**Calculations of Lyapunov Exponents, (LCEs):**

Lyapunov exponents, LCEs, for map (1), calculated for four cases, **Figure 4**, positive LCEs appearing above zero line clearly indicate chaotic motion and those below this line indicate regular motion.

**Topological Entropies:**

*0.5* ≤ *Q* ≤ *3.5 and 1.4* ≤ *Q* ≤ *1.8.*

**Figure 4.**

**Figure 5.**

**189**

*= 5.4 and 0.4* ≤ *Q* ≤ *2.5 & 1.4* ≤ *Q* ≤ *1.9.*

varying parameter Q while keeping parameter A = 5.4.

*Chaos and Complexity Dynamics of Evolutionary Systems*

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

Numerical calculations further proceeded to calculate topological entropies for system (1) and shown in **Figure 5**; where figures of upper row obtained by varying parameter A while keeping parameter Q = 2.76 and those of lower row obtained by

*Topological entropy plots: (a) for upper row Q = 2.76 and 4.0* ≤ *A* ≤ *5.5 & 4.7* ≤ *A* ≤ *5.3; (b) for lower row A*

*Plots of LCEs: (a) for the upper row Q = 2.76, 4.0* ≤ *A* ≤ *5.5 and 5.0* ≤ *A* ≤ *7.0; (b) for the lower row A = 5.4,*

**Figure 2.** *Bifurcation diagrams of map (1) for four cases: when A = 5.4 and parameter Q varies.*

**Figure 3.** *Chaotic time series plots with initial value x0 = 0.5: (a) A = 5.3, Q = 2.76 and (b) A = 5.4, Q = 2.9.*

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

#### **Figure 4.**

*A* ¼ 5*:*4 and varying *Q* in four different ranges, bifurcation diagrams are drawn, **Figure 2**. One observe clearly the appearance of periodic windows within chaotic region of bifurcations as an indication of intermittency and other complex phenomena. Periodic windows become gradually shorter and appearance become more

Both time series plots shown in **Figure 3** are for chaotic evolution of system (1) and correspond to parameters (a) ð Þ *A*, *Q* = (5.3, 2.76), due to which an unstable fixed point obtained as *x*� ¼ 0*:*58531, and parameters (b) ) ð Þ *A*, *Q* = (5.4, 2.9), due to which an unstable fixed point obtained as *x*� ¼ 0*:*572218. For both cases, initial point taken is *x*<sup>0</sup> ¼ 0*:*5 which lies nearby these points and so, also, unstable.

Lyapunov exponents, LCEs, for map (1), calculated for four cases, **Figure 4**, positive LCEs appearing above zero line clearly indicate chaotic motion and those

frequent while moving forward in parameter space.

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

**Calculations of Lyapunov Exponents, (LCEs):**

*Bifurcation diagrams of map (1) for four cases: when A = 5.4 and parameter Q varies.*

*Chaotic time series plots with initial value x0 = 0.5: (a) A = 5.3, Q = 2.76 and (b) A = 5.4, Q = 2.9.*

below this line indicate regular motion.

**Figure 2.**

**Figure 3.**

**188**

*Plots of LCEs: (a) for the upper row Q = 2.76, 4.0* ≤ *A* ≤ *5.5 and 5.0* ≤ *A* ≤ *7.0; (b) for the lower row A = 5.4, 0.5* ≤ *Q* ≤ *3.5 and 1.4* ≤ *Q* ≤ *1.8.*

#### **Topological Entropies:**

Numerical calculations further proceeded to calculate topological entropies for system (1) and shown in **Figure 5**; where figures of upper row obtained by varying parameter A while keeping parameter Q = 2.76 and those of lower row obtained by varying parameter Q while keeping parameter A = 5.4.

#### **Figure 5.**

*Topological entropy plots: (a) for upper row Q = 2.76 and 4.0* ≤ *A* ≤ *5.5 & 4.7* ≤ *A* ≤ *5.3; (b) for lower row A = 5.4 and 0.4* ≤ *Q* ≤ *2.5 & 1.4* ≤ *Q* ≤ *1.9.*

### **Correlation Dimension:**

Extending further the numerical study, correlation dimensions of system (1) calculated for a chaotic attractor by using Mathematica codes, [73].

Consider an orbit *O*ð Þ *x***<sup>1</sup>** = f g *x*1, *x*2, *x*3, *x*<sup>4</sup> … … , of a map *f* : *U* ! *U*, where *U*s an open bounded set in *<sup>R</sup>n*. To compute correlation dimension of *O x*ð Þ<sup>1</sup> , for a given positive real number *r*, we form the correlation integral,

$$\mathbf{C}(\mathbf{r}) = \lim\_{\mathbf{n} \to \infty} \frac{\mathbf{1}}{\mathbf{n} \ (\mathbf{n} + \mathbf{1})} \sum\_{\mathbf{i} \neq \mathbf{j}}^{\mathbf{n}} \mathbf{H} \left(\mathbf{r} - \|\mathbf{x}\_{\mathbf{i}} - \mathbf{x}\_{\mathbf{j}}\|\right) \tag{3}$$

The y-intercept of this straight line is 0*:*687605. Therefore the correlation

b. In a similar way, correlation dimension for second attractor of **Figure 3**, A = 5.4 and Q = 2.9, as *Dc* ¼ 0*:*56. Plots of correlation dimensions against

The population of red blood cells in a healthy human being oscillates within a certain tolerance interval in normal circumstances. But, sometimes, in presence of a disease such as anemia, this behavior fluctuate dramatically. A discrete model of

Let *xn*, *xn*þ<sup>1</sup> representing quantities of cells per unit volume (in millions) at time *n* and *n* þ 1, respectively and *pn*, *dn* are, respectively, the number of cells produced

dn ¼ a xn, a ∈½ � 0, 1

where b, r, s all positive parameters. With these our one-dimensional discrete

The case a ¼ 1 , means that during the time interval under consideration all cells that were alive at time *n* are destroyed. In such a case, above models simply comes as

e�sxn ,

pn <sup>¼</sup> b xð Þ<sup>n</sup> <sup>r</sup>

xnþ<sup>1</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>a</sup> xn <sup>þ</sup> b xð Þ<sup>n</sup> <sup>r</sup>

xnþ<sup>1</sup> <sup>¼</sup> b xð Þ<sup>n</sup> <sup>r</sup>

<sup>0</sup>*:*989813, *<sup>x</sup>*<sup>∗</sup> <sup>2</sup> <sup>¼</sup> <sup>3</sup>*:*53665 obtained for system (6) of which only x<sup>∗</sup>

For *<sup>a</sup>* <sup>¼</sup> <sup>0</sup>*:*8, *<sup>b</sup>* <sup>¼</sup> 10 , *<sup>r</sup>* <sup>¼</sup> 6 and *<sup>s</sup>* <sup>¼</sup> <sup>2</sup>*:*5 , three fixed points *<sup>x</sup>*<sup>∗</sup> <sup>0</sup> <sup>¼</sup> 0, *<sup>x</sup>*<sup>∗</sup> <sup>1</sup> <sup>¼</sup>

and other two are unstable. Chaotic motion observed for values of parameter *a* ¼ 0*:*8, *b* ¼ 10,*r* ¼ 6, *s* ¼ 2*:*5, as shown in the time series plot, **Figure 8**, with initial

*Plots of correlation dimensions: (a) with Q = 2.76 and varying A, (b) with A = 5.4 and varying Q.*

xnþ<sup>1</sup> ¼ xn þ pn � dn (5)

e�s xn (6)

<sup>0</sup> ¼ 0 is stable

e�s xn (7)

dimension of the attractor in this case is DC ¼ 0*:*69.

blood cell populations, Martelli, ([73], p: 35), presented here.

parameters A, Q shown in **Figure 7**.

*Chaos and Complexity Dynamics of Evolutionary Systems*

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

*2.1.2 Dynamics of biological red cells model*

and destroyed during the nth generation then

model for blood cells populations comes as

Then, assuming that

condition *x*<sup>0</sup> ¼ 1*:*5.

**Figure 7.**

**191**

Where,

$$\mathbf{H}(\mathbf{x}) = \begin{cases} \mathbf{0}, & \mathbf{x} < \mathbf{0} \\ \mathbf{1}, & \mathbf{x} \ge \mathbf{0} \end{cases}$$

is the unit-step function, (Heaviside function). The summation indicates number of pairs of vectors closer to *r* when1≤i, j≤n and *i* 6¼ *j*.C rð Þ measures the density of pair of distinct vectors **xi** and **xj** that are closer to *r*.

The correlation dimension Dc of Oð Þ **x1** is then defined as

$$\mathbf{D}\_{\mathbf{c}} = \lim\_{\mathbf{r} \to 0} \frac{\log \mathbf{C}(\mathbf{r})}{\log \mathbf{r}} \tag{4}$$

To obtain Dc, log C rð Þ is plotted against log *r*, **Figure 6**, and then we find a straight line fitted to this curve. The intercept of this straight line on y-axis provides the value of the correlation dimension DC. Correlation dimensions of time series attractors, **Figure 3**, obtained as:

a. For first attractor, Q = 2.76, A = 5.3, a plot of the correlation integral curve is shown in **Figure 6**. Then, the linear fit of the correlation data used in this figure obtained as

$$y = 0.95661x + 0.687605$$

**Figure 6.** *Plot of correlation integral curve for A = 5.3, Q = 2.76 and x0 = 0.5.*

The y-intercept of this straight line is 0*:*687605. Therefore the correlation dimension of the attractor in this case is DC ¼ 0*:*69.

b. In a similar way, correlation dimension for second attractor of **Figure 3**, A = 5.4 and Q = 2.9, as *Dc* ¼ 0*:*56. Plots of correlation dimensions against parameters A, Q shown in **Figure 7**.
