N-coupled chaotic oscillators, (25) x_ i ¼ f ðxi Þ þ Uxi ; i ¼ 1; 2; …; N

} }

FIG. 7. (Color online) Mutually coupled Sprott system (a ¼ 0.225). (a) 2D attractors of oscillator-1 and oscillator-2 (amplified and inverted), (b) plot of x1 vs y1 confirm AS (a ¼ 2), (c) time series of (x1, y1), and (d) plot of ei.

FIG. 9. (Color online) Oscilloscope pictures. Upper row: 2D attractors of one oscillator at left, its amplified and inverted version at the middle; axes in same scale. Output voltages of U1 (500 mV/div.) vs U5 (1 V/div.) at right confirm AS. Lower row: time series measured at U3 (upper) of OSC1 and U7 (below) of OSC2 also confirm AS and amplification.


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where xi ¼ [xi, yi, zi]T is the state vector and f (xi) [ Rn is the flow of the ith oscillator in uncoupled state. The controller Uxi establishes the coupling between the ith and the rest of the oscillators in the network. The error function of the network is defined by exj ¼ ðN 1ÞxN aN

N 1 X xi i

eyj ¼ ðN 1ÞyN aN

N 1 X yi i

ezj ¼ ðN 1ÞzN aN

ai ai

N 1 X zi i

ai

; ;

(26)

;

where e ¼ ½exj ; eyj ; ezj T is the error vector between the jth and the Nth oscillators, j ¼ 1, 2, …, (N 1); ai (i ¼ 1, 2, …, N) is the scaling factor of the ith system. Now, we define the Lyapunov function as usual by Eq. (4) for the network. For a set of mutually coupled N-chaotic oscillators, the stable synchronization state for one of the state variables of the network is defined by x1 x2 xN ¼ ¼ ¼ a1 a2 aN or a2 aN aN x2 ¼ x1 ; x3 ¼ x1 ; …; xN ¼ x1 : a1 a1 a1

}

(27)

The complete definition of the stable synchronized state includes similar relation for other state variables of the network. The desired stable synchronization state is realized _ when the LFS limit VðeÞ < 0 is satisfied. This LFS condition eventually defines the NOLC for N-coupled oscillators. The NOLC scheme is elaborated with an example of N ¼ 4 mutually coupled oscillators. A schematic diagram of the network is shown in Fig. 10(a). To develop the network, we first take two oscillators, OS1 and OS2, and insert the first NOLC based coupling denoted by LFS-1 in a similar way as already explained in the previous example of two mutually coupled oscillators. If we intend to increase the size of the network to three oscillators, we add OS3, then insert the NOLC based coupling denoted by LFS-2 that establish additional coupling between the new pairs of (OS1, OS3) and (OS2, OS3). Next we add a fourth oscillator OS4 and insert LFS-3 to make additional mutual coupling between the pairs (OS1, OS4), (OS2, OS4), and (OS3, OS4). In a similar way, we can proceed step by step to develop NOLC based coupling for a network of N-oscillators. As an example, we describe the LFS based NOLC for a network of four Sprott oscillators. Details are given in the Appendix. We present the numerical results of the network in Figs. 10(b) and 10(c). We first assume that the network has two oscillators, OS1 and OS2, which are only coupled and the rest are uncoupled. As seen in the upper row of Fig. 10(b), OS2 attractor is amplified compared to OS1 as expected for a choice of a1 ¼ 1, a2 ¼ 2. We expect a CS state that is con-

FIG. 10. (Color online) Network of four oscillators. (a) Schematic diagram of four bidirectionally coupled oscillators: OS1 and OS2 in innermost region, OS3 in intermediate region, and OS4 in outermost region, (b) 2D attractors of four Sprott oscillators after synchronization, and (c) synchronization states in 3 3 arrays of four Sprott oscillators.

firmed from the upper row plot of x1 vs x2 in Fig. 10(c). When we add the OS3 to the network and consider (a1 ¼ 1, a2 ¼ 2, a3 ¼ 2), OS3 attractor is expected to be amplified twice the size of OS1 and equal to the size of OS2 attractor but inverted to both since a3 ¼ 2 as shown in the lower row left in Fig. 10(b). The OS1 and OS2 remain in CS state (cf. plot x1 vs x2 in the first row) while OS3 is in AS state with both OS1 and OS2 as shown in plots x1 vs x3 and x2 vs x3 in the second row of Fig. 10(c). Finally, when we add OS4 and consider (a1 ¼ 1, a2 ¼ 2, a3 ¼ 2, a4 ¼ 0.5), we observe that the OS4 attractor is smaller in size (since a4 ¼ 0.5) compared to other attractors in Fig. 10(b). The OS4 maintains a CS state with OS1 and OS2 but in AS state with OS3 as shown in the third row (cf. plots of x1 vs x4, x2 vs x4, and x3 vs x4) of Fig. 10(c). The design of coupling is thus completed to target the desired synchronization regimes in a network of four chaotic oscillators.


013110-9


Chaos 21, 013110 (2011)

VI. CONCLUSIONS

We explored a nonlinear open loop controller (NOLC) based coupling design using Lyapunov function stability (LFS) for engineering synchronization in chaotic oscillators. We described a general scheme applicable for a unidirectional as well as bidirectional coupling mode, which is capable of realizing a desired response such as CS, AS states in chaotic systems. We introduced a scaling factor in the definition of coupling that allows amplification or attenuation of one attractor relative to another. To the best of the knowledge of the authors, the mutual coupling using LFS based controller with a scaling was not explored earlier. We described the theoretical details of the method about how to design the NOLC based coupling and illustrated with numerical examples of several dynamical systems, namely, Lorenz, Ro¨ssler, and Sprott systems. Most importantly, we physically implemented the NOLC based coupling design for both unidirectional and bidirectional modes in electronic circuits using Sprott oscillators. We showed that a physical realization of the NOLC based coupling, although appears difficult, is possible for engineering synchronization in electronic circuits. Finally, we generalized the theory for a network of many oscillators and presented a numerical example of four oscillators.

plification and appropriately select the controllers Uxi, Uyi, _ < 0. Next, we couple a third Uzi for i ¼ 1, 2, so that VðeÞ oscillator OS3 when an additional set of error functions appear

x1 x2 ; þ a1 a2 y1 y2 ey2 ¼ 2y3 a3 ; þ a1 a2 z1 z2 ez2 ¼ 2z3 a3 þ : a1 a2 ex2 ¼ 2x3 a3

Again, we substitute y2 ¼ a2 y1/a1 and y3 ¼ a3 y1/a1 in _ the nonlinear term of VðeÞ and then design the LFS-2 to _ ensure VðeÞ < 0 for stable synchronization in three oscillators. Finally, we add a fourth oscillator OS4 when the error function has one more additional set of error functions

x1 x2 x3 þ þ ; a1 a2 a3 y1 y2 y3 ; þ þ ey3 ¼ 3y4 a4 a1 a2 a3 z1 z2 z3 : ez3 ¼ 3z4 a4 þ þ a1 a2 a3 ex3 ¼ 3x4 a4

ACKNOWLEDGMENTS

This work is partially supported by the Board of Research in Nuclear Sciences/Department of Atomic Energy, India under Grant No. 2009/34/26/BRNS and by the Department of Science and Technology, India under Grant No. SR/S2/HEP-03/2005. APPENDIX: DESIGN OF COUPLING FOR A NETWORK OF OSCILLATORS

z_i ¼ xi þ

y2i

m ðexði1Þ þ aeyði1Þ Þ iði 1Þ X 1 ðexi þ aeyi Þ; ai ði þ 1Þaðiþ1Þ 1i