**3. The proposed coupling scheme**

Two identical unidirectionally coupled chaotic systems can be described by the following system of differential equations:

$$\begin{cases} \dot{\mathbf{x}} = f(\mathbf{x}) + U\_{\chi} \\ \dot{\mathbf{y}} = f(\mathbf{y}) + U\_{\chi} \end{cases} \tag{2}$$

where (*f*(*x*), *f*(y)) ∈ *Rn* are the flows of the systems. Nonlinear controllers (NCs), *UX* and *UY*, define the coupling of the systems, while the error function is given by *e* = *ky* - *lx*, where *k* and *l* are constants [51, 52]. If the Lyapunov function stability (LFS) technique is applied, a stable synchronization state will be obtained when the error function of the coupled system follows the limit:

$$\lim\_{t \to \ast \ast} \|e(t)\| \to 0 \tag{3}$$

so that *lx* = *ky*.

**Figure 11.** Phase portrait of *y* versus *x*, for *a* = 1, *c* = 0.2, *d* = 3 and *B* = 7 (chaotic behaviour).

302 Nonlinear Systems - Design, Analysis, Estimation and Control

**Figure 12.** Phase portrait of *y* versus *x*, for *a* = 1, *c* = 0.2, *d* = 3 and *B* = 9 (periodic behaviour).

The design process of the coupling scheme, is based on the Lyapunov function:

$$V(e) = \frac{1}{2}e^r e$$

where *T* is a transpose of a matrix and *V*(*e*). The Lyapunov function (4) is a positive definite function. Also, for known system's parameters and with the appropriate choice of the controllers *UX* and *UY*, the coupled system has *V*(*e*) < 0. This ensures the asymptotic global stability of synchronization and thereby realizes any desired synchronization state [51, 52].

By using the appropriate NCs functions *UX*, *UY* and error function's parameters *k*, *l*, a bidirectional (mutual) or unidirectional coupling scheme can be implemented. Analytically, while if *UX,Y* ≠ 0 and *k*, *l* ≠ 0, a bidirectional coupling scheme is realized, while if (*UX* = 0, *k* = 1) or (*UY* = 0, *l* = 1), a unidirectional coupling scheme is realized, respectively. The signs of the constants *k*, *l* play a crucial role to the synchronization case (complete synchronization or antisynchronization), which is observed in this work. However, the ratio of *k* over *l* decides the amplification of one oscillator relative to another one.

Next, the simulation results in the unidirectional coupling scheme and for various values of parameters *k* and *l* are presented in details.

## **4. Unidirectional coupling**

In this section, the unidirectional coupling scheme for *UX* = 0, in the case of coupled systems of Eq. (1), is presented. The coupled system is described by the following systems of Eqs. (5) and (6).

Master system:

$$\begin{cases}
\dot{\mathbf{x}}\_1 = a\mathbf{x}\_2 + \mathbf{x}\_1 - c\mathbf{x}\_1\mathbf{x}\_2^2 \\
\dot{\mathbf{x}}\_2 = -\mathbf{x}\_1 - B\cos(dt)
\end{cases} \tag{5}$$

Slave system:

$$\begin{cases}
\dot{\mathbf{y}}\_1 = a\mathbf{y}\_2 + \mathbf{y}\_1 - c\mathbf{y}\_1\mathbf{y}\_2^2 + U\_{Y1} \\
\dot{\mathbf{y}}\_2 = -\mathbf{y}\_1 - B\cos(dt) + U\_{Y2}
\end{cases} \tag{6}$$

where *UY* = [*UY*1, *UY*2] T is the Nonlinear Controller (NC). The error function is defined by *e* = *ky* - *lx*, with *e* = [*e*1, *e*2] T, *x* = [*x*1, *x*2] T and *y* = [*y*1, *y*2] T. So, the error dynamics, by taking the difference of Eqs. (5) and (6), are written as:

$$\begin{cases} \dot{e}\_1 = ae\_2 + e\_1 + lc\mathbf{x}\_1\mathbf{x}\_2^2 - kc\mathbf{y}\_1\mathbf{y}\_2^2 + kU\_{Y1} \\ \dot{e}\_2 = -e\_1 - B(k-l)\cos(dt) + kU\_{Y2} \end{cases} \tag{7}$$

For stable synchronization, *e* → 0 as *t* → ∞. By substituting the conditions in Eq. (7) and taking the time derivative of Lyapunov function

$$\begin{split} \dot{V}(\mathbf{e}) &= \mathbf{e}\_1 \dot{\mathbf{e}}\_1 + \mathbf{e}\_2 \dot{\mathbf{e}}\_2 = \\ &= \mathbf{e}\_1 \left( a \mathbf{e}\_2 + \mathbf{e}\_1 + l \mathbf{c} \mathbf{x}\_1 \mathbf{x}\_2^2 - k \mathbf{c} \mathbf{y}\_1 \mathbf{y}\_2^2 + kU\_{\text{r1}} \right) + \mathbf{e}\_2 \left( -\mathbf{e}\_1 - B(k - l) \cos(\mathbf{d}t) + kU\_{\text{r2}} \right) \end{split} \tag{8}$$

We consider the following NC controllers:

$$\begin{cases} U\_{Y1} = -\frac{1}{k}(ae\_2 + 2e\_1 + lc\mathbf{x}\_1\mathbf{x}\_2^2 - kc\mathbf{y}\_1\mathbf{y}\_2^2) \\\\ U\_{Y2} = -\frac{1}{k}(-e\_1 - B(k-l)\cos(dt) + e\_2) \end{cases} \tag{9}$$

such that

Synchronization Phenomena in Coupled Birkhoff-Shaw Chaotic Systems Using Nonlinear Controllers http://dx.doi.org/10.5772/64811 305

$$\dot{V}(e) = -e\_1^2 - e\_2^2 < 0\tag{10}$$

Eq. (10) ensures the asymptotic global stability of synchronization.

#### **5. Simulation results**

**4. Unidirectional coupling**

304 Nonlinear Systems - Design, Analysis, Estimation and Control

and (6).

Master system:

Slave system:

where *UY* = [*UY*1, *UY*2]

of Eqs. (5) and (6), are written as:


In this section, the unidirectional coupling scheme for *UX* = 0, in the case of coupled systems of Eq. (1), is presented. The coupled system is described by the following systems of Eqs. (5)

> 1 2 1 12 2 1 ( ) *x ax x cx x x x B cos dt* ìï = +- <sup>í</sup> ïî =- -

1 2 1 12 1 2 1 <sup>2</sup> ( )+

*y ay y cy y U y y B cos dt U* ìï = +- + <sup>í</sup>

&

ïî =- -

T and *y* = [*y*1, *y*2]

&

T, *x* = [*x*1, *x*2]

the time derivative of Lyapunov function

11 2 2

We consider the following NC controllers:

+ =

( )=

such that

*V e ee ee*

& & &

&

2 2

*Y*

í

*Y*

*k*

2

2

2 2

For stable synchronization, *e* → 0 as *t* → ∞. By substituting the conditions in Eq. (7) and taking

( ) ( )

( )

<sup>1</sup> ( 2 )

1 2 1 12 12 1 2 1 2

+ + - + + -- - +

1 2 1 12 12

*U ae e lcx x kcy y <sup>k</sup>*

<sup>ì</sup> =- + + - ïï

*U e B k l dt e*

<sup>ï</sup> =- - - - + ïî

2 1 2

<sup>1</sup> ( )cos( )

*= e ae e lcx x kcy y kU e e B k l cos dt kU*

1 2 1 12 12 1 2 1 <sup>2</sup> ( )cos( )

*e ae e lcx x kcy y kU e e B k l dt kU* ìï = ++ - + <sup>í</sup> ïî =- - - +

*Y Y*

T is the Nonlinear Controller (NC). The error function is defined by *e* = *ky*

*Y Y*

( ) () *Y Y*

2 2

& (7)

& (5)

& (6)

T. So, the error dynamics, by taking the difference

(8)

(9)

In this section, the simulation results, with the unidirectional coupling scheme, in three different cases are presented.

**Figure 13.** The phase portrait of *y*1 versus *x*1, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

#### **5.1. The case for** *k* **=** *l* **= 1**

As it is mentioned, the phenomenon of complete synchronization is achieved for every value of *k, l*. Especially for *k* = *-l* = 1, the two coupled systems are in the chaotic state, due to the chosen values of system's parameters (*a* = 1, *B* = 7, *c* = 0.2 and *d* = 3) and initial conditions (*x*1, *x*2, *y*1, *y*2) = (3, 2, –1, –5). The goal of complete synchronization is achieved as it is shown from the plots of *y*1 versus *x*1, the time-series of *x*2, *y*2 and the errors *ei* in **Figures 13**–**15**.

**Figure 14.** The time-series of *x*2, *y*2, in regards to the external periodic signal, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

**Figure 15.** The time-series of errors *e*1, *e*2, with *k* = *l* = 1, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

#### **5.2. The case for** *k* **= l = 1**

In the second case, by using opposing values for the parameters *k* = –*l* = 1 and for the same values of system's parameters (*a* = 1, *B* = 7, *c* = 0.2 and *d* = 3), the phenomenon of antisynchronization is achieved. This conclusion is derived from the phase portrait of *y*1 versus *x*<sup>1</sup> (**Figure 16**), as well as from the time series of *x*2, *y*2 (**Figure 17**). Also, the plot of errors *ei* = *yi* + *xi* in **Figure 18** confirms the antisynchronization of the coupled system.

**Figure 16.** The phase portrait of *y*1 versus *x*1, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

**Figure 14.** The time-series of *x*2, *y*2, in regards to the external periodic signal, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

306 Nonlinear Systems - Design, Analysis, Estimation and Control

**Figure 15.** The time-series of errors *e*1, *e*2, with *k* = *l* = 1, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

In the second case, by using opposing values for the parameters *k* = –*l* = 1 and for the same values of system's parameters (*a* = 1, *B* = 7, *c* = 0.2 and *d* = 3), the phenomenon of antisynchronization is achieved. This conclusion is derived from the phase portrait of *y*1 versus *x*<sup>1</sup>

**5.2. The case for** *k* **= l = 1**

**Figure 17.** The time-series of -*x*2, *y*2, in regard to the external periodic signal, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

**Figure 18.** The time-series of errors *e*1, *e*2, with *k* = *l* = 1, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

**Figure 19.** The phase portraits of *x2* versus *x*1(black colour) and *y2* versus *y*1 (red colour), for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

#### **5.3. The case for** *k* **= 1,** *l* **= 2**

In this case, the parameters of the error functions are chosen as *k* = 1 and *l* = 2. By choosing the systems' parameters as *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3 the chaotic attractor of the second system is enlarged by two times, as it is shown with red colour in **Figure 19**, as well as by the timeseries of signals *x*2 and *y*<sup>2</sup> (**Figure 21**). The *y*1 versus *x*1 plot in **Figure 20** confirms that the coupled system is in complete synchronization state independently of the values of the error's parameters *k*, *l*. The error plot *ei* = *yi* - 2*xi* (*i* = 1, 2) in **Figure 22** shows the exponential convergence to zero that confirms the realization of system's complete synchronization state.

**Figure 20.** The phase portrait of *y*1 versus *x*1, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

**Figure 18.** The time-series of errors *e*1, *e*2, with *k* = *l* = 1, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

308 Nonlinear Systems - Design, Analysis, Estimation and Control

**5.3. The case for** *k* **= 1,** *l* **= 2**

**Figure 19.** The phase portraits of *x2* versus *x*1(black colour) and *y2* versus *y*1 (red colour), for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

In this case, the parameters of the error functions are chosen as *k* = 1 and *l* = 2. By choosing the systems' parameters as *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3 the chaotic attractor of the second system is enlarged by two times, as it is shown with red colour in **Figure 19**, as well as by the time-

**Figure 21.** The time-series of 2*x*2, *y*2, in regard to the external periodic signal, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

**Figure 22.** The time-series of errors *e*1, *e*2, with *k* = 1, *l* = 2, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

## **6. Circuit's implementation of the coupling scheme**

The circuit implementation of the proposed synchronization coupling scheme, with the electronic simulation package Cadense OrCAD, for *k* = *l* = 1, is presented in this section, in order to prove the feasibility of the proposed method. The coupling system's circuitry design consists of three sub-circuits, which are the master circuit, the coupling circuit and the slave circuit. Also, the circuit is realized by using common electronic components.

**Figure 23** shows the schematic of the master circuit, which has two integrators (*U*1 and *U*2) and one differential amplifier (*U*3), which are implemented with the TL084, as well as two signals multipliers (*U*4, *U*5) by using the AD633. By applying Kirchhoff's circuit laws, the corresponding circuital equations of designed master circuit can be written as:

$$\begin{cases}
\dot{\mathbf{x}}\_1 = \frac{1}{RC} \left(\mathbf{x}\_2 + \mathbf{x}\_1 - \frac{R}{100R\_1} \mathbf{x}\_1 \mathbf{x}\_2^2\right) \\
\dot{\mathbf{x}}\_2 = \frac{1}{RC} \left(-\mathbf{x}\_1 - V\_0 \cos(\alpha t)\right)
\end{cases} \tag{11}$$

Synchronization Phenomena in Coupled Birkhoff-Shaw Chaotic Systems Using Nonlinear Controllers http://dx.doi.org/10.5772/64811 311

**Figure 23.** The schematic representation of the master circuit.

**Figure 22.** The time-series of errors *e*1, *e*2, with *k* = 1, *l* = 2, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3.

310 Nonlinear Systems - Design, Analysis, Estimation and Control

**6. Circuit's implementation of the coupling scheme**

circuit. Also, the circuit is realized by using common electronic components.

ing circuital equations of designed master circuit can be written as:

1

*RC*

î

&

&

<sup>ï</sup> = -- <sup>ï</sup>

The circuit implementation of the proposed synchronization coupling scheme, with the electronic simulation package Cadense OrCAD, for *k* = *l* = 1, is presented in this section, in order to prove the feasibility of the proposed method. The coupling system's circuitry design consists of three sub-circuits, which are the master circuit, the coupling circuit and the slave

**Figure 23** shows the schematic of the master circuit, which has two integrators (*U*1 and *U*2) and one differential amplifier (*U*3), which are implemented with the TL084, as well as two signals multipliers (*U*4, *U*5) by using the AD633. By applying Kirchhoff's circuit laws, the correspond-

( )

1 2 1 1 2

<sup>ì</sup> æ ö <sup>ï</sup> = +- ç ÷ <sup>ï</sup> è ø <sup>í</sup>

*<sup>R</sup> x x x xx RC <sup>R</sup>*

<sup>1</sup> cos( )

2 10

*x xV t*

2

(11)

1

w

100

where *xi* (*i* = 1, 2) are the voltages in the outputs of the operational amplifiers *U*3 and *U*2. Normalizing the differential equations of system (18) by using τ = *T*/*RC* we could see that this system is equivalent to the system (12). The circuit components have been selected as: *R* = 10 kΩ, *R*1 = 500 Ω, *C* = 10 nF, *V*0 = 7 V and *f* = 4777 Hz, while the power supplies of all active devices are ±17 VDC. For the chosen set of components the master system's parameters are: *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3. In **Figure 24**, the chaotic attractor, which is obtained from Cadence OrCAD in (*x*1, *x*2) phase plane, is proved to be in a very good agreement with the respective phase portrait from system's numerical simulation process (**Figure 11**). So, the proposed circuit emulates very well the master system.

**Figure 24.** The chaotic attractor produced by the designed master circuit, obtained from Cadence OrCAD in the (*x*1, *x*2) phase plane.

In **Figure 25**, the schematic of the slave circuit, which is similar to the master circuit, is shown. The difference of this circuit in comparison to the previous one are the signals *u*1 and *mu*2, where *u*1 is the control signal *UY*1 and *mu*2 is the opposite, due to the integrator, of the signal *UY*2, of system (6). So, for *k* = *l* = 1, the signal *mu*2 is given as

$$
mu\_2 = -e\_1 + e\_2 \tag{12}$$

**Figure 25.** The schematic representation of the slave circuit.

The dynamics of the slave circuit is described by the following set of differential equations.

$$\begin{cases} \dot{\mathcal{y}}\_1 = \frac{1}{RC} \left( \mathcal{y}\_2 + \mathcal{y}\_1 - \frac{R}{100R\_1} \mathcal{y}\_1 \mathcal{y}\_2^2 + \mathcal{u}\_1 \right) \\\\ \dot{\mathcal{y}}\_2 = \frac{1}{RC} \left( -\mathcal{y}\_1 - V\_0 \cos(\alpha t) - m\mathcal{u}\_2 \right) \end{cases} \tag{13}$$

Finally, the units from which the coupling circuit is consisted, are shown in the schematic of **Figure 26**, in which *ei* , (*i* = 1, 2) are the difference signals (*ei* = *kyi* - *lxi* , *i* = 1, 2), with *k* = *l* = 1 and *me*2 is the opposite of *e*2. Also, the resistors *R*2 = 5 kΩ and *R*3 = 50 kΩ have been used for achieving the desired values of system's parameters.

Synchronization Phenomena in Coupled Birkhoff-Shaw Chaotic Systems Using Nonlinear Controllers http://dx.doi.org/10.5772/64811 313

**Figure 26.** The schematic representation of the coupling circuit.

In **Figure 25**, the schematic of the slave circuit, which is similar to the master circuit, is shown. The difference of this circuit in comparison to the previous one are the signals *u*1 and *mu*2, where *u*1 is the control signal *UY*1 and *mu*2 is the opposite, due to the integrator, of the signal

The dynamics of the slave circuit is described by the following set of differential equations.

1 2 1 12 1

*y y y yy u RC <sup>R</sup>*

<sup>ì</sup> æ ö <sup>ï</sup> = +- + ç ÷ <sup>ï</sup> è ø <sup>í</sup>

1

2 10 2

<sup>1</sup> cos( )

*y y V t mu RC*

<sup>ï</sup> = -- - <sup>ï</sup>

, (*i* = 1, 2) are the difference signals (*ei*

( )

Finally, the units from which the coupling circuit is consisted, are shown in the schematic of

*me*2 is the opposite of *e*2. Also, the resistors *R*2 = 5 kΩ and *R*3 = 50 kΩ have been used for achieving

100

*R*

1

w

2

 = *kyi* - *lxi* (13)

, *i* = 1, 2), with *k* = *l* = 1 and

*mu e e* 2 12 =- + (12)

*UY*2, of system (6). So, for *k* = *l* = 1, the signal *mu*2 is given as

312 Nonlinear Systems - Design, Analysis, Estimation and Control

**Figure 25.** The schematic representation of the slave circuit.

î

the desired values of system's parameters.

**Figure 26**, in which *ei*

&

&

**Figure 27.** The phase portrait of *y*1 vs. *x*1, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3, obtained from Cadence OrCAD.

**Figures 27** and **28** depict the phase portraits in (*xi* , *yi* ) phase planes, with *i* = 1, 2, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3, obtained from Cadence OrCAD. These figures confirm the achievement of complete synchronization in the case of unidirectionally coupled circuits with the proposed method.

**Figure 28.** The phase portrait of *y*2 versus *x*2, for *a* = 1, *B* = 7, *c* = 0.2 and *d* = 3, obtained from Cadence OrCAD.
