*2.2.1 Two-Gene Andrecut-Kauffman System*

Chaos and complexity study of a discrete two-dimensional map for two-gene system, proposed by Andrecut and Kaufmann, investigated recently, [35, 71, 92]. The map used to investigate the dynamics of two-gene system for chemical

#### **Figure 9.**

*Bifurcation plots of Blood Cell model for* <sup>¼</sup> <sup>8</sup> *, b = 1.1* � *<sup>10</sup><sup>6</sup> then for (a) s= 16 and 0* <sup>≤</sup> *<sup>a</sup>* <sup>≤</sup> *1 and for (b) a = 0.1 and* 3*:*5≤*s*≤16*:*

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

**Figure 10.** *Bifurcation of Blood Cell model when* <sup>3</sup>*:*<sup>0</sup> <sup>≤</sup>s≤7*:*<sup>5</sup> *and* <sup>a</sup> <sup>¼</sup> <sup>0</sup>*:*8, r <sup>¼</sup> <sup>8</sup>*,* <sup>b</sup> *= 1.1* � *106*

*.*

**Figure 11.**

Interesting bifurcations observed for this map: For *<sup>b</sup>* = 1.1 � <sup>10</sup><sup>6</sup>

*Chaotic time series plot of map (6) for a = 0.8, b = 10, r = 6, s = 2.5 and x*<sup>0</sup> ¼ 1*:*5*.*

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

bifurcation diagrams are drawn; (a) in one for *s* =16 and 0 ≤*a*≤1, and (b) in another for *a* ¼ 0*:*8 and 3*:*5≤*s* ≤16*:*0 and shown in **Figure 9**. In former case one finds initially period doubling bifurcation followed by loops before emergence of chaos. In later case, one finds some typical type of bifurcation showing chaos adding, folding and the bistability like phenomena. A magnification of right figure,

**Figure 10**, for smaller range, 4*:*5≤*s*≤ 8*:*5, justifying chaos adding behavior. Regular and chaotic motion experienced through bifurcation diagrams, **Figures 9** and **10**, again confirmed by plots of Lyapunov exponents, **Figure 11**. This system, bears enough complexity and, as its measure, plot of topological entropies,

**Figure 12**, obtained for values *<sup>r</sup>* <sup>¼</sup> 6, *<sup>s</sup>* <sup>¼</sup> 16 and *<sup>b</sup>* = 1.1 � <sup>10</sup><sup>6</sup> and 0≤*a*≤1. Fluctuations in increase of topological entropies appear, approximately, in the

The correlation dimension of its chaotic attractor for values *a* ¼ 0*:*78, when

Chaos and complexity study of a discrete two-dimensional map for two-gene system, proposed by Andrecut and Kaufmann, investigated recently, [35, 71, 92]. The map used to investigate the dynamics of two-gene system for chemical

*Bifurcation plots of Blood Cell model for* <sup>¼</sup> <sup>8</sup> *, b = 1.1* � *<sup>10</sup><sup>6</sup> then for (a) s= 16 and 0* <sup>≤</sup> *<sup>a</sup>* <sup>≤</sup> *1 and for*

region 0*:*25≤*a*≤0*:*95 indicate existence of complexity.

*<sup>r</sup>* <sup>¼</sup> 6, *<sup>s</sup>* <sup>¼</sup> 16 and *<sup>b</sup>* = 1.1 � 106 is obtained as *Dc* ffi <sup>0</sup>*:*253.

**2.2 Two-dimensional models**

**Figure 8.**

**Figure 9.**

**192**

*(b) a = 0.1 and* 3*:*5≤ *s*≤16*:*

*2.2.1 Two-Gene Andrecut-Kauffman System*

, *r* = 8, two

*LCE Plots for* <sup>¼</sup> <sup>6</sup> *, s* <sup>¼</sup> <sup>16</sup> *and b = 1.1* � *106 , negative and positive values of LCEs, respectively, below and above the zero line show the regular and chaotic zones of parameter space.*

**Figure 12.** *Topological entropy plot for r* <sup>¼</sup> 6, *<sup>s</sup>* <sup>¼</sup> <sup>16</sup> *and b = 1.1* � *106 and* <sup>0</sup>≤*<sup>a</sup>* <sup>≤</sup>1*.*

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:

$$\mathbf{x}\_{\mathbf{n}+1} = \frac{\mathbf{a}}{\mathbf{1} + (\mathbf{1} - \mathbf{b})\ \mathbf{x}\_{\mathbf{n}}^{\mathbf{t}} + \mathbf{b}\ \mathbf{y}\_{\mathbf{n}}^{\mathbf{t}}} + \mathbf{c}\ \mathbf{x}\_{\mathbf{n}}$$

$$\mathbf{y}\_{\mathbf{n}+1} = \frac{\mathbf{a}}{\mathbf{1} + (\mathbf{1} - \mathbf{b})\ \mathbf{y}\_{\mathbf{n}}^{\mathbf{t}} + \mathbf{b}\ \mathbf{x}\_{\mathbf{n}}^{\mathbf{t}}} + \mathbf{d}\mathbf{y}\_{\mathbf{n}}\tag{8}$$

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

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 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.

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 and complexity.
