Chaos and Complexity Dynamics of Evolutionary Systems

Chaotic phenomena and presence of complexity in various nonlinear dynamical systems extensively discussed in the context of recent researches. Discrete as well as continuous dynamical systems both considered here. Visualization of regularity and chaotic motion presented through bifurcation diagrams by varying a parameter of the system while keeping other parameters constant. In the processes, some perfect indicator of regularity and chaos discussed with appropriate examples. Measure of chaos in terms of Lyapunov exponents and that of complexity as increase in topological entropies discussed. The methodology to calculate these explained in details with exciting examples. Regular and chaotic attractors emerging during the study are drawn and analyzed. Correlation dimension, which provides the dimensionality of a chaotic attractor discussed in detail and calculated for different systems. Results obtained presented through graphics and in tabular form. Two techniques of chaos control, pulsive feedback control and asymptotic stability analysis, discussed and applied to control chaotic motion for certain cases. Finally, a brief discussion held for the concluded investigation.


Introduction
, [1], was first to acknowledge the possible existence of chaos in nonlinear systems while studying a 3-body problem comprising Sun, Moon and Earth. He noticed the dynamics of the system turned to be sensitive towards initial conditions, which was later termed as chaos. His results based on theoretical analysis and he could not demonstrate it because computers were not available at that time. Lorenz, a weather scientist, demonstrated existence of chaos by using a computer in 1963, [2], and in this way supported chaos theory of Poincaré. Thus, Lorenz provided the foundation of chaos theory and inspired a fundamental reappraisal of systems of nonlinearity in many disciplines of science, engineering, biological and medical sciences, atmospheric science, economics, social sciences and where not? In our everyday life, chaos happened frequently in various form like cyclone, tsunami, tornado, epidemics/pandemics etc. Spread of any uncontrollable form of disease in medical science is nothing but a chaotic and contagious nature of disease. Systematic studies in various areas resulting in numerous articles on chaos and nonlinear dynamics appeared in many well-reputed scientific journals, [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].

Dynamics of laser map
A highly simplified type discrete nonlinear model for laser system, arising from Laser Physics, described in articles, [12,50,[89][90][91]. The model describes evolution of certain Fabry-Perot cavity containing a saturable absorber and driven by an external laser represented by Here Q is the normalized input field and A is a parameter depends on the specifics of the parameters and A > 0. The fixed points of the map are the real root of equation This equation has either three real roots or one real and a pair of complex conjugate roots depending on parameter space A, Q ð Þ. Stability occur in the form of stability and bistability, [89]. 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 frequent while moving forward in parameter space.
Both time series plots shown in Figure 3 are for chaotic evolution of system (1) and correspond to parameters (a) A, Q ð Þ= (5.3, 2.76), due to which an unstable fixed point obtained as x Ã ¼ 0:58531, and parameters (b) ) A, Q ð Þ= (5.4, 2.9), due to which an unstable fixed point obtained as x Ã ¼ 0:572218. For both cases, initial point taken is x 0 ¼ 0:5 which lies nearby these points and so, also, unstable.

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:
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. Where, is the unit-step function, (Heaviside function). The summation indicates number of pairs of vectors closer to r when1 ≤ i, j ≤ n and i 6 ¼ j. C r ð Þ measures the density of pair of distinct vectors x i and x j that are closer to r.
The correlation dimension D c of O x 1 ð Þ is then defined as To obtain D c , log C r ð 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 D C . 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 ¼ 0:95661x þ 0:687605 The y-intercept of this straight line is 0:687605. Therefore the correlation dimension of the attractor in this case is D C ¼ 0:69 .
b. In a similar way, correlation dimension for second attractor of Figure 3, A = 5.4 and Q = 2.9, as D c ¼ 0:56. Plots of correlation dimensions against parameters A, Q shown in Figure 7.

Dynamics of biological red cells model
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 blood cell populations, Martelli, ( [73], p: 35), presented here. Let x n , x nþ1 representing quantities of cells per unit volume (in millions) at time n and n þ 1, respectively and p n , d n are, respectively, the number of cells produced and destroyed during the n th generation then Then, assuming that d n ¼ a x n , a ∈ 0, 1 ½ where b, r, s all positive parameters. With these our one-dimensional discrete model for blood cells populations comes as 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 For a ¼ 0:8, b ¼ 10 , r ¼ 6 and s ¼ 2:5 , three fixed points x * 0 ¼ 0, x * 1 ¼ 0:989813, x * 2 ¼ 3:53665 obtained for system (6) of which only x * 0 ¼ 0 is stable 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 condition x 0 ¼ 1:5. Interesting bifurcations observed for this map: For b = 1.1 Â 10 6 , r = 8, two 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, The correlation dimension of its chaotic attractor for values a ¼ 0:78, when r ¼ 6, s ¼ 16 and b = 1.1 Â 10 6 is obtained as D c ffi 0:253.

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     reactions corresponding to gene expression and regulation. The discrete dynamic variables x n and y n describe the evolutions of the concentration levels of transcription factor proteins. The map represented by following pair of difference equations: .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 6 ¼ d of evolutions showing irregularities due to presence of chaos and complexity.
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: 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 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.
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 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 The y-intercept of this straight line is 0.580866. Therefore the correlation dimension of the attractor in this case is, approximately, D c = 0.581.
Computation of correlation dimension carried out for more cases for different set of values of parameters as shown in Table 1.

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][94][95], extended here for 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 x n and y n of both firms are evolving in discrete time steps.
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.
Following two chaos controlling technique discussed here:

Asymptotic Stability Method
Asymptotic stability analysis to stabilize unstable fixed point and to control chaotic motion appeared in some recent researches, [83][84][85]. Though this method has some limitations, it is perfect way to control chaos in models where it can be applicable.

Description of the Method:
Dynamics of the actual map X n + 1 and that of the desired map Y n + 1 can be explained by following mapping: Also, the neighborhood dynamics of X n + 1 and Y n + 1 can be represented by the relation: Matrices A R , A D , B R , B D can be obtained from the following: Here, Figure 28. Figure 26.

Plots of LCEs of chaotic attractors of model (1) for values of c. Other parameters are same as in
Let a, b are two parameters of the system and (x n , y n ) 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 The control input parameter matrix p*can be given by Then, using (11)-(13), one obtains the following error equation: And e n = X n -Y n. Note that in equation (13) and (14) the coefficient matrices C R , C D and C M are to be determined so that if the error vector e n = X n -Y n is initialized as e 0 = 0, then it will be zero for all n future times. For asymptotic stability, we must have e n ! 0 as n ! ∞, then equation (14) implies And The necessary and sufficient condition for e n !0 as n!∞ is From these, one can obtain matrices C M , C D , C R and then control parameter matrix P* from (13).
A necessary and sufficient condition for the existence of matrices C M , C D , C R, given by:

Chaos Control in a 2-Dimensional Prey-Predator map
Considered a prey-predator model where both species evolve with logistic rule and also influencing each other, [30], written as x nþ1 ¼ a x n 1 À x n ð ÞÀb x n y n y nþ1 ¼ c y n 1 À y n À Á þ b x n y n

Food chain model
Next, we have considered three dimensional food chain model, [23], written as x nþ1 ¼ a x n 1 À x n ð ÞÀb x n y n y nþ1 ¼ c x n y n À d y n z n Figure 29.
In the process of stabilizing the desired point (0.5, 0.3, 0.2), calculations performed to replace parameters a = 4.1, d = 3.5 and r = 3.8 to earlier case of map (18). After obtaining all concerned matrices, replacement matrix obtained as  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 controlled and the system returns to regularity.
As an application of this technique let us consider a simple 2dimension discrete time Burger's map

Controlling Chaos in 2-D Burger's Map
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 (À0.9, 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: Þy n þ x n y n þ ∈ y À 0:948683 À Á Repeating stability analysis for system (2) with the fixed point (À0.9, 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.
Applying the method of pulsive feedback, and re-writing eq. (10) as Then, using stability analysis, for stabilize the above unstable point P*, one obtains the parameter ε = À0.45.

Discussions
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][64][65][66][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, 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.