**Numerical Simulations:**

Drawing bifurcation diagrams and calculating Lyapunov exponents, topological entropy and correlation dimensions of the system for different cases have investigated performing numerical simulations. For values of the control parameters following ranges proposed: *a*∈ ½ � 0, 50 , *c*∈½ � �0*:*4, 0*:*4 , *b* ¼ 0*:*1, *d* ¼ 0*:*5, *t* ¼ 3, 4, 5.

Case 1: Taking *c* ¼ *d*, bifurcation diagrams are drawn along the directions *x* and *y*, by varying *c* for cases *t* = 3, 4, 5 and certain fixed values of other parameters as shown in **Figure 13**. Then, plots of attractors have been obtained for parameters *a* ¼ 25, *b* ¼ 0*:*1, *t* ¼ 3 and (i) for regular case *c* ¼ *d* ¼ 0*:*32 and (ii) for chaotic case *c* ¼ *d* ¼ 0*:*18 and shown in **Figure 14**. In each case when *t* = 3, 4, 5, bifurcations show period doubling leading to chaos and then to regularity. Also, bistability and folding nature of phenomena are appearing here.

## **Lyapunov Exponents & Topological Entropies**:

For chaotic evolution, when *a* ¼ 25, *b* ¼ 0*:*1, *t* ¼ 3,*c* ¼ *d* ¼ 0*:*18, Lyapunov exponents are obtained shown in **Figure 15**. Numerical investigations further proceeded for calculation of topological entropies. In **Figure 16**, plots of topological entropies are presented for *t* = 3, 4, 5 and for different ranges of parameter *c:* Analysis of these plots, gives an impression that for the case *t* = 3, system shows enough complexity in the range 0.05 ≤ *c* ≤ 0.23. For the case *t* = 4, the system shows high complexity in the range 0 ≤ *c* ≤ 0.22 and in case *t* = 5, high complexity appears in 0 ≤ *c* ≤ 0.44.

an efficient and practical method in comparison to others, like box counting etc. The procedure to obtain correlation dimension follows from steps of calculations in [73]: For case *t* = 3 and *a* = 25, *b* = 0.1, *c* = 0.28, *d* = 0.12, correlation integral data calculated and its plot is obtained, **Figure 20**. The linear fit of correlation integral

*Three cases of bifurcation scenarios of map (8) for parameters c* ¼ *d: (a) t= 3, a* ¼ 25, *b* ¼ 0*:*1 *and 0* ≤ *c* ≤

*0.5; (b) t = 4, a* ¼ 35, *b* ¼ 0*:*1 *and 0* ≤ *c* ≤ *0.65; (c) t = 5, a* ¼ 25, *b* ¼ 0*:*1 *and 0* ≤ *c* ≤ *0.5.*

*Y* ¼ 0*:*0581323*x* � 0*:*580866

Computation of correlation dimension carried out for more cases for different

Investigation on microeconomic chaotic disturbances and certain measure to control chaos appeared in some recent articles, [72, 93–95], extended here for

The y-intercept of this straight line is 0.580866. Therefore the correlation

dimension of the attractor in this case is, approximately, *Dc* = 0.581.

*2.2.2 Complexities in micro-economic Behrens Feichtinger model*

set of values of parameters as shown in **Table 1**.

*Chaos and Complexity Dynamics of Evolutionary Systems*

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data obtained as

**195**

**Figure 13.**

Case II: When c and d are different, bifurcation diagrams, **Figure 17**, shows clear picture of complex nature of the system.

In **Figure 18**, plots of Lyapunov exponents, (LCE's), for chaotic evolution for different cases discussed above are shown in the upper row and plots of topological entropies are shown in the lower row for these cases. For all the plots, parameters *a* = 25 and *b* = 0.1 are common. Here, topological entropy plots are drawn for different ranges of parameter c.

When parameters c and d both were allowed to vary, one gets 3D plots for topological entropies as shown here in **Figure 19**.

#### **Correlation dimensions**:

Being one of the characteristic invariants of nonlinear system dynamics, the correlation dimension provides measure of dimensionality for the underlying attractor of the system. A statistical method used to determine correlation dimension. It is

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

**Figure 13.**

reactions corresponding to gene expression and regulation. The discrete dynamic variables xn and yn describe the evolutions of the concentration levels of transcription factor proteins. The map represented by following pair of difference equations:

> <sup>n</sup> þ b yt n

<sup>n</sup> þ b xt n

With parameter values *a* ¼ 25, *b* ¼ 0*:*1, *c* ¼ *d* ¼ 0*:*18 and *t* ¼ 3, one obtains four different fixed points with coordinates (2.30409, 2.30409), (�2.52688, 2.44162),

For *c* 6¼ *d* and when *a* ¼ 25, *b* ¼ 0*:*1,*c* ¼ 0*:*18, *d* ¼ 0*:*42, and *t* ¼ 3, again, four

We intend to investigate certain dynamic behavior of system (8) for cases when *c* ¼ *d* and when *c* 6¼ *d* of evolutions showing irregularities due to presence of chaos

Drawing bifurcation diagrams and calculating Lyapunov exponents, topological

Case 1: Taking *c* ¼ *d*, bifurcation diagrams are drawn along the directions *x* and *y*, by varying *c* for cases *t* = 3, 4, 5 and certain fixed values of other parameters as shown in **Figure 13**. Then, plots of attractors have been obtained for parameters *a* ¼ 25, *b* ¼ 0*:*1, *t* ¼ 3 and (i) for regular case *c* ¼ *d* ¼ 0*:*32 and (ii) for chaotic case *c* ¼ *d* ¼ 0*:*18 and shown in **Figure 14**. In each case when *t* = 3, 4, 5, bifurcations show period doubling leading to chaos and then to regularity. Also, bistability and folding nature

For chaotic evolution, when *a* ¼ 25, *b* ¼ 0*:*1, *t* ¼ 3,*c* ¼ *d* ¼ 0*:*18, Lyapunov exponents are obtained shown in **Figure 15**. Numerical investigations further proceeded for calculation of topological entropies. In **Figure 16**, plots of topological entropies are presented for *t* = 3, 4, 5 and for different ranges of parameter *c:* Analysis of these plots, gives an impression that for the case *t* = 3, system shows enough complexity in the range 0.05 ≤ *c* ≤ 0.23. For the case *t* = 4, the system shows high complexity in the

Case II: When c and d are different, bifurcation diagrams, **Figure 17**, shows clear

In **Figure 18**, plots of Lyapunov exponents, (LCE's), for chaotic evolution for different cases discussed above are shown in the upper row and plots of topological entropies are shown in the lower row for these cases. For all the plots, parameters *a* = 25 and *b* = 0.1 are common. Here, topological entropy plots are drawn for

When parameters c and d both were allowed to vary, one gets 3D plots for

Being one of the characteristic invariants of nonlinear system dynamics, the correlation dimension provides measure of dimensionality for the underlying attractor of the system. A statistical method used to determine correlation dimension. It is

range 0 ≤ *c* ≤ 0.22 and in case *t* = 5, high complexity appears in 0 ≤ *c* ≤ 0.44.

þ c xn

þ dyn (8)

xnþ<sup>1</sup> <sup>¼</sup> <sup>a</sup>

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

ynþ<sup>1</sup> <sup>¼</sup> <sup>a</sup>

(2.44162, �2.52866), (�2.39464, �2.39464 ) and all are unstable.

and complexity.

*d* ¼ 0*:*5, *t* ¼ 3, 4, 5.

**Numerical Simulations:**

of phenomena are appearing here.

picture of complex nature of the system.

topological entropies as shown here in **Figure 19**.

different ranges of parameter c.

**Correlation dimensions**:

**194**

**Lyapunov Exponents & Topological Entropies**:

1 þ ð Þ 1 � b xt

1 þ ð Þ 1 � b yt

unstable fixed points exists as (2.2832, 2.5413), (�2.5458, 2.6566), (2.4613, �2.7288), (�2.3744, �2.61705).Therefore, for all these the cases, orbit with initial point taken nearby any of the fixed points be unstable and may be chaotic also.

entropy and correlation dimensions of the system for different cases have investigated performing numerical simulations. For values of the control parameters following ranges proposed: *a*∈ ½ � 0, 50 , *c*∈½ � �0*:*4, 0*:*4 , *b* ¼ 0*:*1,

*Three cases of bifurcation scenarios of map (8) for parameters c* ¼ *d: (a) t= 3, a* ¼ 25, *b* ¼ 0*:*1 *and 0* ≤ *c* ≤ *0.5; (b) t = 4, a* ¼ 35, *b* ¼ 0*:*1 *and 0* ≤ *c* ≤ *0.65; (c) t = 5, a* ¼ 25, *b* ¼ 0*:*1 *and 0* ≤ *c* ≤ *0.5.*

an efficient and practical method in comparison to others, like box counting etc. The procedure to obtain correlation dimension follows from steps of calculations in [73]:

For case *t* = 3 and *a* = 25, *b* = 0.1, *c* = 0.28, *d* = 0.12, correlation integral data calculated and its plot is obtained, **Figure 20**. The linear fit of correlation integral data obtained as

$$Y = 0.0581323x - 0.580866$$

The y-intercept of this straight line is 0.580866. Therefore the correlation dimension of the attractor in this case is, approximately, *Dc* = 0.581.

Computation of correlation dimension carried out for more cases for different set of values of parameters as shown in **Table 1**.

#### *2.2.2 Complexities in micro-economic Behrens Feichtinger model*

Investigation on microeconomic chaotic disturbances and certain measure to control chaos appeared in some recent articles, [72, 93–95], extended here for

**Figure 14.**

*Figures (a), (b), (c) correspond to time series, phase plane attractors and Lyapunov exponents; upper row is for regular case and the lower row is for chaotic case of map (8). Parameters values are taken as a* ¼ 25, *b* ¼ 0*:*1, *t* ¼ 3 *and (i) for regular case c* ¼ *d* ¼ 0*:*32 *and (ii) for chaotic case c* ¼ *d* ¼ 0*:*18*.*

**Figure 15.**

*Plots of Lyapunov exponents for chaotic evolution of map (8). Parameters are a* ¼ 25, *b* ¼ 0*:*1, *t* ¼ 3,*c* ¼ *d* ¼ 0*:*18 *and when evolving from initial point (2.1, 2.1).*

**Figure 16.**

*Plots of topological entropy for map (8) when parameter c* ¼ *d. From left: (i) t = 3, a* ¼ 25, *b* ¼ 0*:*1 *and 0* ≤ *c* ≤ *0.5; (ii) t = 4, a* ¼ 35, *b* ¼ 0*:*1 *and 0* ≤ *c* ≤ *0.65; (iii) t = 5, a* ¼ 25, *b* ¼ 0*:*1 *and 0* ≤ *c* ≤ *0.8.*

complexity analysis. The problem proposed as an micro economic model of two firms X and Y competing on the same market of goods having asymmetric strategies. The sales *xn* and *yn* of both firms are evolving in discrete time steps.

$$\mathbf{x}\_{\mathbf{n}+1} = \left(\mathbf{1} - \alpha\right)\mathbf{x}\_{\mathbf{n}} + \frac{\mathbf{a}}{\mathbf{1} + \mathbf{e}^{\left[-\mathbf{c}\cdot\left(\mathbf{x}\_{\mathbf{n}} - \mathbf{y}\_{\mathbf{n}}\right)\right]}}$$

$$\mathbf{y}\_{\mathbf{n}+1} = \left(\mathbf{1} - \beta\right)\mathbf{y}\_{\mathbf{n}} + \frac{\mathbf{b}}{\mathbf{1} + \mathbf{e}^{\left[-\mathbf{c}\cdot\left(\mathbf{X}\_{\mathbf{n}} - \mathbf{Y}\_{\mathbf{n}}\right)\right]}}\tag{9}$$

where *α*, *β* (0 <*α*, *β* < 1) are the time rates at which the sales of both firm decays in the absence of investments. Parameters *a*, *b* describe the investment effectiveness of both the firms. Parameter *c* is an "elasticity" measure of the investment strategies. For parameter values *α* = 0.46¸ *β* = 0.7, *a* = 0.16, *b* = 0.9, *c* = 105, we have

*Bifurcation plots when c* 6¼ *d for different ranges of parameter c. Cases (a), (b), (c), corresponds to t = 3, t = 4,*

*t = 5. Parameters are a* ¼ 25, *b* ¼ 0*:*1 *and d* ¼ 0*:*20 *for plots (a) & (c) and d = 0.30 for plot (b).*

Bifurcation diagrams for system (9) obtained for *α* = 0.46¸ *β* = 0.7, *a* = 0.16, *b* = 0.9 and by varying parameter *c*, 8 ≤ *c* ≤ 160 and in close range, 6 ≤ *c* ≤ 8, **Figure 21**. Then, again it obtained for values *α* = 0.46¸ *β* = 0.7, *a* = 0.16, *b* = 0.6, *c* = 110 and 0 ≤ *a* ≤ 0.4, **Figure 22**. Appearance of period doubling followed by chaos

Time series plots and a plot of chaotic attractor obtained for values *a* = 0.16, *b* = 0.9, *c* = 105, α = 0.46, β = 0.7 of system (9) shown in **Figure 23**. Plots shown in

**Topological Entropies:** Topological entropies calculated numerically and plot-

ted. These are shown in **Figure 25**. One finds significant increase topological

observed the chaotic attractor of this model.

*Chaos and Complexity Dynamics of Evolutionary Systems*

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**Figure 24** are of LCEs for the chaotic motion.

**Bifurcation Diagram**:

visible from these figures.

Attractors:

**197**

**Figure 17.**

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**Figure 17.**

*Bifurcation plots when c* 6¼ *d for different ranges of parameter c. Cases (a), (b), (c), corresponds to t = 3, t = 4, t = 5. Parameters are a* ¼ 25, *b* ¼ 0*:*1 *and d* ¼ 0*:*20 *for plots (a) & (c) and d = 0.30 for plot (b).*

where *α*, *β* (0 <*α*, *β* < 1) are the time rates at which the sales of both firm decays in the absence of investments. Parameters *a*, *b* describe the investment effectiveness of both the firms. Parameter *c* is an "elasticity" measure of the investment strategies. For parameter values *α* = 0.46¸ *β* = 0.7, *a* = 0.16, *b* = 0.9, *c* = 105, we have observed the chaotic attractor of this model.

### **Bifurcation Diagram**:

Bifurcation diagrams for system (9) obtained for *α* = 0.46¸ *β* = 0.7, *a* = 0.16, *b* = 0.9 and by varying parameter *c*, 8 ≤ *c* ≤ 160 and in close range, 6 ≤ *c* ≤ 8, **Figure 21**. Then, again it obtained for values *α* = 0.46¸ *β* = 0.7, *a* = 0.16, *b* = 0.6, *c* = 110 and 0 ≤ *a* ≤ 0.4, **Figure 22**. Appearance of period doubling followed by chaos visible from these figures.

Attractors:

Time series plots and a plot of chaotic attractor obtained for values *a* = 0.16, *b* = 0.9, *c* = 105, α = 0.46, β = 0.7 of system (9) shown in **Figure 23**. Plots shown in **Figure 24** are of LCEs for the chaotic motion.

**Topological Entropies:** Topological entropies calculated numerically and plotted. These are shown in **Figure 25**. One finds significant increase topological

complexity analysis. The problem proposed as an micro economic model of two firms X and Y competing on the same market of goods having asymmetric strategies. The sales *xn* and *yn* of both firms are evolving in discrete time steps.

≤ *0.5; (ii) t = 4, a* ¼ 35, *b* ¼ 0*:*1 *and 0* ≤ *c* ≤ *0.65; (iii) t = 5, a* ¼ 25, *b* ¼ 0*:*1 *and 0* ≤ *c* ≤ *0.8.*

*Plots of topological entropy for map (8) when parameter c* ¼ *d. From left: (i) t = 3, a* ¼ 25, *b* ¼ 0*:*1 *and 0* ≤ *c*

*Plots of Lyapunov exponents for chaotic evolution of map (8). Parameters are a* ¼ 25, *b* ¼ 0*:*1, *t* ¼ 3,*c* ¼ *d* ¼

*Figures (a), (b), (c) correspond to time series, phase plane attractors and Lyapunov exponents; upper row is for regular case and the lower row is for chaotic case of map (8). Parameters values are taken as a* ¼ 25, *b* ¼ 0*:*1, *t* ¼ 3 *and (i) for regular case c* ¼ *d* ¼ 0*:*32 *and (ii) for chaotic case c* ¼ *d* ¼ 0*:*18*.*

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

a <sup>1</sup> <sup>þ</sup> <sup>e</sup> �c xn�<sup>y</sup> ½ � ð Þ<sup>n</sup>

b 1 þ e½ � �c Xð Þ <sup>n</sup>�Yn

(9)

xnþ<sup>1</sup> ¼ ð Þ 1 � α xn þ

**Figure 15.**

**Figure 14.**

**Figure 16.**

**196**

0*:*18 *and when evolving from initial point (2.1, 2.1).*

ynþ<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>β</sup> yn <sup>þ</sup>

**Cases (t)/Parameters a b c d Approximate Dc** t = 3 25 0.1 0.28 0.12 0.581 t = 4 25 0.1 0.18 0.18 0.645 t = 5 25 0.1 0.18 0.18 0.703 t = 4 25 0.1 0.28 0.12 0.676 t = 5 25 0.1 0.28 0.12 0.772 t = 3 35 0.1 0.2 0.2 0.877 t = 4 35 0.1 0.2 0.2 0.618 t = 5 35 0.1 0.2 0.2 1.264

*Bifurcation diagrams of system (9) with respect to coordinates x and y. Lower plots are correspond to bifurcations in close range to indicate the appearance of periodic windows within bifurcation. α = 0.46¸ β = 0.7,*

*Bifurcation of map (9) α = 0.46¸ β = 0.7, a = 0.16, b = 0.6, c = 110 and 0* ≤ *a* ≤ *0.4*

**Table 1.**

**Figure 21.**

**Figure 22.**

**199**

*a = 0.16, b = 0.9, 8* ≤ *c* ≤ *160 & 6* ≤ *c* ≤ *8.*

*Correlation Dimensions for different sets of parameters.*

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#### **Figure 18.**

*Upper row plots are for LCE's and lower row plots are for topological entropies. Plots with (a), (b), (c) are respectively corresponds to the cases t = 3, 4, 5. Parameters a = 25, b = 0.1 are common for all the plots. Then, for (b) & (c) LCE's plots, c = 0.2, d = 0.15 and that for plot (c) , c = 0.28, d = 0.12. For lower row topological entropy plots, except parameter t, parameters a = 25, b = 0.1, d = 0.15 are common for all.*

**Figure 19.**

*3D plots for topological entropy variations. Parameters values are taken as a = 25, b = 0.1 and then 0* ≤ c ≤ *0.5 & 0* ≤ d ≤ *0.5.*

entropy where the system shows regularity, (e.g., 20 ≤ *c* ≤ 75), and for values *α* = 0*.*46*, β* = 0*.*7*, a* = 0*.*16 and *b* = 0*.*9. This shows presence of complexities though there is no chaos.

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


### **Table 1.**

*Correlation Dimensions for different sets of parameters.*

#### **Figure 21.**

*Bifurcation diagrams of system (9) with respect to coordinates x and y. Lower plots are correspond to bifurcations in close range to indicate the appearance of periodic windows within bifurcation. α = 0.46¸ β = 0.7, a = 0.16, b = 0.9, 8* ≤ *c* ≤ *160 & 6* ≤ *c* ≤ *8.*

**Figure 22.** *Bifurcation of map (9) α = 0.46¸ β = 0.7, a = 0.16, b = 0.6, c = 110 and 0* ≤ *a* ≤ *0.4*

entropy where the system shows regularity, (e.g., 20 ≤ *c* ≤ 75), and for values *α* = 0*.*46*, β* = 0*.*7*, a* = 0*.*16 and *b* = 0*.*9. This shows presence of complexities though there

*Plot of correlation integral curve for t = 3 and a = 25, b = 0.1, c = 0.28, d = 0.12.*

*3D plots for topological entropy variations. Parameters values are taken as a = 25, b = 0.1 and then 0* ≤ c ≤ *0.5*

*Upper row plots are for LCE's and lower row plots are for topological entropies. Plots with (a), (b), (c) are respectively corresponds to the cases t = 3, 4, 5. Parameters a = 25, b = 0.1 are common for all the plots. Then, for (b) & (c) LCE's plots, c = 0.2, d = 0.15 and that for plot (c) , c = 0.28, d = 0.12. For lower row topological*

*entropy plots, except parameter t, parameters a = 25, b = 0.1, d = 0.15 are common for all.*

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

is no chaos.

**198**

**Figure 20.**

**Figure 19.**

**Figure 18.**

*& 0* ≤ d ≤ *0.5.*

dy

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**Figure 28**.

**Figure 26.**

**Figure 27.**

*Figure 26.*

**201**

doubling phenomena.

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

dz

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

Bifurcation diagram for predator z while varying prey parameter b shown there,

Plots of time series for x(t), for cases of chaos, are given in **Figure 27** and that of Lyapunov exponents, (LCEs), of chaotic attractors shown in last two plots in

In conclusion, one observes that the system (10) evolve into chaos after period

*Periodic bifurcations and chaotic attractor formations of Volterra – Petzoldt model for different values of c fixed*

*Plots of time series curves for x(t) for chaotic evolutions for values of c. Other parameters are same as in*

*parameters a = 1, b = 1, α <sup>1</sup> = 0.205, α <sup>2</sup> = 1, k1 = 0.05, k2 = 0, w = 0.006.*

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

Petzoldt [86], is interesting. Periodic bifurcations and chaotic attractor of this model for different parameter space are presented in the figure, **Figure 26**.

y z 1 þ k2y

(10)

y z 1 þ k2y

**Figure 23.**

*Time series plots and chaotic attractor of the system (9) for a = 0.16, b = 0.9, c = 105, α = 0.46, β = 0.7 and initial condition (0.1, 0.1).*

**Figure 24.**

*Plots of Lyapunov exponents for chaotic evolution of the system (9) for a = 0.16, b = 0.9, c = 105, α = 0.46, β = 0.7.*

#### **Figure 25.**

*Plots of topological entropies: (a) left 2D plot is obtained for 12* ≤ c ≤ *170 and values of* a *= 0*.*16*, b *= 0*.*9*, α *= 0*.*46 and* β *= 0*.*7 and (b) right 3D plot is for 120* ≤ c ≤ *150 and 0* ≤ a ≤ *0*.*4 keeping same values for* α *and* β*.*

## **Correlation dimension:**

Following steps used for map (8), correlation dimension of chaotic the attractor for values α = 0.46, β = 0.7, *a* = 0.16, *b* = 0.9, *c* = 105, obtained as *Dc* = 0.064

### *2.2.3 Continuous Volterra-Petzoldt Model*

A continuous 2-dimensional Lotka – Volterra type predator� prey model of constant period chaotic amplitude, (UPCA model), proposed by Petzoldt, [96] based on works, [97, 98], written as

$$\frac{d\mathbf{x}}{dt} = \mathbf{a}\,\mathbf{x} - \alpha\_1 \frac{\mathbf{x}\,\mathbf{y}}{1 + \mathbf{k}\_1 \mathbf{x}}$$

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$$\frac{\text{dy}}{\text{dt}} = -\mathbf{b}\,\,\mathbf{y} + \alpha\_1 \frac{\mathbf{x}\,\,\mathbf{y}}{\mathbf{1} + \mathbf{k}\_1 \mathbf{x}} - \alpha\_2 \frac{\mathbf{y}\,\,\mathbf{z}}{\mathbf{1} + \mathbf{k}\_2 \mathbf{y}}$$

$$\frac{\text{dz}}{\text{dt}} = -\mathbf{c}(\mathbf{z} - \mathbf{w}) + \alpha\_2 \frac{\mathbf{y}\,\,\mathbf{z}}{\mathbf{1} + \mathbf{k}\_2 \mathbf{y}}\tag{10}$$

Bifurcation diagram for predator z while varying prey parameter b shown there, Petzoldt [86], is interesting. Periodic bifurcations and chaotic attractor of this model for different parameter space are presented in the figure, **Figure 26**.

Plots of time series for x(t), for cases of chaos, are given in **Figure 27** and that of Lyapunov exponents, (LCEs), of chaotic attractors shown in last two plots in **Figure 28**.

In conclusion, one observes that the system (10) evolve into chaos after period doubling phenomena.

**Figure 26.**

**Correlation dimension:**

**Figure 23.**

**Figure 24.**

**Figure 25.**

**200**

*β = 0.7.*

*initial condition (0.1, 0.1).*

*2.2.3 Continuous Volterra-Petzoldt Model*

based on works, [97, 98], written as

Following steps used for map (8), correlation dimension of chaotic the attractor

*Plots of topological entropies: (a) left 2D plot is obtained for 12* ≤ c ≤ *170 and values of* a *= 0*.*16*, b *= 0*.*9*, α *= 0*.*46 and* β *= 0*.*7 and (b) right 3D plot is for 120* ≤ c ≤ *150 and 0* ≤ a ≤ *0*.*4 keeping same values for* α *and* β*.*

*Time series plots and chaotic attractor of the system (9) for a = 0.16, b = 0.9, c = 105, α = 0.46, β = 0.7 and*

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*Plots of Lyapunov exponents for chaotic evolution of the system (9) for a = 0.16, b = 0.9, c = 105, α = 0.46,*

A continuous 2-dimensional Lotka – Volterra type predator� prey model of constant period chaotic amplitude, (UPCA model), proposed by Petzoldt, [96]

> x y 1 þ k1x

for values α = 0.46, β = 0.7, *a* = 0.16, *b* = 0.9, *c* = 105, obtained as *Dc* = 0.064

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

dx

*Periodic bifurcations and chaotic attractor formations of Volterra – Petzoldt model for different values of c fixed parameters a = 1, b = 1, α <sup>1</sup> = 0.205, α <sup>2</sup> = 1, k1 = 0.05, k2 = 0, w = 0.006.*

**Figure 27.** *Plots of time series curves for x(t) for chaotic evolutions for values of c. Other parameters are same as in Figure 26.*

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

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And **en = Xn-Yn.**

matrix **P\*** from (13).

**3.2 Applications**

given by:

**203**

n ! ∞, then equation (14) implies

**J** ¼

*Chaos and Complexity Dynamics of Evolutionary Systems*

*∂* **f** *∂***x**

0

BB@

*∂***g** *∂***x**

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

*∂* **f** *∂***y**

1

CCA , **<sup>J</sup> <sup>∗</sup>** <sup>¼</sup>

**en**þ**<sup>1</sup>** ¼ ð Þ **AR**–**BD CR en** þ f g **AR**–**AD** þ **BD**ð Þ **CD**–**CR Yn** þ ð Þ **BR**–**BDCM p** (14)

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

From these, one can obtain matrices **CM, CD, CR** and then control parameter

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

**Rank B**ð Þ¼ **<sup>D</sup> Rank B**ð Þ¼ **<sup>D</sup>**, **AD**–**AR Rank B**ð Þ **<sup>D</sup>**,**BR**

Considered a prey-predator model where both species evolve with logistic rule

xnþ<sup>1</sup> ¼ a xnð Þ� 1 � xn b xn yn

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,

ynþ<sup>1</sup> <sup>¼</sup> c yn <sup>1</sup> � yn

**AR**–**AD** þ **BD** ð Þ¼ **CD**–**CR 0** ¼ **> BD**ð Þ¼ **CD**–**CR AD** � **AR** (15)

And **BR**–**BD CM** ¼ **0** ¼ **> BDCM** ¼ **BR** (16)

**AR**–**BD CR** ¼ �**I** (17)

� � <sup>þ</sup> b xn yn (18)

*∂* **f** *∂***a**

0

BB@

*∂***g** *∂***a** *∂* **f** *∂***b** 1

CCA

*∂***g** *∂***b**

**<sup>P</sup><sup>∗</sup>** <sup>¼</sup> **CR Xn** <sup>þ</sup> **CM <sup>p</sup>**–**CDYn** (13)

*∂***g** *∂***y**

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

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

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

and also influencing each other, [30], written as

**Figure 28.** *Plots of LCEs of chaotic attractors of model (1) for values of c. Other parameters are same as in Figure 26.*
