Generalized analytical solutions and experimental confirmation of complete synchronization in a class of mutually coupled simple nonlinear electronic circuits

Chaos, Solitons and Fractals 113 (2018) 294–307

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

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Generalized analytical solutions and experimental confirmation of complete synchronization in a class of mutually coupled simple nonlinear electronic circuits G. Sivaganesh a, A. Arulgnanam b,∗, Seethalakshmi c a

Department of Physics, Alagappa Chettiar College of Engineering & Technology, Karaikudi 630 004, Tamilnadu, India Department of Physics, St. John’s College, Palayamkottai 627002, Tamilnadu, India, Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli 627012, Tamilnadu, India c Department of Physics, M. D. T Hindu College, Tirunelveli 627010, Tamilnadu, India, Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli 627012, Tamilnadu, India b

a r t i c l e

i n f o

Article history: Received 20 October 2017 Revised 5 March 2018 Accepted 6 June 2018

2010 MSC: 05.45.Xt 05.45.-a Keywords: Mutual coupling Synchronization Master stability function

a b s t r a c t A novel generalized analytical solution for the normalized state equations of a class of mutually coupled chaotic systems is presented. To the best of our knowledge, for the first time synchronization dynamics of mutually coupled chaotic systems is studied analytically. The coupled dynamics obtained through the analytical solutions has been validated by numerical simulation results. Furthermore, we provide a suitable condition for the occurrence of synchronization in mutually coupled, second-order, non-autonomous chaotic systems through the analysis of the difference system on the stability of fixed points. The bifurcation of the eigenvalues of the difference system as a function of the coupling parameter in each of the piecewise-linear regions, revealing the existence of stable synchronized states, is presented. The stability of synchronized states in each coupled system discussed in this article is studied using the Master Stability Function. Finally, the electronic circuit experimental results confirming the analytical and numerical results are presented.

1. Introduction Synchronization of chaotic systems offers encouraging applications in secure transmission of signals [1–3]. Several chaotic systems have been studied for synchronization following the MasterSlave method introduced by Pecora and Carroll [4]. A complete study on the different types of synchronization was presented by Boccaletti et al. [5]. The stability of the synchronized states becomes equally important as compared to the type of synchronization observed in the coupled system. The Master Stability Function(MSF) approach introduced by Pecora and Carroll provided the necessary condition for synchronization of coupled identical chaotic systems [6]. The MSF approach has been used to study the stability of the synchronized states in certain unidirectionally coupled simple chaotic systems [7–9]. Chaos synchronization has been observed in electronic circuit experiments for numerous chaotic systems [10–16].

∗

Corresponding author. E-mail address: [email protected] (A. Arulgnanam).

https://doi.org/10.1016/j.chaos.2018.06.001 0960-0779/© 2018 Elsevier Ltd. All rights reserved.

© 2018 Elsevier Ltd. All rights reserved.

The existence of chaotic attractors in a low-dimensional circuit system was observed by Murali et al. [11]. The Chua’s diode, a three-segmented piecewise-linear nonlinear element, has been the only nonlinear element used in the circuit. Several simple circuit systems with significant chaotic dynamics have been introduced subsequently [17–20]. Owing to the mathematical simplicity of these state equations, explicit analytical solutions explaining their chaotic dynamics have been obtained [18,19]. Furthermore, explicit analytical solutions for the state equations of unidirectionally coupled simple chaotic systems and strange non-chaotic systems have been reported recently in the literature [8,9,21]. However, explicit analytical solutions explaining the synchronization dynamics observed in mutually coupled chaotic systems have not been reported in the literature. In this article, we aim to provide an explicit analytical solution to the state equations of mutually coupled simple chaotic systems. Analytical solutions for mutually coupled simple identical chaotic systems have been developed and their synchronization dynamics is studied through phase-portraits. Furthermore, a simple method to identify the stability of synchronization in mutually coupled chaotic systems is proposed. The method is based on a study of the eigenvalues of the difference systems corresponding to the mutually coupled systems. A simi-

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295

Fig. 1. Schematic circuit realization of the sinusoidally forced series LCR circuit with the nonlinear element NR connected parallel to the capacitor.

lar study has been carried out in unidirectionally - coupled chaotic systems to identify the phenomenon of complete synchronization [9,21,22]. However, in this article we confirm that the study on the eigenvalues of the difference system of mutually coupled chaotic systems can provide a suitable condition for the synchronization of the systems. The reliability of the results thus obtained has been confirmed by the Master Stability Functions obtained for the corresponding systems which provides the necessary condition for complete synchronization. Furthermore, the stability of the fixed points corresponding to the difference system indicating the convergence of the trajectories within the synchronization manifold is studied through phase plots. We discuss the complete synchronization phenomenon in four different simple circuit systems, namely the Murali–Lakshmanan–Chua, the Variant of Murali–Lakshmanan– Chua, series LCR circuit with a simplified nonlinear element and the parallel LCR circuit with simplified nonlinear element circuits. The circuit systems are categorized under two classes, the series LCR and Parallel LCR circuit systems, each with two different nonlinear elements. The following are discussed in this article. In Section 5.2 we explain the two different nonlinear elements and their role in the series and parallel LCR circuits. The normalized state equations of the circuit systems which are mutually coupled are also presented. In Section 3 we present generalized explicit analytical solutions to the mutually coupled series and parallel LCR chaotic systems exhibiting complete synchronization in their dynamics. Furthermore, the unsynchronized and synchronized states of the coupled systems are studied using the analytical solutions obtained. Section 4 deals with the numerical study on complete synchronization and its stability using the MSF approach for series LCR circuits with different nonlinear elements while Section 5 with that of the parallel LCR circuits. The experimental results are presented for each case in Section 4 and 5. Finally, a discussion on the suitable condition for synchronization in mutually coupled second-order chaotic systems based on the eigenvalues of the difference system is presented in Section 6.

Fig. 2. (a) Schematic realization of the Chua’s diode constructed from two operational amplifiers and six linear resistors and their (b) (v − i ) characteristics with one negative inner slope Ga = −0.76 mS, two negative outer slopes Gb = −0.41 mS and the break points B p = ± 1.0 V, respectively.

ment called as the Murali–Lakshmanan–Chua circuit is given as

C

dv = i L − g( v ) , dt

(1a)

L

diL = −RiL − Rs iL − v + F sin(t ), dt

(1b)

where g(v) is the mathematical form of the voltage-controlled, piecewise-linear resistor given by



2. Circuit equations A sinusoidally forced series LCR circuit with a piecewise-linear nonlinear element NR connected parallel to the capacitor is shown in Fig. 1. The nonlinear element NR can be of two types as shown in Fig. 2. The first type of nonlinear element shown in Fig. 2(a) is the Chua’s diode introduced by Chua et al. and is constructed using two operational amplifiers and six linear resistors. The corresponding (v )-(i ) characteristics shown in Fig. 2(b) represent two negative outer slope regions and one negative inner slope region. The state equations of the circuit with the Chua’s diode as the nonlinear ele-

g( v ) =

Gb v + (Ga − Gb ) ifv ≥ 1 Ga v if|v| ≤ 1 Gb v − (Ga − Gb ) ifv ≤ −1

(2)

The second type of nonlinear element called the simplified nonlinear element shown in Fig. 3, introduced by Arulgnanam et al. has been constructed using only one operational amplifier and three linear resistors. This nonlinear element has been identified to be the simplest piecewise-linear element constructed using fewer circuit elements. The state equations of a series LCR circuit with a simplified nonlinear element connected parallel to the capacitor is

296

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Fig. 4. Schematic circuit realization of the sinusoidally forced parallel LCR circuit with the nonlinear element NR connected parallel to the capacitor.

The piecewise-linear function g(x) representing the threesegmented piecewise nonlinear element is given by



bx + (a − b) if$x ≥ 1$ ax if$|x| ≤ 1$ bx − (a − b) if$x ≤ −1$

g( x ) =

(5)

where σ = (β + νβ ) and β = (C/LG2 ), ν = GRs , a = Ga /G, b = Gb /G, f1 = (F1 β /B p ), ω1 = (1C/G ), G = 1/R. The values of the normalized parameters of the system depend on the circuit parameters. For the circuit with a simplified nonlinear element, ν = 0 and hence σ = β . The circuit said above can be mutually coupled with another circuit having the same nonlinear element. In this case, the normalized state variables of the two systems can be represented as   (x, y)) and (x , y ), respectively. The normalized state equations of the mutually coupled systems under coupling of the x variables is given as 

Fig. 3. (a) Schematic realization of the simplified nonlinear element constructed using one operational amplifier and three linear resistors and their (b) (v − i ) characteristics with a negative inner slope Ga = −0.56 mS, two positive outer slopes Gb = +2.5 mS and the break points B p = ± 3.8 V, respectively.

given by

C

L

dv = i L − g( v ) , dt diL = −RiL + F sin(t ), dt

x˙ = y − g(x ) +  (x − x ),

(6a)

y˙ = −σ y − β x + f1 sin(θ ),

(6b)

θ˙ = ω1 ,

(6c)

   x˙  = y − g(x ) +  (x − x ),

(6d)

   y˙  = −σ y − β x + f2 sin(θ ),

(6e)

(3a)

θ˙  = ω2 ,

(6f) 

(3b)

The function g(v) is represented the same as given in Eq. (2) except that the circuit parameters Ga and Gb are different for the two elements. Thus the nonlinear element NR determines the chaotic dynamics of the circuit for a proper choice of the other circuit parameters. The state equations of the series LCR circuits with different nonlinear elements given in Eqs. (1) and (3) can be written in a general form as

where g(x ) is the piecewise-linear function of the second system given by





g( x ) =





bx + (a − b) if$x ≥ 1$   ax if$|x | ≤ 1$   bx − (a − b) if$x ≤ −1$

(7)

A sinusoidally forced parallel LCR circuit with a nonlinear element NR connected parallel to the capacitor is shown in Fig. 4. The nonlinear element could be either a Chua’s diode or a simplified nonlinear element. The generalized state equations for a parallel LCR circuit with a three-segmented nonlinear element can be written as

x˙ = y − g(x ),

(4a)

y˙ = −σ y − β x + f sin(θ ),

(4b)

C

dv 1 = (F sin(t ) − v ) − iL − g(v ), dt R

(8a)

θ˙ = ω.

(4c)

L

diL = v, dt

(8b)

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297

√ −(A )± (A2 −4B ) m1,2 = corresponding to the D0 region may be ei2 ther two real values or a pair of complex conjugates. When the roots are a pair of complex conjugates, the state variables y(t) and x(t) are

y(t ) = eut (C1 cosvt + C2 sinvt ) + E1 + E2 sinω1 t +E3 cosω1 t, Fig. 5. Analytically obtained (a) one-band and (b) double-band chaotic attractors of the Murali–Lakshmanan–Chua circuit at the control parameter values f = 0.1 and f = 0.14, respectively.

where the nonlinear function g(v) is as given in Eq. (2). The normalized state equations are given as

x˙ = f sin(θ ) − x − y − g(x ), y˙ =

(9a)

β x,

(9b)

θ˙ = ω.

(9c)

The normalized circuit parameters of the circuit are different when different nonlinear elements are used. The normalized state equations of the mutually coupled system under x-coupling is 

x˙ = f1 sin(θ ) − x − y − g(x ) +  (x − x ),

(10a)

β x,

(10b)

θ˙ = ω1 ,

(10c)

y˙ =

x(t ) =

1

β

(11a)

(−σ y − y˙ + f1 sinω1 t ), √

(11b)

(4B−A2 )

where, u = −A and m1,2 = u ± iv. When the roots are 2 , v= 2 two real values then the state variables are

y(t ) = C1 em1 t + C2 em2 t + E1 + E2 sin ω1 t +E3 cos ω1 t, x(t ) =

1

β

(12a)

(y˙ ).

(12b)

The fixed points (k1 , k2 ) corresponding to the outer regions D ± 1 where the function g(x ) = bx ± (a − b) are, k1 = ±((a − b)σ / β + bσ ) and k2 = ±(β (b − a ) / β + bσ ). When the roots m3,4 = √ −(A )± (A2 −4B ) are a pair of complex conjugates, the state variables 2 are

y(t ) = eut (C3 cosvt + C4 sinvt ) + E3 sin(ω1 t ) +E4 cos(ω1 t )± , x(t ) =

1

β

(13a)

(y˙ ).

(13b)

When the roots are two real values, the state variables are      x˙  = f2 sin(θ ) − x − y − g(x ) +  (x − x ),

y˙  =



βx ,

θ˙ = ω2 . 

(10d)

y(t ) = C3 em3 t + C4 em4 t + E3 sin ω1 t +E4 cos ω1 t ± ,

(10e) x(t ) = (10f)

3. Explicit analytical solutions In this section, we present the generalized analytical solutions for the generalized state equations of the x-coupled series and parallel LCR circuits shown in Eqs. (6) and (10). It has to be noted that the chaotic dynamics of the mutually coupled systems evolves independently with time, controlling the dynamics of each other with increase in the coupling strength from zero. Hence, the state equations of the coupled systems can be solved explicitly with the consideration that identical piecewise-linear regions control each other. The method of obtaining explicit analytical solutions by framing difference systems evolving with time is explained in this section. The analytical solutions thus obtained were used to generate the synchronization dynamics of the coupled systems. 3.1. Series LCR circuits An explicit analytical solution can be presented for the normalized state equations of the series LCR circuits given in Eq. (4) for each of the piecewise-linear regions D0 and D ± 1 . In both the nonlinear elements, the central region has been taken as D0 and the outer regions in the positive and negative voltage regime as D+ and D− , respectively. For both type of nonlinear elements, the fixed point corresponding to the central region is (0,0). The roots

1

β

(14a)

(y˙ ),

(14b)

where = (b − a ) and + , − correspond to D+1 and D−1 regions, respectively. When the coupling parameter  = 0, the state equations of the   two systems with state variables (x, y) and (x , y ) become identical. Hence, in the uncoupled state the explicit analytical solutions to the normalized state equations of the two systems remain the same. The solutions to the state variables x, y given from   Eqs. (11) to (14) hold good for the state variables x , y except that the second system operates with a different set of initial conditions. For the values of the coupling parameter  > 0, the x-coupled systems represented by Eq. (6) are mutually bound and the dynamics of each system is controlled by the other. Now, we present explicit analytical solutions for the mutually coupled system for coupling strengths greater than zero. Because the normalized state equations are piecewise-linear, we couple each piecewise-linear identical region of the first and the second system to form a new set of equations corresponding to the difference system. The difference system obtained by coupling the two set of equations given in Eq. (6) is as follows: 

x˙∗ = y∗ − (g(x ) − g(x )) − 2 x∗ ,

(15a)

y˙∗ = −σ y∗ − β x∗ + f1 sin(ω1 t ) − f2 sin(ω2 t ). 



(15b) 

where = ( x − x ), = (y − y ) and g(x ) − g(x ) = take the values ax∗ or bx∗ depending on the corresponding region of x∗

y∗

g( x ∗ )

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operation of the two systems. From these state variables (x∗ , y∗ ), the state variables of the second system evolve as 

x = x − x∗ ,

(16a)



y = y − y∗ .

(16b)

Now, the state variables x(t), y(t) of the first system can be obtained by arriving at another difference system given as  x˙† = y† − (g(x ) − g(x )) − 2 x† ,

(17a)

y˙† = −σ y† − β x† + f2 sin(ω2 t ) − f1 sin(ω1 t ), 



(17b)



where x† = (x − x ), y† = (y − y ) and g(x ) − g(x ) = g(x† ) take the values ax† or bx† depending on the corresponding region of operation of the two systems. However, the dynamics of the second system evolves with initial conditions given in Eq. (16), enabling the coupling of the two systems. From these state variables (x†, y†), the state variables of the first system evolve as 

x = x − x† ,

(18a)



y = y − y† .

(18b)

The state variables of the second and the first systems can be obtained by using Eqs. (16) and (18) through an estimation of the variables of the difference system (x∗ , y∗ ) and (x†, y†), respectively. One can easily establish that a unique equilibrium point (x0(∗,†) , y0(∗,†) ) exists for Eqs. (15) and (17) in each of the following three subsets (∗,† ) D+1 = {(x(∗,† ) , y(∗,† ) )|x(∗,† ) > 1}|P+(∗,† ) (∗,† )

(∗,† )

= ( 0, 0 ),

(∗,† ) D−1 = {(x(∗,† ) , y(∗,† ) )|x(∗,† ) < −1}|P−(∗,† )

= ( 0, 0 ).

D0

= { (x

(∗,† )

,y

(∗,† )

)||x

(∗,† )

⎫ ⎬

= (0, 0 ),⎪

| < 1}|O

⎪ ⎭

(19)

Since the two systems differ only by their initial conditions, the origin (0,0) becomes the fixed points for all the three regions of the difference systems, as expected. The stability of the fixed points given in Eq. (19) can be calculated from the stability ma  trices. In the first case, g(x) and g(x ) take the values ax and ax respectively, corresponding to the central region in the (v − i ) characteristics of the nonlinear element, which has been taken as the (∗,† ) D0 region of the difference systems. The stability-determining (∗,† )

eigenvalues in the D0 matrix (∗,† )

J0



=

− ( a + 2 )

1

−β

−σ

region are calculated from the stability

(20) 

J±



=

− ( b + 2 )

1

−β

−σ



(21)

The eigenvalues of the difference systems are thus determined by the coupling parameters in all the three piecewiselinear regions. We first present an explicit analytical solution for the difference system given in Eq. (15) for each of the piecewise-linear regions. A similar solution can be obtained for the difference system given in Eq. (17). The solutions of those equations are, [x∗ (t; t0 , x∗0 , y∗0 ), y∗ (t; t0 , x∗0 , y∗0 )]T and †

†

†

†

[x† (t; t0 , x0 , y0 ), y† (t; t0 , x0 , y0 )]T for which the initial conditions are written as (t,

x∗ ,

y∗ )

=

3.1.1. Region : D∗0

(t0 , x∗0 , y∗0 )

and (t, x†, y†) =

(t0 , x†0 , y†0 ),





In this region g(x) and g(x ) take the values ax and ax , respectively. Hence the normalized equations obtained from Eq. (15) are

x˙∗ = y∗ − (a + 2 )x∗

(22a)

y˙∗ = −σ y∗ − β x∗ + f1 sin(ω1 t ) − f2 sin(ω2 t )

(22b)

Differentiating Eq. (22b) with respect to time and using Eqs. (22a) and (22b) in the resultant equation, we obtain

y¨ ∗ + Ay˙ ∗ + By∗ = (a + 2 ) f1 sinω1 t + f1 ω1 cosω1 t −

(a + 2 ) f2 sinω2 t − f2 ω2 cosω2t

(23)

A = σ + a + 2 and B = β √ + σ (a + 2 ). The roots of −(A )± (A2 −4B ) Eq. (23) are given by m1,2 = . Since the roots m1, 2 2 depend on the coupling parameters, the orientation of the trajectories around the fixed point changes as the coupling parameter is varied. For (A2 > 4B), the roots m1 and m2 are real and distinct. The general solution to Eq. (23) can be written as where,

y∗ (t ) = C1 em1 t + C2 em2 t + E1 sin(ω1 t ) + E2 cos(ω1 t ) +E3 sin(ω2 t ) + E4 cos(ω2 t )

(24)

where C1 and C2 are integration constants and

E1 =

f 1 ω1 2 ( A − a −  ) + f 1 B ( a +  ) A2 ω1 2 + ( B − ω1 2 )2

(25a)

E2 =

f1 ω1 ((B − ω1 2 ) − A(a +  )) A2 ω1 2 + ( B − ω1 2 )2

(25b)

E3 = −

f 2 ω2 2 ( A − a −  ) + f 2 B ( a +  ) A2 ω2 2 + ( B − ω2 2 )2

(25c)

E4 = −

f2 ω2 ((B − ω2 2 ) − A(a +  )) A2 ω2 2 + ( B − ω2 2 )2

(25d)

Differentiating Eq. (24) and using it in Eq. (22b), we get

x∗ (t ) = (



In the second case, g(x) and g(x ) take the values bx ± (a − b) and  (∗,† ) bx ± (a − b) respectively. Hence in the D±1 region, the stabilitydetermining eigenvalues are calculated from the stability matrix (∗,† )

respectively. From the solution x∗ (t), y∗ (t) and x†(t), y†(t) thus ob  tained the state variables x (t ), y (t ) and x(t), y(t) can be found using Eqs. (16) and (18), respectively. The explicit analytical solutions for the piecewise-linear regions of the difference system corresponding to Eq. (15) are as follows:

1

β

)(y˙∗ − σ y∗ + f1 sin(ω1 t ) − f2 sin(ω2 t ))

(26)

The constants C1 and C2 are given as

C1 =

e−m1 t0 {(−σ y∗0 − β x∗0 − m2 y∗0 ) m1 − m2 −(ω1 E1 − m2 E2 )cosω1 t0 +( f1 + ω1 E2 + m2 E1 )sinω1 t0 −(ω2 E3 − m2 E4 )cosω2 t0

+(ω2 E4 + m2 E3 − f2 )sinω2 t0 } e−m2 t0 C2 = {(−σ y∗0 − β x∗0 − m1 y∗0 ) m2 − m1 −(ω1 E1 − m1 E2 )cosω1 t0 +( f1 + ω1 E2 + m1 E1 )sinω1 t0 −(ω2 E3 − m1 E4 )cosω2 t0 +(ω2 E4 + m1 E3 − f2 )sinω2 t0 } From the results of y∗ (t), x∗ (t) and y(t), x(t) obtained from Eqs. (24),   (26) and Eq. (12), x (t ) and y (t ) can be obtained from Eq. (16).

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299

For (A2 < 4B), the roots m1 and m2 are a pair of complex con√ (4B−A2 ) jugates given as m1,2 = u ± iv, with u = −A and v = . The 2 2 general solution to Eq. (23) can be written as

For C2 < 4D, the roots are a pair of√ complex conjugates given (4D−C 2 ) as m3,4 = u ± iv, with u = −C . Hence the general 2 and v = 2 solution to Eq. (30) can be written as

y∗ (t ) = eut (C3 cosvt + C4 sinvt ) + E5 sinω1 t + E6 cosω1 t

y∗ (t ) = eut (C3 cosvt + C4 sinvt ) + E5 sinω1 t + E6 cosω1 t

+E7 sinω2 t + E8 cosω2 t

where the constants E5 , E6 , E7 , and E8 are the same as that of E1 , E2 , E3 , and E4 given in Eq. (24), respectively. Differentiating Eq. (27) and using it in Eq. (22b) we get,

x∗ (t ) = (

1

β

)(−y˙∗ − σ y∗ + f1 sin(ω1 t ) − f2 sin(ω2 t ))

(28)

The constants C3 and C4 are given as

C3 =

C4 =

e−ut0

{((σ + β + )sinvt0 + v )cosvt0 v +((ω1 E5 − uE6 )sinvt0 − vE6 cosvt0 )cosω1 t0 −(( f1 + ω1 E6 + uE5 )sinvt0 + vE5 cosvt0 )sinω1 t0 +((ω2 E7 − uE8 )sinvt0 − vE8 cosvt0 )cosω2 t0 −((ω2 E8 + uE7 − f2 )sinvt0 + vE7 cosvt0 )sinω2 t0 } y∗0

x∗0

uy∗0

y∗0

e−ut0

{((−σ y∗0 − β x∗0 − uy∗0 )cosvt0 + vy∗0 )sinvt0 v −((ω1 E5 − uE6 )cosvt0 + vE6 sinvt0 )cosω1 t0 +(( f1 + ω1 E6 + uE5 )cosvt0 − vE5 sinvt0 )sinω1 t0 −((ω2 E7 − uE8 )cosvt0 + vE8 sinvt0 )cosω2 t0 −((ω2 E8 + uE7 − f2 )cosvt0 − vE7 sinvt0 )sinω2 t0 }

From the results of y∗ (t), x∗ (t) and y(t), x(t), obtained from   Eqs. (27), (28) and (11), x (t ) and y (t ) can be obtained from Eq. (16). 3.1.2. Region : D∗±1





In the D∗±1 region, g(x), g(x ) take the values bx ± (a − b), bx ± (a − b), respectively and hence g(x∗ ) = bx∗ . The normalized state equations can be written as

x˙∗ = y∗ − (b + 2 )x∗

(29a)

y˙∗ = −σ y∗ − β x∗ + f1 sin(ω1 t ) − f2 sin(ω2 t )

(29b)

Differentiating Eq. (29b) with respect to time and using Eqs. (29a) and (29b) in the resultant equation, we obtain

y¨ ∗ + C y˙ ∗ + By∗ = (b + 2 ) f1 sinω1 t + f1 ω1 cosω1 t −

(b + 2 ) f2 sinω2t − f2 ω2 cosω2t where C = σ + b + 2 and D = β + σ (b + 2 ). For

(30) C2

>√4D, the (C 2 −4D ) . 2

−C ±

roots are real and distinct, and are given by m3,4 = Hence the general solution to Eq. (30) can be written as

y∗ (t ) = C1 em3 t + C2 em4 t + E1 sin(ω1 t ) + E2 cos(ω1 t ) +E3 sin(ω2 t ) + E4 cos(ω2 t )

(31)

where C1 , C2 , E1 , E2 , E3 , and E4 are the same as their counterparts in the D∗0 region except that the constants A and B are replaced with the constants C and D. Differentiating Eq. (31) and using it in Eq. (29b), we get

x∗ (t ) = (

1

β

+E7 sinω2 t + E8 cosω2 t

(27)

)(−y˙∗ − σ y∗ + f1 sin(ω1 t ) − f2 sin(ω2 t ))

(32)

From the results of y∗ (t), x∗ (t) and y(t), x(t) obtained from Eqs. (31),   (32) and (14), x (t ) and y (t ) can be obtained from Eq. (16).

(33)

The constants C3 , C4 , E5 , E6 , E7 , and E8 are the same as given in D∗0 region except that the constants A and B are replaced with the constants C and D. Differentiating Eq. (33) and using it in Eq. (29b), we get

x∗ (t ) = (

1

β

)(−y˙∗ − σ y∗ + f1 sin(ω1 t ) − f2 sin(ω2 t ))

(34)

From the results of y∗ (t), x∗ (t) and y(t), x(t) obtained from Eqs. (33),   (34) and Eq. (13), x (t ) and y (t ) can be obtained from Eqs. 16. The difference system given by Eq. (17) can be solved in a similar way explained above to obtain the state variables (x†(t), y†(t)). The state variables (x(t), y(t)) of the first system can then be obtained from Eq. (18). It has to be noted that the two chaotic systems evolving independently is controlled by each other as the coupling strength is increased, at every instant of time, simultaneously. Now let us briefly explain how the solution can be generated  in the (x − x ) phase space, to observe complete synchronization in the mutually coupled system. With time (t) being considered as an independent variable, the state variables of the difference system evolves within each coupled piecewise-linear region depending on its initial values. If we start with the initial conditions x∗ (t = 0 ) = x∗0 , y∗ (t = 0 ) = y∗0 in the D∗0 region at time t = 0, the arbitrary constants C1 and C2 get fixed. Thus x∗ (t) evolves as given  by Eq. (26) up to either t = T1 , when x∗ (T1 ) = 1 or t = T1 when 

x∗ (T1 ) = −1. The next region of operation (D∗+1 or D∗−1 ) thus depends on the value of x∗ in the D∗0 region at that instant of time. Similarly, the state variables of the first system evolves at the same   instant of time as x(t ) = x (t ) − x† (t ) and y(t ) = y (t ) − y† (t ), respectively. As the trajectory enters into the next region of interest, the arbitrary constant corresponding to that region could be evaluated, with the initial conditions to that region being either   (x∗0 (T1 ), y∗0 (T1 )) or (x∗0 (T1 ), y∗0 (T1 )), which is determined by the result of the second difference system given by Eq. (18). During each region of operation, the state variables of the second system   evolves as x (t ) = x(t ) − x∗ (t ) and y (t ) = y(t ) − y∗ (t ), respectively. Hence, the dynamical state of each system at every instant of time depends on each other. The procedure can be continued for each successive crossing. In this way, explicit solutions can be obtained in each of the piecewise-linear regions of the two systems. The solution obtained in each region has been matched across the boundaries and used to generate the synchronization dynamics. From the analytical solutions obtained above, the synchronization dynamics of mutually coupled series LCR circuits, under xcoupling can be studied. First we present the complete synchronization phenomenon exhibited by the coupled-circuits with a Chua’s diode as the nonlinear element, called the MLC circuit. The MLC circuit exhibits two prominent chaotic attractors at the amplitude of the external force f = 0.1 and f = 0.14, respectively, as shown in Fig. 6. The mutually coupled systems, which are initially unsynchronized for the coupling strength  = 0, become completely synchronized at higher values of the coupling strength. The trajectories of the two systems enter into a stable synchronization manifold and exist thereafter. Fig. 7 shows the unsynchronized state for  = 0, the completely synchronized state for  = 0.1 and  their corresponding time series plots of the state variable x, x , respectively. A similar study can be made for the coupled systems each with a simplified nonlinear element. The forced series LCR circuit with a

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Fig. 6. Analytically obtained synchronization dynamics of the MLC circuit for cou pled double-band chaotic attractors; a(i) unsynchronized state in (x − x ) phase  plane and (ii) the corresponding time series of the signals x (blue line) and x (red  line) for  = 0; b(i) synchronized state in the (x − x ) phase plane and (ii) the cor responding time series of the signals x (blue line) and x (red line) for  = 0.1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. Analytically obtained synchronization dynamics of the forced series LCR circuit with a simplified nonlinear element circuit: a (i) unsynchronized state of the  coupled systems in x − x phase plane and (ii) time series of the signals x (blue   line) and x (red line) for  = 0; b(i) synchronized state in x − x phase plane and  (ii) time series of the signals x (blue line) and x (red line) for  = 0.12. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. Analytically obtained chaotic attractors at the control parameter values (a) f = 0.39 and (b) f = 0.411 for the Variant of Murali–Lakshmanan–Chua circuit.

Fig. 7. Analytically obtained chaotic attractor of the series LCR circuit with a simplified nonlinear element for the control parameter value f = 0.31.

simplified nonlinear element exhibits a prominent chaotic attractor at the value of the forcing amplitude f = 0.31 as shown in Fig. 7. The synchronization dynamics of the x-coupled system obtained from the analytical solutions is shown in Fig. 8. The unsynchronized state for  = 0 and the complete synchronization state for  = 0.12 along with the time series of the signals indicating synchronization are shown in Fig. 8. 3.2. Parallel LCR circuits An explicit analytical solution to the x-coupled, normalized state equations of the mutually coupled parallel LCR circuits given in Eq. (10) can be obtained in a similar way as that of the series LCR circuits. The explicit analytical solutions to the state equations of the uncoupled system given in Eq. (9) can be summarized as follows. In the D0 region, √ g(x ) = ax with (0,0) as the fixed point. The − ( A )±

Fig. 10. Analytically obtained synchronization dynamics of the Variant of Murali– Lakshmanan–Chua circuit. a (i) Unsynchronized state of the coupled systems in the   x − x phase plane and (ii) time series of the signals x (blue line) and x (red line)  for  = 0; b(i) synchronized state in the x − x phase plane and (ii) time series of  the signals x (blue line) and x (red line) for  = 0.09. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(A2 −4B )

roots m1,2 = are either two real values or a pair of 2 complex conjugates. When the roots are a pair of complex conjugates, the state variables y(t) and x(t) are

y(t ) = eut (C1 cos vt + C2 sin vt ) + E1 sin ω1 t +E2 cos ω1 t,

x(t ) =

1

β

(y˙ ).

(35a)

(35b)

Fig. 11. Analytically obtained chaotic attractors at the control parameter values (a) f = 0.695 and (b) f = 0.855 for the forced parallel LCR circuit with a simplified nonlinear element.

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301

−g(x† ) − 2 x† , y˙† =

(40a)

β x† ,

(40b) 







where = ( x − x ), = ( y − y ), = (x − x ) and = (y −   y ). The factors g(x∗ ) = g(x ) − g(x ) and g(x† ) = g(x ) − g(x ) take the values ax∗, † or bx∗, † depending on the type of the difference system. The state variables of the two systems can be obtained from Eqs. (16) and (18). The equilibrium points corresponding to all the three regions of the difference systems are as given in Eq. (19). Hence, the origin acts as the fixed point of for all the piecewiselinear regions of the difference systems. The stability of the fixed points i.e. the origin, can be calculated from the stability matrices. ∗,† ∗,† The stability determining eigenvalues in the D0 and D±1 regions are determined from the stability matrices x∗

Fig. 12. Analytically obtained synchronization dynamics of the parallel LCR circuit with a simplified nonlinear element circuit. a (i) Unsynchronized state of the coupled  systems in the x − x phase plane and (ii) time series of the signals x (blue line) and   x (red line) for  = 0; b(i) synchronized state in the x − x phase plane and (ii) time  series of the signals x (blue line) and x (red line) for  = 0.12. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

√ (4B−A2 ) where, u = −A . When the roots are two real val2 and v = 2 ues then the state variables are

y(t ) = C1 em1 t + C2 em2 t + E1 + E2 sin ω1 t +E3 cos ω1 t, x(t ) =

1

β

(36a)

(y˙ ).

(36b)

In outer regions D ± 1 , we have g(x ) = bx ± (a − b) with k1 = 0, k2 = ±(a − b) as the fixed point. When√the roots are a pair of complex conjugates given by m3,4 = y(t) and x(t) are

− ( A )±

(A2 −4B )

2

+E4 cos(ω1 t )± , x(t ) =

1

β

(y˙ ).

(37a)

(37b)

are

y(t ) = C3 em3 t + C4 em4 t + E3 sin ω1 t

x(t ) =

1

β

(y˙ ).

=

and ∗,† J±1 =

(38a)

(38b)

x†



−(a + 2 + 1 ) −1 , β 0



y†

− ( b + 2 + 1 )

−1

β

0

(41)

 .

(42)

The explicit analytical solutions obtained for the mutually coupled parallel LCR circuits can be obtained by solving the difference system equations given by Eqs. (39) and (40). The explicit analytical solutions for the difference system given by Eq. (39) is as follows. 3.2.1. Region : D∗0





In this region, g(x) and g(x ) take the values ax and ax , respectively. Hence the normalized equations obtained from Eqs. (39) are

x˙∗ = −(a + 2 + 1 )x∗ − y∗ + f1 sin(ω1 t )

y˙∗ =

When the roots are two real values then the state variables

+E4 cos ω1 t ± ,



J0∗,†

, the state variables

y(t ) = eut (C3 cos vt + C4 sin vt ) + E3 sin(ω1 t )

y∗

− f2 sin(ω2 t ),

(43a)

β x∗ .

(43b)

Differentiating Eq. (43b) with respect to time and using Eqs. (43a) and (43b) in the resulting equation, we obtain

y¨ ∗ + Ay˙ ∗ + By∗ = f1 sin(ω1 t ) − f2 sin(ω2 t ),

(44)

where, A = a√+ 2 + 1 and B = β . The roots of Eq. (44) is given by − ( A )±

(A2 −4B )

m1,2 = . When (A2 < 4B), the roots m1 and m2 are a 2 pair of complex conjugates given as m1,2 = u ± iv, with u = −A 2 and √ (4B−A2 ) v= . Hence, the general solution to Eq. (44) can be writ2 ten as

y∗ (t ) = eut (C1 cosvt + C2 sinvt ) + E1 sin ω1 t + E2 cos ω1 t +E3 sin ω2 t + E4 cos ω2 t,

(45)

Where = (b − a ) and + , − corresponds to D+1 and D−1 regions, respectively. The solutions given above are also valid for   the state variables of the second system (x , y ) except that the two systems operate with different set of initial conditions. The difference systems obtained from Eq. (10) are

where C1 and C2 are integration constants and

E1 =

f 1 β ( B − ω1 2 ) , A2 ω1 2 + ( B − ω1 2 )2

(46a)

x˙∗ = f1 sin(ω1 t ) − f2 sin(ω2 t ) − x∗ − y∗

E2 =

−Aω1 f1 β , A2 ω1 2 + ( B − ω1 2 )2

(46b)

E3 =

− f 2 β ( B − ω2 2 ) , A2 ω2 2 + ( B − ω2 2 )2

(46c)

E4 =

Aω 2 f 2 β . A2 ω2 2 + ( B − ω2 2 )2

(46d)

−g(x∗ ) − 2 x∗ , y˙∗ =

βx , ∗

(39a)

(39b)

and

x˙† = f2 sin(ω2 t ) − f1 sin(ω1 t ) − x† − y†

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Differentiating Eq. (45) and using it in Eq. (43b), we get

x∗ (t ) =

1

β

(y˙∗ ).

(47)

pair of complex conjugates given as m3,4 = u ± iv, with u = −C 2 and √ 2 v = (4D2−C ) . The state variables y∗ (t), x∗ (t) can be written as

y∗ (t ) = eut (C3 cosvt + C4 sinvt ) + E5 sin ω1 t + E6 cos ω1 t +E7 sin ω2 t + E8 cos ω2 t,

The constants C1 and C2 are

C1 =

C2 =

{(vy∗0 cos vt0 − (β x∗0 − ux∗0 ) sin vt0 ) v +((ω1 E1 − uE2 )sinvt0 − vE2 cosvt0 )cosω1 t0 −((ω1 E2 + uE1 )sinvt0 + vE1 cosvt0 ) sin ω1 t0 +((ω2 E3 − uE4 )sinvt0 − vE4 cosvt0 )cosω2 t0 −((ω2 E4 + uE3 )sinvt0 + vE3 cosvt0 ) sin ω2 t0 },

x∗ (t ) =

{((β x∗0 − ux∗0 ) cos vt0 + vy∗0 sin vt0 ) v −((ω1 E1 − uE2 ) cos vt0 + vE2 sin vt0 )cosω1 t0 +((ω1 E2 + uE1 ) cos vt0 − vE1 sin vt0 ) sin ω1 t0 −((ω2 E3 − uE4 ) cos vt0 + vE4 sin vt0 )cosω2 t0 +((ω2 E4 + uE3 ) cos vt0 − vE3 sin vt0 ) sin ω2 t0 }. 

x (t ) =

β

( ). y˙∗

(49)

The constants C1 and C2 are

C1 =

e−m2 t0 {(β x∗0 − m1 y∗0 ) + (m1 E2 − ω1 E1 ) cos ω1t0 m2 − m1 +(ω1 E2 + m1 E1 ) sin ω1 t0 + (m2 E4 − ω2 E3 ) cos ω2 t0 +(ω2 E4 + m2 E3 ) sin ω2 t0 },

C2 =

e−m1 t0 {(β x∗0 − m2 y∗0 ) + (m2 E2 − ω1 E1 ) cos ω1t0 m1 − m2 +(ω1 E2 + m2 E1 ) sin ω1 t0 + (m1 E4 − ω2 E3 ) cos ω2 t0 

Using the results of y∗ (t), x∗ (t) given in Eqs. (48) and (49), y (t )  and x (t ) can be obtained from Eq. (16).



In this region, g(x) and g(x ) take the values bx ± (a − b) and bx ± (a − b), respectively. Hence the normalized state equations obtained from Eq. (39) are 

x˙∗ = −(b + 2 + 1 )x∗ − y∗ + f1 sin(ω1 t )

y˙∗ =

− f2 sin(ω2 t ),

(50a)

β x∗ .

(50b)

Differentiating Eq. (50b) with respect to time and using Eqs. (50a) and (50b) in the resulting equation, we obtain

y¨ ∗ + C y˙ ∗ + Dy∗ = f1 sin(ω1 t ) − f2 sin(ω2 t ),

1

β

(y˙∗ ).

(53b)

The constants C3 , C4 , E5 , E6 , E7 , and E8 are the same as the constants C1 , C2 , E1 , E2 , E3 , and E4 in the D∗0 region except that the constants A and B are replaced with C and D, respectively. The solutions thus obtained can be used to generate the state variables as explained in the previous section. 4. Mutually coupled series LCR circuits: numerical and experimental results In this section, we present numerical and experimental results for the synchronization dynamics observed in mutually coupled series LCR circuits with two different nonlinear elements. Numerical  studies are performed for x ↔ x coupling of state variables of the systems represented by Eq. (6). Linear stability analysis indicating the bifurcation of fixed points in each piecewise-linear region of the difference systems is presented. The stability of synchronization in each circuit system is studied using the Master Stability Function. Finally, experimental results are presented to confirm the analytical and numerical simulation results. 4.1. Murali–Lakshmanan–Chua circuits

+(ω2 E4 + m1 E3 ) sin ω2 t0 }.

3.2.2. Region : D∗±1

x∗ (t ) =

(53a)

(48)

where C1 and C2 are the integration constants and the constants E1 , E2 , E3 , and E4 are the same as Eq. (46). Differentiating Eq. (48) and using it in Eq. (43b), we get

1

(52b)

+E7 sin(ω2 t ) + E8 cos(ω2 t ),

y∗ (t ) = C1 em1 t + C2 em2 t + E1 sin(ω1 t ) + E2 cos(ω1 t ) +E3 sin(ω2 t ) + E4 cos(ω2 t ),

(y˙∗ ).

y∗ (t ) = C3 em3 t + C4 em4 t + E5 sin(ω1 t ) + E6 cos(ω1 t )

Using the results of given in Eqs. (45) and (47), y (t )  and x (t ) can be obtained from Eqs. (16). When (A2 > 4B), the roots m1 and m2 are real and distinct. The general solution to Eq. (44) can be written as x ∗ ( t)

1

β

The constants C3 , C4 , E5 , E6 , E7 , and E8 are the same as the constants C1 , C2 , E1 , E2 , E3 , and E4 in the D∗0 region except that the constants A and B are replaced with C and D, respectively. When (C2 > 4D), the roots m3 and m4 are real and distinct. The state variables y∗ (t), x∗ (t) can be written as

e−ut0

y∗ (t),

∗

(52a)

e−ut0

(51)

where, C = b√ + 2 + 1 and D = β . The roots of Eq. (51) are given by −(C )± (C 2 −4D ) m3,4 = . When (C2 < 4D), the roots m3 and m4 are a 2

A sinusoidally forced series LCR circuit with the Chua’s diode as the nonlinear element called the Murali–Lakshmanan–Chua circuit was introduced by Murali et al. [23]. The normalized circuit parameters take the values β = 1, ν = 0.015, ω = 0.72, a = −1.02, and b = −0.55. The circuit exhibits a period doubling sequence to chaos and has two prominent chaotic attractors, the one-band and the double-band chaotic attractors, at the control parameter values f = 0.1 and f = 0.14, respectively. A linear stability analysis is carried out to analyze the stability of the fixed points of the difference system in their corresponding synchronization manifold. From Eq. (18) we observe that the origin (0,0) acts as the fixed point of the difference system for all the three regions D∗0 and D∗±1 . The Jacobian matrix of the difference system obtained from the state equations of the coupled system  for the x ↔ x coupling in the D∗0 and D∗±1 regions can be written as

⎛

J0∗,±1

⎞



a, $|x∗ | < 1$ −2 − b, $|x∗ | > 1$ =⎝ −β

1 −σ

⎠.

(54)

In the D∗0 region, the origin transforms into a stable spiral from a saddle through the evolution of a stable node. The origin is a saddle for  < 0.01739 and is a stable node for 0.01739 ≤  ≤ 0.01749 and

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Fig. 13. (a) Stable (green line), unstable manifolds (red line) and trajectories (blue lines) indicating the saddle and (b) stable focus nature of the origin corresponding to the D∗0 region of the difference system for mutually coupled Murali–Lakshmanan– Chau circuits. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 15. Numerically obtained synchronization dynamics of the Murali– Lakshmanan–Chua circuit. a (i) Unsynchronized state of the coupled systems   in the x − x phase plane and (ii) time series of the signals x (blue line) and x  (magenta line) for  = 0; b(i) synchronized state in the x − x phase plane and (ii)  time series of the signals x (blue line) and x (magenta line) for  = 0.03. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 14. (a) Bifurcation of eigenvalues as a function of coupling parameter in the D∗0 region; b(i) MSF for coupled one-band (blue line) and double-band (red line) chaotic attractors and (b) 3-D plot indicating the variation of MSF as functions of f and  . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

becomes a stable spiral for  > 0.01749. Fig. 13 shows the behavior of the origin for different values of the coupling parameter. It has to be noted that the origin acts as a stable node within a smaller range of  and also that one of the eigenvalues is nearly equal to zero. Phase plots in the (x∗ − y∗ ) phase planes indicating the saddle and stable spiral nature of the origin are shown in Fig. 13(a) and (b). The bifurcation of the eigenvalues as a function of the coupling parameter is as shown in Fig. 14(a). In the D∗±1 region, the origin is a stable spiral for all values of the coupling strength. The stable spiral nature of the origin in all the three regions of the difference system leads to asymptotic convergence of the trajectories around the origin in the synchronization sub-space. Now we discuss the stability of the synchronized states observed in the coupled chaotic system using the Master stability Function analysis. The phenomenon of complete synchronization (CS) is studied for two types of chaotic attractors observed at the control parameter values f = 0.1 and f = 0.14,  respectively. Fig. 14b(i) showing the MSF obtained for x ↔ x coupling indicates that the coupled one-band (red line) and double-band chaotic attractors (blue line) entering into stable synchronized states and remaining thereafter for the coupling strengths  ≥ 0.02 and  ≥ 0.04, respectively. For coupled oneband chaotic attractors the Lyapunov exponents indicating hyperchaoticity are given as λ1 = 0.0709, λ2 = 0.0395, λ3 = 0, λ4 = −0.4063, λ5 = −0.4476, and λ6 = 0 for  = 0.005. For coupled double-band chaotic attractors the Lyapunov exponents obtained for the coupling strength  = 0.015 are given as λ1 = 0.1599, λ2 = 0.0517, λ3 = 0, λ4 = −0.3798, λ5 = −0.4561, and λ6 = 0. The existence of the x-coupled system in the stable synchronized states can be attributed to the stable spiral nature of the origin in both regions D∗0 and D ± 1 , respectively. A 3D plot indicating the variation of MSF as functions of the coupling parameter  and control parameter f is shown in Fig. 14b(ii). The numerical simulation results representing the unsynchronized and synchronized states of mu-

Fig. 16. Forced series LCR circuit with a simplified nonlinear element. (a) Bifurcation of eigenvalues as a function of coupling parameter in the D∗0 region; (b) MSF for mutually coupled chaotic attractors obtained at f = 0.31 as a function of the coupling parameter  .

tually coupled Murali–Lakshmanan–Chua circuits for the x-coupled case are shown in Fig. 15. 4.2. Forced series LCR circuit with simplified nonlinear element We present in this section, the synchronization dynamics of the mutually coupled forced series LCR circuits having the simplified nonlinear element as their nonlinear element. The sinusoidally forced series LCR circuit with a simplified nonlinear element was introduced by Arulgnanam et al. [19]. The normalized circuit parameters of this circuit take the values β = 0.9865, ω = 0.7084, a = −1.148, and b = 5.125. The circuit exhibits a prominent chaotic attractor at the control parameter value f = 0.31. A linear stability analysis may be carried out for the difference system in each of the piecewise-linear regions based on the Ja-

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Fig. 17. Numerically obtained synchronization dynamics of the series LCR circuit with a simplified nonlinear element circuit. a (i) Unsynchronized state of the coupled  systems in x − x phase plane and (ii) time series of the signals x (blue line) and   x (magenta line) for  = 0; b(i) synchronized state in the x − x phase plane and  (ii) time series of the signals x (blue line) and x (magenta line) for  = 0.03. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

cobian matrix given in Eq. (55). In the D∗0 region, the fixed point (0,0) acting as a saddle for  < 0.074 becomes an unstable star in the range 0.074001 ≤  ≤ 0.074002, and transforms into an unstable focus in the range 0.074023 ≤  ≤ 0.0807 and becomes a stable focus for  ≥ 0.0808 as the coupling parameter is increased. The bifurcation of the eigenvalues as a function of the coupling parameter is shown in Fig. 17(a). In the D∗±1 region, the origin is a stable node for all values of  . The MSF corresponding to coupling of identical chaotic attractors obtained at f = 0.31 indicates a transition of the coupled system into a stable synchronized state for  > 0.04 as shown in Fig. 17(b). The numerical simulation results representing the unsynchronized and synchronized states of mutually coupled systems are shown in Fig. 18. The electronic circuit experimental results, confirming complete synchronization observed through analytical and numerical studies is shown in Fig. 17. The linear circuit element values are fixed as C = 10.32 nF, L = 42.6 mH, R = 1990  and ν = 5.5 kHz. The voltage and the current signals of the drive and the response systems   are represented as (v, iL ) and (v , iL ), respectively. The two systems are initially kept in the chaotic state observed at F = 1.7 V. For larger values of the coupling resistance RC , i.e., for RC > 22 k, the coupling between the drive and response system becomes weak and hence they are unsynchronized. The unsynchronized state of  the coupled system in the (v − v ) phase plane and the corre sponding time series of the signals v(t) (green line) and v (t ) (yellow line) are shown in Fig. 17a(i) and a(ii), respectively. As the coupling resistance is decreased, the coupling between the two systems become stronger enough to get completely synchronized. Fig. 17b(i) and b(ii) shows the phase portraits and time series plots of the completely synchronized state of the coupled system for the coupling resistance RC = 2.

5. Mutually coupled parallel LCR circuits: numerical and experimental results In this section, we present the synchronization dynamics observed in mutually coupled parallel LCR circuits. Two types of circuit systems differing only by their constituent nonlinear element  is discussed for x ↔ x coupling of the state variables.

Fig. 18. Experimental observation of CS in mutually coupled series LCR circuits  with simplified nonlinear elements. Phase portraits in the (v − v ) phase plane indicating the a(i) unsynchronized and b(i) synchronized states for RC = 22.23 k and RC = 2 , respectively. The corresponding time series of the signals v(t) (green line)  and v (t ) (yellow line) for a(ii) unsynchronized and b(ii) synchronized states. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

5.1. Variant of Murali–Lakshmanan–Chua Circuits The Variant of Murali–Lakshmanan–Chua circuit is a forced parallel LCR circuit with a Chua’s diode as the nonlinear element was introduced by Thamilmaran et al. [17,18]. The normalized circuit parameters take the values β = 0.05, ω = 0.105, a = −1.121, and b = −0.6047. This circuit exhibits significant chaotic attractors at two values of the control parameter f. The chaotic attractor at f = 0.39 is obtained through the destruction of an invariant torus while that at f = 0.411 is obtained through a reverse perioddoubling sequence. The mutually coupled system is studied for the complete synchronization of these two types of identical chaotic attractors.  Under x ↔ x coupling, the Jacobian matrix of the difference system corresponding to the D∗0 region is

⎛

J ∗0,±1



a, $|x∗ | < 1$ −1 − 2 − b, $|x∗ | > 1$ =⎝

β

⎞

−1⎠ . 0

(55)

A linear stability analysis of the difference system corresponding to the D∗0 region shows that the origin is an unstable spiral for  < 0.0605, a stable spiral for 0.0605 ≤  ≤ 0.284 and becomes a stable node for  ≥ 0.285. In the D∗±1 region, the origin transforms into a stable node from a stable spiral for  ≥ 0.026. Fig. 19(a) and (b) show the trajectories of the difference system in the (x∗ − y∗ ) phase plane for different values of the coupling parameter. The phase plot in Fig. 19(a) shows the transformation of the unstable spiral into a stable spiral through the evolution of a centre. Fig. 19(b) shows the trajectories of the stable node nature of the origin in

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Fig. 19. The (a) unstable focus (red line), elliptic (blue line) and stable focus and (b) stable node nature of the origin corresponding to the D∗0 region of the difference system for mutually coupled Variant of Murali–Lakshmanan–Chau circuit. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 21. Numerically obtained synchronization dynamics of the Variant of MLC cir cuit. a (i) Unsynchronized state of the coupled systems in the x − x phase plane  and (ii) time series of the signals x (blue line) and x (magenta line) for  = 0; b(i)  synchronized state in the x − x phase plane and (ii) time series of the signals x  (blue line) and x (magenta line) for  = 0.05. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 20. Bifurcation of eigenvalues as a function of coupling parameter in the (a) D∗0 and (b) D∗±1 regions, respectively; b(i) MSF for coupled chaotic attractors corresponding to f = 0.39 (blue line) and f = 0.411 (red line), respectively, and (b) the 3-D plot indicating the variation of MSF as functions of f and  .

the (x∗ − y∗ ) phase plane. The bifurcation of the eigenvalues as a function of the coupling parameter in the D∗0 and D∗±1 regions are 

shown in Fig. 20(a) and 20(b), respectively. The MSFs for the x ↔ x  coupled systems are shown in Fig. 20(c). Under x ↔ x coupling, both the coupled identical chaotic attractors belonging to the control parameter values f = 0.39 (red line) and f = 0.411 (blue line) enters into stable synchronized states at the coupling strengths  = 0.0045 and  = 0.004, respectively and remain thereafter in the synchronized state as shown in Fig. 20c(i). The variation of MSF as functions of the coupling parameter  and f are shown in Fig. 20c(ii). The numerical simulation results of the x-coupled systems indicating the unsynchronized states are as shown in Fig. 21. 5.2. Forced parallel LCR circuit with simplified nonlinear element The forced parallel LCR circuit with a simplified nonlinear element exhibiting chaotic dynamics was introduced by Arulgnanam et al. [19,20]. The synchronization dynamics of the unidirectionally coupled circuit has been studied analytically, numerically and experimentally [9]. The normalized state equations of the circuit are as given in Eq. (6) with the system parameters taking the values a = −1.148, b = 5.125, β = 0.2592 and ω = 0.2402. The system exhibits significant chaotic attractors at two values of the control parameter f = 0.695 and f = 0.855, respectively. The Jacobian  matrix of the difference system corresponding to x ↔ x coupling is as given in Eq. (55). A linear stability analysis in the D∗0 region reveals the transformation of the origin from an unstable spiral into a stable spiral with the evolution of a centre/elliptic in the range 0 ≤  ≤ 0.583 and becoming a stable node for  > 0.583. The bifurcation of the eigenvalues as a function of the coupling parameter is as shown in Fig. 22(a). However, in the D∗±1 region, the origin acts as a stable node for all values of the coupling strength. The MSF  in the x ↔ x coupled case shows the chaotic attractors pertaining

Fig. 22. Forced parallel LCR circuit with a simplified nonlinear element. (a) Bifurca tion of eigenvalues as a function of coupling parameter in the D∗0 region for x ↔ x coupling; b(i) MSF for coupled chaotic attractors corresponding to f = 0.695 (blue line) and f = 0.855 (red line), respectively, and (b) the 3-D plot indicating the variation of MSF as functions of f and  . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

to f = 0.695 and f = 0.855 enters into stable synchronized states at the coupling strengths  = 0.04 and  = 0.06, respectively, as indicated by the negative values of MSF shown in Fig. 22b(i). Beyond this lower bound of synchronization the coupled systems exist in the synchronized state forever, as the coupling strength is increased. The variation of MSF as functions of f and  is shown in Fig. 22b(ii). The phenomenon of complete synchronization observed in the x-coupled systems through numerical simulation is shown in Fig. 23. The experimental results confirming complete synchronization is shown in Fig. 24. The linear circuit element values are fixed as C = 10.36 nF, L = 168 mH, R = 2036  and ν = 1.69 kHz. The voltage and the current signals of the drive and the response sys  tems are represented as (v, iL ) and (v , iL ), respectively. The two systems are initially kept in the chaotic state observed at F = 7.8 V. For larger values of the coupling resistance RC , i.e., for RC > 22.23 k, the coupling between the drive and response system becomes weak and hence they are unsynchronized. The unsynchronized  state of the coupled system in the (v − v ) phase plane and the  corresponding time series of the signals v(t) (green line) and v (t ) (red line) are shown in Fig. 24a(i) and a(ii), respectively. As the coupling resistance is decreased, the coupling between the two systems become stronger enough to get completely synchronized. Fig. 24b(i) and b(ii) shows the phase portraits and timeseries plots

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Fig. 23. Numerically obtained synchronization dynamics of the paralell LCR circuit with a simplified nonlinear element circuit. a (i) Unsynchronized state of the coupled  systems in the x − x phase plane and (ii) time series of the signals x (blue line) and   x (magenta line) for  = 0; b(i) synchronized state in the x − x phase plane and (ii)  time series of the signals x (blue line) and x (magenta line) for  = 0.1.

Explicit analytical solutions for the forced series and parallel LCR circuit systems with different nonlinear elements were presented and their synchronization dynamics were studied using the solutions. The analytical results thus obtained have been validated through numerical simulation results. The stability of the synchronized states were analyzed using the MSF approach. By analyzing the eigenvalues of the difference system in each of the piecewiselinear region, we could arrive at a sufficient condition for complete synchronization in the mutually coupled simple second-order chaotic systems. The existence of stable fixed points in any one of the piecewise-linear regions of the difference system guarantee synchronization of the coupled systems. Further, the numerical and analytical results are confirmed through electronic circuit experimental results for each of the circuit systems. The phenomenon of complete synchronization in mutually coupled systems have been reported through analytical results in the literature of the first time. Following the solutions to the normalized state equations of quasi periodically forced and unidirectionally coupled simple chaotic systems, the analytical solutions to mutually coupled systems is developed which could be further developed for analyzing simple chaotic networks involved in secure transmission of signals. Acknowledgment One of the authors A. Arulgnanam gratefully acknowledges Dr.K. Thamilmaran, Centre for Nonlinear Dynamics, Bharathidasan University, Tiruchirapalli, for his help and permission to carry out the experimental work during his doctoral programme. References

Fig. 24. Experimental observation of CS in mutually coupled parallel LCR circuits  with simplified nonlinear elements. Phase portraits in the (v − v ) phase plane indicating the a(i) unsynchronized and b(i) synchronized states for RC = 22.23 k and RC = 2 , respectively. The corresponding time series of the signals v(t) (green line)  and v (t ) (red line) for a(ii) unsynchronized and b(ii) synchronized states. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

of the completely synchronized state of the coupled system for RC = 2. 6. Summary and conclusions We have developed in this article, explicit generalized analytical solutions for a class of second-order, non-autonomous simple electronic circuit systems with two different nonlinear elements.

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