**3. Chaos control technique**

As nonlinear systems are hardly comparable in the sense that behavior of one nonlinear system hardly match with another nonlinear system so the chaotic evolutions. So controlling chaos to bring any chaotic system to regularity may differ from one nonlinear system to another nonlinear system. Different types of controlling chaos technique discussed in recent literatures, [75–88].

Following two chaos controlling technique discussed here:

## **3.1 Asymptotic Stability Method**

Asymptotic stability analysis to stabilize unstable fixed point and to control chaotic motion appeared in some recent researches, [83–85]. Though this method has some limitations, it is perfect way to control chaos in models where it can be applicable.

### **Description of the Method:**

Dynamics of the actual map **Xn+1** and that of the desired map **Yn+1** can be explained by following mapping:

$$\mathbf{X\_{n+1}} = \mathbf{F} \left( \mathbf{x\_n}, \mathbf{p} \right) \tag{11}$$

$$\mathbf{Y\_{n+1}} = \mathbf{F} \left( \mathbf{y\_n}, \mathbf{p}\* \right) \tag{12}$$

Also, the neighborhood dynamics of **Xn+1** and **Yn+1** can be represented by the relation:

$$\mathbf{X\_{n+1}} = \mathbf{A\_R}\mathbf{X\_n} + \mathbf{B\_R}\mathbf{p}$$

$$\mathbf{Y\_{n+1}} = \mathbf{A\_D}\mathbf{Y\_n} + \mathbf{B\_D}\mathbf{p}\*$$

Matrices **AR, AD, BR, BD** can be obtained from the following:

$$\mathbf{A}\_{\mathbf{R}} = \mathbf{D}\_{\mathbf{Xn}} \mathbf{F} \left( \mathbf{X\_n}, \mathbf{p} \right), \mathbf{A\_D} = \mathbf{D}\_{\mathbf{Yn}} \mathbf{F} \left( \mathbf{Y\_n}, \mathbf{p}^\* \right)$$
 
$$\mathbf{B\_R} = \mathbf{D} \mathbf{p} \, \mathbf{F} \left( \mathbf{X\_n}, \mathbf{p} \right), \mathbf{B\_D} = \mathbf{D} \mathbf{p} \ast \mathbf{F} \left( \mathbf{Y\_n}, \mathbf{p}^\* \right)$$

Here,

$$\mathbf{X}\_{\mathbf{n}+\mathbf{1}} = \begin{pmatrix} \mathbf{x}\_{\mathbf{n}+1} \\ \mathbf{y}\_{\mathbf{n}+1} \end{pmatrix} \\ \mathbf{Y}\_{\mathbf{n}+\mathbf{1}} = \begin{pmatrix} \mathbf{x}\_{\mathbf{n}+1}^{\*} \\ \mathbf{y}\_{\mathbf{n}+1}^{\*} \end{pmatrix}.$$

*Chaos and Complexity Dynamics of Evolutionary Systems DOI: http://dx.doi.org/10.5772/intechopen.94295*

Let *a*, *b* are two parameters of the system and (*xn*, *y<sup>n</sup>* ) be any unstable fixed point of above system for given values of *a* and *b*. Then, our objective is to obtain two new values for *a* and *b* so that this unstable point becomes stable. For this, we need the Jacobian matrices defined by

$$\mathbf{J}^\* = \begin{pmatrix} \frac{\partial \mathbf{f}}{\partial \mathbf{x}} & \frac{\partial \mathbf{f}}{\partial \mathbf{y}}\\ \frac{\partial \mathbf{g}}{\partial \mathbf{x}} & \frac{\partial \mathbf{g}}{\partial \mathbf{y}} \end{pmatrix}, \text{ J}^\* = \begin{pmatrix} \frac{\partial \mathbf{f}}{\partial \mathbf{a}} & \frac{\partial \mathbf{f}}{\partial \mathbf{b}}\\ \frac{\partial \mathbf{g}}{\partial \mathbf{a}} & \frac{\partial \mathbf{g}}{\partial \mathbf{b}} \end{pmatrix}$$

The control input parameter matrix p\*can be given by

$$\mathbf{P}^\* = \mathbf{C}\_{\mathsf{R}} \, \mathbf{X}\_{\mathsf{n}} + \mathbf{C}\_{\mathsf{M}} \, \mathbf{p} - \mathbf{C}\_{\mathsf{D}} \mathbf{Y}\_{\mathsf{n}} \tag{13}$$

Then, using (11)-(13), one obtains the following error equation:

$$\mathbf{e\_{n+1}} = \left(\mathbf{A\_{R}} - \mathbf{B\_{D}}\,\mathbf{C\_{R}}\right)\mathbf{e\_{n}} + \left\{\mathbf{A\_{R}} - \mathbf{A\_{D}} + \mathbf{B\_{D}}(\mathbf{C\_{D}} - \mathbf{C\_{R}})\right\}\mathbf{Y\_{n}} + \left(\mathbf{B\_{R}} - \mathbf{B\_{D}}\,\mathbf{C\_{M}}\right)\mathbf{p} \tag{14}$$

And **en = Xn-Yn.**

**3. Chaos control technique**

**3.1 Asymptotic Stability Method**

**Description of the Method:**

explained by following mapping:

applicable.

**Figure 28.**

relation:

Here,

**202**

chaos technique discussed in recent literatures, [75–88].

*A Collection of Papers on Chaos Theory and Its Applications*

Following two chaos controlling technique discussed here:

As nonlinear systems are hardly comparable in the sense that behavior of one nonlinear system hardly match with another nonlinear system so the chaotic evolutions. So controlling chaos to bring any chaotic system to regularity may differ from one nonlinear system to another nonlinear system. Different types of controlling

*Plots of LCEs of chaotic attractors of model (1) for values of c. Other parameters are same as in Figure 26.*

Asymptotic stability analysis to stabilize unstable fixed point and to control chaotic motion appeared in some recent researches, [83–85]. Though this method has some limitations, it is perfect way to control chaos in models where it can be

Dynamics of the actual map **Xn+1** and that of the desired map **Yn+1** can be

Also, the neighborhood dynamics of **Xn+1** and **Yn+1** can be represented by the

**Xn**þ**<sup>1</sup>** ¼ **ARXn** þ **BR p Yn**þ**<sup>1</sup>** ¼ **ADYn** þ **BD p ∗**

**AR** <sup>¼</sup> **DXn F X**ð Þ **<sup>n</sup>**, **<sup>p</sup>** , **AD** <sup>¼</sup> **DYn F Yn**, **<sup>p</sup><sup>∗</sup>** ð Þ **BR** <sup>¼</sup> **Dp F X**ð Þ **<sup>n</sup>**, **<sup>p</sup>** , **BD** <sup>¼</sup> **Dp <sup>∗</sup> F Yn**, **<sup>p</sup><sup>∗</sup>** ð Þ

**Yn**þ**<sup>1</sup>** <sup>¼</sup> <sup>x</sup><sup>∗</sup>

nþ1 y∗ nþ1

!

Matrices **AR, AD, BR, BD** can be obtained from the following:

**Xn**þ**<sup>1</sup>** <sup>¼</sup> xnþ<sup>1</sup>

ynþ<sup>1</sup>

!

**Xn**þ**<sup>1</sup>** ¼ **F x**ð Þ **<sup>n</sup>**, **p** (11) **Yn**þ**<sup>1</sup>** <sup>¼</sup> **F yn**, **<sup>p</sup> <sup>∗</sup>** � � (12)

Note that in equation (13) and (14) the coefficient matrices **CR**, **CD** and **CM** are to be determined so that if the error vector **en = Xn-Yn** is initialized as **e0 = 0**, then it will be zero for all n future times. For asymptotic stability, we must have **en** ! **0** as n ! ∞, then equation (14) implies

$$\mathbf{A}\_{\mathbf{R}} - \mathbf{A}\_{\mathbf{D}} + \mathbf{B}\_{\mathbf{D}} \left( \mathbf{C}\_{\mathbf{D}} - \mathbf{C}\_{\mathbf{R}} \right) = \mathbf{0} = \ \ \ \mathbf{B}\_{\mathbf{D}} (\mathbf{C}\_{\mathbf{D}} - \mathbf{C}\_{\mathbf{R}}) = \mathbf{A}\_{\mathbf{D}} - \mathbf{A}\_{\mathbf{R}} \tag{15}$$

$$\text{And } \mathbf{B}\_{\mathsf{R}} \mathbf{-B}\_{\mathsf{D}} \mathbf{C}\_{\mathsf{M}} = \mathbf{0} = \ \mathsf{B}\_{\mathsf{D}} \mathbf{C}\_{\mathsf{M}} = \mathbf{B}\_{\mathsf{R}} \tag{16}$$

The necessary and sufficient condition for **en**!**0** as **n**!∞ is

$$\mathbf{A}\_{\mathsf{R}} - \mathbf{B}\_{\mathsf{D}} \; \mathbf{C}\_{\mathsf{R}} = -\mathbf{I} \tag{17}$$

From these, one can obtain matrices **CM, CD, CR** and then control parameter matrix **P\*** from (13).

A necessary and sufficient condition for the existence of matrices **CM, CD, CR,** given by:

$$\mathbf{Rank} \left( \mathbf{B\_D} \right) = \mathbf{Rank} \left( \mathbf{B\_D}, \mathbf{A\_D} - \mathbf{A\_R} \right) = \mathbf{Rank} \left( \mathbf{B\_D}, \mathbf{B\_R} \right)$$

#### **3.2 Applications**

### *3.2.1 Chaos Control in a 2–Dimensional Prey-Predator map*

Considered a prey-predator model where both species evolve with logistic rule and also influencing each other, [30], written as

$$\mathbf{x\_{n+1}} = \mathbf{a}\,\mathbf{x\_n}(\mathbf{1} - \mathbf{x\_n}) - \mathbf{b}\,\mathbf{x\_n}\,\mathbf{y\_n}$$

$$\mathbf{y\_{n+1}} = \mathbf{c}\,\mathbf{y\_n}(\mathbf{1} - \mathbf{y\_n}) + \mathbf{b}\,\mathbf{x\_n}\,\mathbf{y\_n} \tag{18}$$

For *a* = 3.7, *b* = 3.5, *c* = 0.2, one obtains four fixed points obtained as: (0, 0), (0, �4.0), (0.72973, 0) & (0.25712, 0.49961) of which (0.25712, 0.49961) is unstable. So, the orbits originating nearby it would also be unstable and unpredictable & may be chaotic. Nearby this unstable fixed point, we assume a desired initial point as (0.3, 0.5). With this as initial point together with parameters *a* = 3.7, *b* = 3.5,

znþ<sup>1</sup> ¼ r ynzn (19)

For values *a* = 4.1, *b* = 3.7, *c* = 3, *d* = 3.5, *r* = 3.8 five fixed points exist for system (19) given by: P0(0, 0, 0), P1(0, 0.2632, 0.2857), P2(0.518614, 0.263158, 0.158812),

P4(0.3333, 0.4685, 0) are unstable. Then, taking nearby P2, a desired initial point P\*

In the process of stabilizing the desired point (0.5, 0.3, 0.2), calculations performed to replace parameters *a* = 4.1, *d* = 3.5 and *r* = 3.8 to earlier case of map (18). After obtaining all concerned matrices, replacement matrix obtained as

> *a d r*

1

CA <sup>¼</sup>

0

B@

4*:*1035 1*:*05194 1*:*02707 1

CA

0

B@

*Phase plot and LCE plot of controlled system when c = 0.2, a = 3.91525, b = 2.99538.*

P3(0.7561, 0, 0) and P4(0.3333, 0.4685, 0). Then, by stability analysis it has

obtained that the fixed points P2(0.518614, 0.263158, 0.158812) and

*<sup>p</sup>*<sup>∗</sup> <sup>¼</sup>

(0.5, 0.3, 0.2), chaotic attractors drawn, **Figure 31**.

*Chaos and Complexity Dynamics of Evolutionary Systems*

*DOI: http://dx.doi.org/10.5772/intechopen.94295*

**Figure 31.**

**205**

**Figure 30.**

*Time series and attractors of unstable system.*

**Figure 29.** *Time series graphs, attractor and LCE plots of the unstable system.*

*c* = 0.2, time series, attractor and LCE plots are obtained and shown by **Figure 29**. Clearly the system (18) is showing chaos at (0.3, 0.5) with *a* = 3.7, *b* = 3.5, *c* = 0.2.

Then, applying asymptotic stability discussed above for the map (18). For fixed value c = 0.2, unstable fixed point obtained as (0.25712, 0.49961). Nearby this point . When above-mentioned

take initial point (0.3, 0.5) and *<sup>p</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>a</sup> b* <sup>¼</sup> **<sup>3</sup>***:***<sup>7</sup> 3***:***5** method applied, one obtains matrices:

$$\begin{aligned} \mathbf{A\_R} &= \begin{bmatrix} \mathbf{0.048652} & -\mathbf{0.899924} \\ \mathbf{1.74865} & \mathbf{0.900078} \end{bmatrix} & \mathbf{A\_D} &= \begin{bmatrix} -\mathbf{0.27} & -\mathbf{1.05} \\ \mathbf{1.75} & \mathbf{1.05} \end{bmatrix} \\ \mathbf{B\_R} &= \begin{bmatrix} \mathbf{0.19101} & -\mathbf{0.128462} \\ \mathbf{0} & \mathbf{0.128462} \end{bmatrix} & \mathbf{B\_D} &= \begin{bmatrix} \mathbf{0.21} & -\mathbf{0.15} \\ \mathbf{0} & \mathbf{0.15} \end{bmatrix} \\ \mathbf{C\_M} &= \begin{bmatrix} \mathbf{0.90957} & \mathbf{0} \\ \mathbf{0} & \mathbf{0.85641} \end{bmatrix} & \mathbf{C\_R} &= \begin{bmatrix} 3.79669 & -4.76117 \\ \mathbf{11.6577} & -\mathbf{0.66615} \end{bmatrix} \\ \mathbf{C\_D} &= \begin{bmatrix} 2.28571 & -4.7619 \\ \mathbf{11.6667} & \mathbf{0.333333} \end{bmatrix} & \mathbf{p}\* &= \begin{pmatrix} 3.91525 \\ 2.99538 \end{pmatrix} \end{aligned}$$

For the case when *c* = 0.2; new values of *a* and *b*; *a* = 3.91525, *b* = 2.99538 along with initial point (0.3, 0.5)a phase plot and a plot of Lyapunov exponents (LEC), are given in **Figure 30**.

#### *3.2.2 Food chain model*

**Next,** we have considered three dimensional food chain model, [23], written as

$$\mathbf{x\_{n+1}} = \mathbf{a}\,\mathbf{x\_n}(\mathbf{1} - \mathbf{x\_n}) - \mathbf{b}\,\mathbf{x\_n}\mathbf{y\_n}$$

$$\mathbf{y\_{n+1}} = \mathbf{c}\,\mathbf{x\_n}\,\mathbf{y\_n} - \mathbf{d}\,\mathbf{y\_n}\,\mathbf{z\_n}$$

*Chaos and Complexity Dynamics of Evolutionary Systems DOI: http://dx.doi.org/10.5772/intechopen.94295*

$$\mathbf{z\_{n+1}} = \mathbf{r} \,\mathbf{y\_n}\mathbf{z\_n} \tag{19}$$

For values *a* = 4.1, *b* = 3.7, *c* = 3, *d* = 3.5, *r* = 3.8 five fixed points exist for system (19) given by: P0(0, 0, 0), P1(0, 0.2632, 0.2857), P2(0.518614, 0.263158, 0.158812), P3(0.7561, 0, 0) and P4(0.3333, 0.4685, 0). Then, by stability analysis it has obtained that the fixed points P2(0.518614, 0.263158, 0.158812) and P4(0.3333, 0.4685, 0) are unstable. Then, taking nearby P2, a desired initial point P\* (0.5, 0.3, 0.2), chaotic attractors drawn, **Figure 31**.

In the process of stabilizing the desired point (0.5, 0.3, 0.2), calculations performed to replace parameters *a* = 4.1, *d* = 3.5 and *r* = 3.8 to earlier case of map (18). After obtaining all concerned matrices, replacement matrix obtained as

$$p^\* = \begin{pmatrix} a \\ d \\ r \end{pmatrix} = \begin{pmatrix} 4.1035 \\ 1.05194 \\ 1.02707 \end{pmatrix}$$

**Figure 30.** *Phase plot and LCE plot of controlled system when c = 0.2, a = 3.91525, b = 2.99538.*

**Figure 31.** *Time series and attractors of unstable system.*

*c* = 0.2, time series, attractor and LCE plots are obtained and shown by **Figure 29**. Clearly the system (18) is showing chaos at (0.3, 0.5) with *a* = 3.7, *b* = 3.5, *c* = 0.2. Then, applying asymptotic stability discussed above for the map (18). For fixed value c = 0.2, unstable fixed point obtained as (0.25712, 0.49961). Nearby this point

> *b*

For the case when *c* = 0.2; new values of *a* and *b*; *a* = 3.91525, *b* = 2.99538 along with initial point (0.3, 0.5)a phase plot and a plot of Lyapunov exponents (LEC),

**Next,** we have considered three dimensional food chain model, [23], written as

xnþ<sup>1</sup> ¼ a xnð Þ� 1 � xn b xnyn ynþ<sup>1</sup> <sup>¼</sup> c xn yn � d yn zn

<sup>¼</sup> **<sup>3</sup>***:***<sup>7</sup> 3***:***5** 

. When above-mentioned

**AD** <sup>¼</sup> �**0***:***<sup>27</sup>** �**1***:***<sup>05</sup> 1***:***75 1***:***05** 

**BD** <sup>¼</sup> **<sup>0</sup>***:***<sup>21</sup>** �**0***:***<sup>15</sup> 0 0***:***15** 

**CR** <sup>¼</sup> **<sup>3</sup>***:***<sup>79669</sup>** �**4***:***<sup>76117</sup>**

**<sup>p</sup> <sup>∗</sup>** <sup>¼</sup> **<sup>3</sup>***:***<sup>91525</sup>**

**11***:***6577** �**0***:***66615** 

> **2***:***99538**

take initial point (0.3, 0.5) and *<sup>p</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>a</sup>*

**AR** <sup>¼</sup> **<sup>0</sup>***:***<sup>048652</sup>** �**0***:***<sup>899924</sup> 1***:***74865 0***:***900078** 

*Time series graphs, attractor and LCE plots of the unstable system.*

*A Collection of Papers on Chaos Theory and Its Applications*

**BR** <sup>¼</sup> **<sup>0</sup>***:***<sup>19101</sup>** �**0***:***<sup>128462</sup>**

**0 0***:***85641** 

**CD** <sup>¼</sup> **<sup>2</sup>***:***<sup>28571</sup>** �**4***:***<sup>7619</sup> 11***:***6667 0***:***333333** 

**0 0***:***128462** 

method applied, one obtains matrices:

**Figure 29.**

**CM** <sup>¼</sup> **<sup>0</sup>***:***90957 0**

are given in **Figure 30**.

*3.2.2 Food chain model*

**204**

**Figure 32.** *Phase plot and LCE plot of map (19) showing regular motion and chaos is controlled.*

At these new parameter values of *a*, *d* and *r*, the phase plot and the plot of Lyapunov exponents of map (19) obtained, **Figure 32**. These show chaotic motion controlled and the system returns to regularity.
