*3.2.2.1 Pulsive Feedback Technique to Chaos Control*

Pulsive chaos control technique is discussed in detail in recent articles, [86–88]. As an application of this technique let us consider a simple 2 – dimension discrete time Burger's map

### *3.2.3 Controlling Chaos in 2-D Burger's Map*

$$\mathbf{x\_{n+1}} = (\mathbf{1} - \mathbf{a})\ \mathbf{x\_n} - \mathbf{y\_n^2}$$

$$\mathbf{y\_{n+1}} = (\mathbf{1} + \mathbf{b})\ \mathbf{y\_n} + \mathbf{x\_n}\ \mathbf{y\_n} \tag{20}$$

Applying the method of pulsive feedback, and re-writing eq. (10) as

x y

<sup>1</sup> <sup>þ</sup> k1x <sup>þ</sup> <sup>∈</sup> ð Þ <sup>x</sup> � <sup>19</sup>*:*<sup>5374</sup>

<sup>þ</sup> <sup>∈</sup> <sup>y</sup> � <sup>9</sup>*:*<sup>64328</sup>

þ ∈ð Þ z � 1*:*02602 (22)

y z 1 þ k2y

y z 1 þ k2y

Then, using stability analysis, for stabilize the above unstable point P\*, one

Regular and chaotic evolutions observed in some 1-3 dimensional discrete and continuous nonlinear models, which have applications in different areas of science. Presence of complexity in these systems viewed by indications of significant increase in topological entropies in certain parameter spaces. More increase in topological entropy in a system signified the system is more complex. Bifurcation phenomena for different systems show interesting properties like bistability, folding, intermittency, chaos adding etc. which are not common to all nonlinear systems. Proper numerical simulations performed for each system to obtain regular and chaotic attractors, Lyapunov exponents (LCEs) as a measure of chaos, (evolution is regular if LCE < 0 and chaotic if LCE > 0), topological entropies and correlation dimensions for chaotic attractors. It appears from the plots of topological entropies that obtained for discrete models that complexity exists even in absence of chaos. Correlation dimensions obtained for chaotic attractors are nonintegers because these attractors bear fractal properties. A chaotic attractor is composed of complex pattern and so, in a variety of nonlinear evolving systems mea-

To control chaotic motion, techniques of asymptotic stability analysis and that of

pulsive feedback control applied here. Pulsive control technique applied to Volterra-Petzoldt map (10) and to Burger's map (20), show chaos successfully controlled and systems returned to regularity, **Figures 34** and **35**. Application of Pulsive control method perfectly controlled chaotic motions in systems (10), (20) shown here. Chaos is also controlled by this method for system (10), [72]. Asymptotic stability analysis method applied to a prey-predator system and to a food chain model, respectively, to maps (18) and (19), and chaos effectively controlled shown,

dx

*Chaos and Complexity Dynamics of Evolutionary Systems*

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

dt ¼ �b y <sup>þ</sup> <sup>α</sup><sup>1</sup>

dz

dy

*Chaos in Burgerger's map for a = 1, b = 0.9.*

obtains the parameter ε = �0.45.

**4. Discussions**

**207**

**Figure 33.**

dt <sup>¼</sup> a x � <sup>α</sup><sup>1</sup>

dt ¼ �c zð Þþ � <sup>w</sup> <sup>α</sup><sup>2</sup>

surement of topological entropy is equally important, [63–67].

x y <sup>1</sup> <sup>þ</sup> k1x � <sup>α</sup><sup>2</sup>

where *a* and *b* are non-zero parameters . This map evolve chaotically when *a*= 0.9, *b*=0.856. To control chaotic motion we have used pulsive feedback control technique, Litak et al. [86] by

Here (�0.9, 0.948683) is an unstable fixed point of the original Burger's map. It has been observed that above chaotic motion is controlled and display regular behavior after re-writing equations (1) as follows:

$$\mathbf{x\_{n+1}} = (\mathbf{1} - \mathbf{a})\,\mathbf{x\_n} - \mathbf{y\_n^2} + \mathbf{E} \,(\mathbf{x} + \mathbf{0.9})$$

$$\mathbf{y\_{n+1}} = (\mathbf{1} + \mathbf{b})\,\mathbf{y\_n} + \mathbf{x\_n}\,\mathbf{y\_n} + \mathbf{E} \,(\mathbf{y} - \mathbf{0.948683})\tag{21}$$

Repeating stability analysis for system (2) with the fixed point (�0.9, 0.948683), one finds this point be stable if ε < 0.45. So, taking ε = 0.435, phase plot obtained as shown in **Figure 34**, indicates chaotic motion, **Figure 33**, is now controlled.

#### *3.2.4 Controlling Chaos in Volterra-Petzoldt Map*

Evolution of Volterra-Petzoldt map already discussed in Section 2, Eq. (10). For parameters *a* = 1, *b* = 1, *c* = 9.7, *α*<sup>1</sup> = 0.205, *α*<sup>2</sup> = 1, *k*<sup>1</sup> = 0.05, *k*<sup>2</sup> =0, *w* = 0.006, this map shows chaotic motion. An unstable equilibrium solution P\* (19.5374, 9.64328, 1.02602) exists in this case.

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

**Figure 33.** *Chaos in Burgerger's map for a = 1, b = 0.9.*

Applying the method of pulsive feedback, and re-writing eq. (10) as

$$\frac{\text{d}\mathbf{x}}{\text{d}\mathbf{t}} = \mathbf{a}\,\mathbf{x} - \alpha\_1 \frac{\mathbf{x}\,\mathbf{y}}{1 + \mathbf{k}\_1 \mathbf{x}} + \in (\mathbf{x} - 19.5374)$$

$$\frac{\text{d}\mathbf{y}}{\text{d}\mathbf{t}} = -\mathbf{b}\,\mathbf{y} + \alpha\_1 \frac{\mathbf{x}\,\mathbf{y}}{1 + \mathbf{k}\_1 \mathbf{x}} - \alpha\_2 \frac{\mathbf{y}\,\mathbf{z}}{1 + \mathbf{k}\_2 \mathbf{y}} + \in (\mathbf{y} - 9.64328)$$

$$\frac{\text{d}\mathbf{z}}{\text{d}\mathbf{t}} = -\mathbf{c}(\mathbf{z} - \mathbf{w}) + \alpha\_2 \frac{\mathbf{y}\,\mathbf{z}}{1 + \mathbf{k}\_2 \mathbf{y}} + \in (\mathbf{z} - 1.02602) \tag{22}$$

Then, using stability analysis, for stabilize the above unstable point P\*, one obtains the parameter ε = �0.45.
