**4. Discussions**

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

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

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

xnþ<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>a</sup> xn � <sup>y</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

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

xnþ<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>a</sup> xn � y2

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

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,

n

<sup>n</sup> þ ∈ ð Þ x þ 0*:*9

ynþ<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>b</sup> yn <sup>þ</sup> xn yn <sup>þ</sup> <sup>∈</sup> <sup>y</sup> � <sup>0</sup>*:*<sup>948683</sup> (21)

ynþ<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>b</sup> yn <sup>þ</sup> xn yn (20)

controlled and the system returns to regularity.

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

technique, Litak et al. [86] by

1.02602) exists in this case.

controlled.

**206**

time Burger's map

**Figure 32.**

*3.2.2.1 Pulsive Feedback Technique to Chaos Control*

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

behavior after re-writing equations (1) as follows:

*3.2.4 Controlling Chaos in Volterra-Petzoldt Map*

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

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,

**Figure 34.** *Plot of regular attractor for a = 1, b = 0.9 and ε = 0.435.*

**Figure 35.** *Plots of chaotic attractor changing into regular attractor by application of pulsive feedback technique.*

respectively, through figures, **Figures 30** and **32**. Asymptotic stability analysis technique has some limitations explained in the articles where this method proposed, [83, 84]. Though there are many ways to control chaos in dynamical systems, [74], both the techniques applied here are perfect and very effective in controlling chaos, especially in real systems.

**Author details**

Lal Mohan Saha

**209**

Department of Mathematics, Shiv Nadar University, Gautam Budha Nagar, India

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: lmsaha.msf@gmail.com

*Chaos and Complexity Dynamics of Evolutionary Systems*

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

provided the original work is properly cited.
