Open access peer-reviewed chapter

Chaos and Complexity Dynamics of Evolutionary Systems

Written By

Lal Mohan Saha

Submitted: July 27th, 2020 Reviewed: October 2nd, 2020 Published: December 2nd, 2020

DOI: 10.5772/intechopen.94295

Chapter metrics overview

373 Chapter Downloads

View Full Metrics


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.


  • chaos
  • Lyapunov exponents
  • chaos indicator
  • bifurcation
  • topological entropy
  • correlation dimension

1. Introduction

Henri Poincaré, (1892–1908), [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, Lorenzprovided 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].

Most biological systems exhibit enormous diversity and structurally multicomponent resulting in ecological imbalance and disorder/disharmony in environment. Inspired by articles of Lotka, Volterra, and Allee, numerous articles appeared with diversity in assumptions depending of species and their living environmental conditions in predator-prey models, [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44].

Real systems are mostly nonlinear and many of them are with multicomponent structure. Their individual elements possess individual properties. Such systems are termed as the complex system.

During evolution, a complex system exhibits chaos in some parameter space but also some other phenomena called complexity. This complexity is due to the interaction among multiple agents within the system displayed in the form of coexistence of multiple attractors, bistability, intermittency, cascading effects, exhibit of hysteresis properties etc. Thus, complexity can viewed as its systematic nonlinear properties and it is due to the interaction among multiple agents within the system. Foundation work and elaborate descriptions on complexity can viewed from some pioneer articles on complexity in nonlinear dynamics presented in [45, 46, 47, 48, 49, 50, 51]. Study of complexity means to know the results that emerging from a collection of interacting parts.

A dynamical system be chaotic then it must be (i) sensitive to initial conditions, (ii) topologically mixing and (iii) its periodic orbits must be dense. In chaotic systems, there exists a strange attractor, a chaotic set, which has fractal structure. Complex systems are also sensitive to their initial conditions and two complex systems that are initially very close together in terms of their various elements and dimensions can end up in distinctly different places. Wide discussions on complex system may found in some pioneer literatures, [14, 18, 45, 46, 48, 50, 51].

Chaos measured by Lyapunov exponents, (also called Lyapunov characteristic components or LCEs); LCE > 0 indicates existence of chaos and LCE < 0 indicates regularity, [52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62]. A complex system can better understood by measuring (i) chaos, (ii) Topological entropies and (iii) correlation dimension. Topological entropy, a non-negative number, provides a perfect way to measure complexity of a system. More topological entropy in any system signifies more complexity in it. Actually, it measures the evolution of distinguishable orbits over time, thereby providing an idea of how complex the orbit structure of a system is, [48, 49, 50, 61, 62, 63, 64, 65, 66, 67, 68, 69]. A system may be chaotic with zero topological entropy. In addition, a significant increase in topological entropy does not justify that it is chaotic. The book by Nagashima and Baba, [62], gives a very clear definition of topological entropy. The correlation dimension provides the dimensionality of the chaotic attractor. Correlation dimensions are non-integers and this is one reasons besides self-similarity that chaotic sets have fractal structure, [60, 68, 69, 70, 71, 72, 73].

It emerges from a good number of recent researches that chaos appearing in dynamical system be controlled and suggested number techniques to control chaos, [74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88]. These techniques have some limitations depending on the models and nature of nonlinearity.

Objective of this article is to investigate the emergence of chaos and complexity in nonlinear dynamical systems through examples of nonlinear models. Numerical simulations carried out for bifurcation analysis, plotting of LCEs and topological entropies for different systems. Numerical calculations extended to obtain correlation dimensions for certain chaotic attractors emerging in different systems. The study further extended to explain different types of chaos controlling technique. Studies confined to one, two and three-dimensional systems only.


2. Dynamic models with chaos and complexity

2.1 One dimensional discrete models

2.1.1 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 AQ. Stability occur in the form of stability and bistability, [89].

Fixed Points and Bifurcations:

For Qfixed, Q=2.76, and A<4.3793, only one stable steady state solution exits and stable two cycle starts when Aexceeds this value. Thus, approximately, A=4.3793, is the bifurcation point. At value A=4.3, the stable steady state solution is x* = 0.720533.

Keeping Q=2.76and varying parameter A, bifurcation diagrams are drawn, Figure 1 , for four different ranges of values of A. Similarly, keeping Afixed, A=5.4and varying Qin 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.

Figure 1.

Bifurcation diagrams of map(1)for four cases: when Q = 2.76 and parameter A varies.

Figure 2.

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

Both time series plots shown in Figure 3 are for chaotic evolution of system (1) and correspond to parameters (a) AQ= (5.3, 2.76), due to which an unstable fixed point obtained as x*=0.58531, and parameters (b) ) AQ= (5.4, 2.9), due to which an unstable fixed point obtained as x*=0.572218. For both cases, initial point taken is x0=0.5which lies nearby these points and so, also, unstable.

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.

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.

Figure 4.

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 Ox1= x1x2x3x4, of a map f:UU, where Us an open bounded set in Rn. To compute correlation dimension of Ox1, for a given positive real number r, we form the correlation integral,




is the unit-step function, (Heaviside function). The summation indicates number of pairs of vectors closer to rwhen1i, jnand ij. Crmeasures the density of pair of distinct vectors xiand xjthat are closer to r.

The correlation dimension Dcof Ox1is then defined as


To obtain Dc, logCris 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:

  1. 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


    The y-intercept of this straight line is 0.687605. Therefore the correlation dimension of the attractor in this case is DC=0.69.

  2. In a similar way, correlation dimension for second attractor of Figure 3 , A = 5.4 and Q = 2.9, as Dc=0.56. Plots of correlation dimensions against parameters A, Q shown in Figure 7 .

Figure 6.

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

Figure 7.

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

2.1.2 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 xn,xn+1representing quantities of cells per unit volume (in millions) at time nand n+1, respectively and pn,dnare, respectively, the number of cells produced and destroyed during the nth generation then


Then, assuming that


where b,r,sall positive parameters. With these our one-dimensional discrete model for blood cells populations comes as


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


For a=0.8, b=10, r=6and s=2.5, three fixed points x0=0,x1=0.989813,x2=3.53665obtained for system (6) of which only x0=0is stable and other two are unstable. Chaotic motion observed for values of parameter a=0.8,b=10,r=6,s=2.5, as shown in the time series plot, Figure 8 , with initial condition x0=1.5.

Figure 8.

Chaotic time series plot of map(6)fora= 0.8,b= 10,r= 6,s= 2.5 andx0=1.5.

Interesting bifurcations observed for this map: For b= 1.1 × 106, r= 8, two bifurcation diagrams are drawn; (a) in one for s=16 and 0a1, and (b) in another for a=0.8and 3.5s16.0and 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, 4.5s8.5, justifying chaos adding behavior.

Figure 9.

Bifurcation plots of Blood Cell model for=8,b= 1.1 × 106 then for (a)s= 16 and 0 ≤a≤ 1 and for (b)a= 0.1 and3.5s16.

Figure 10.

Bifurcation of Blood Cell model when3.0s7.5anda=0.8,r=8,b= 1.1 × 106.

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 r=6, s=16and b= 1.1 × 106 and 0a1. Fluctuations in increase of topological entropies appear, approximately, in the region 0.25a0.95indicate existence of complexity.

Figure 11.

LCE Plots for=6,s=16andb= 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 forr=6,s=16andb= 1.1 × 106 and0a1.

The correlation dimension of its chaotic attractor for values a=0.78,when r=6, s=16and b= 1.1 × 106 is obtained as Dc0.253.

2.2 Two-dimensional models

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 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:


With parameter values a=25, b=0.1, c=d=0.18and t=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 cdand when a=25,b=0.1,c=0.18,d=0.42, and t=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=dand when cdof 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:a050, c0.4,0.4, b=0.1, d=0.5,t=3,4,5.

Case 1: Taking c=d, bifurcation diagrams are drawn along the directions xand y, by varying cfor 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=25,b=0.1,t=3and (i) for regular case c=d=0.32and (ii) for chaotic case c=d=0.18and 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.

Figure 13.

Three cases of bifurcation scenarios of map(8)for parametersc=d: (a)t= 3,a=25,b=0.1and 0 ≤c≤ 0.5; (b)t= 4,a=35,b=0.1and 0 ≤c≤ 0.65; (c)t= 5,a=25,b=0.1and 0 ≤c≤ 0.5.

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 asa=25,b=0.1,t=3and (i) for regular casec=d=0.32and (ii) for chaotic casec=d=0.18.

Lyapunov Exponents & Topological Entropies:

For chaotic evolution, when a=25,b=0.1,t=3,c=d=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.

Figure 15.

Plots of Lyapunov exponents for chaotic evolution of map(8). Parameters area=25,b=0.1,t=3,c=d=0.18and when evolving from initial point (2.1, 2.1).

Figure 16.

Plots of topological entropy for map(8)when parameterc=d. From left: (i)t= 3,a=25,b=0.1and 0 ≤c≤ 0.5; (ii)t= 4,a=35,b=0.1and 0 ≤c≤ 0.65; (iii)t= 5,a=25,b=0.1and 0 ≤c≤ 0.8.

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

Figure 17.

Bifurcation plots whencdfor different ranges of parameterc. Cases (a), (b), (c), corresponds tot= 3,t= 4,t= 5. Parameters area=25,b=0.1andd=0.20for plots (a) & (c) andd= 0.30 for plot (b).

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.

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 casest= 3, 4, 5. Parametersa= 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, parametersa= 25,b= 0.1,d= 0.15 are common for all.

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

Figure 19.

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

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

Figure 20.

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


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 .

Cases (t)/ParametersabcdApproximate Dc
t = 3250.
t = 4250.
t = 5250.
t = 4250.
t = 5250.
t = 3350.
t = 4350.
t = 5350.

Table 1.

Correlation Dimensions for different sets of parameters.

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, 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 xnand ynof both firms are evolving in discrete time steps.


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,bdescribe the investment effectiveness of both the firms. Parameter cis 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.

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


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.

Figure 23.

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

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

Figure 25.

Plots of topological entropies: (a) left 2D plot is obtained for 12 ≤c≤ 170 and values ofa= 0.16, b= 0.9, α= 0.46 andβ= 0.7 and (b) right 3D plot is for 120 ≤c≤ 150 and 0 ≤a≤ 0.4 keeping same values forαandβ.

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


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 .

Figure 26.

Periodic bifurcations and chaotic attractor formations of Volterra – Petzoldt model for different values of c fixed parametersa= 1,b= 1, α 1 = 0.205, α 2 = 1,k1 = 0.05,k2 = 0,w= 0.006.

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 .

Figure 27.

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

Figure 28.

Plots of LCEs of chaotic attractors of model(1)for values ofc. Other parameters are same as inFigure 26.

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


3. Chaos control technique

As nonlinear systems are hardly comparable in the sense that behavior of one nonlinear system hardly match with another nonlinear system so the chaotic evolutions. So controlling chaos to bring any chaotic system to regularity may differ from one nonlinear system to another nonlinear system. Different types of controlling chaos technique discussed in recent literatures, [75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88].

Following two chaos controlling technique discussed here:

3.1 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 Xn + 1and that of the desired map Yn + 1can be explained by following mapping:


Also, the neighborhood dynamics of Xn + 1and Yn + 1can be represented by the relation:


Matrices AR, AD, BR, BDcan be obtained from the following:




Let a, bare two parameters of the system and (xn, yn) be any unstable fixed point of above system for given values of aand b. Then, our objective is to obtain two new values for aand bso 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 en= Xn-Yn.

Note that in equation (13) and (14) the coefficient matrices CR, CDand CMare to be determined so that if the error vector en= Xn-Ynis initialized as e0= 0, then it will be zero for all n future times. For asymptotic stability, we must have en0as n → ∞, then equation (14) implies


The necessary and sufficient condition for en→0as n→is


From these, one can obtain matrices CM, CD, CRand then control parameter matrix P*from (13).

A necessary and sufficient condition for the existence of matrices CM, CD, CR,given by:


3.2 Applications

3.2.1 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


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, c= 0.2, time series, attractor and LCE plots are obtained and shown by Figure 29 . Clearly the system (18) is showing chaos at (0.3, 0.5) with a= 3.7, b= 3.5, c= 0.2.

Figure 29.

Time series graphs, attractor and LCE plots of the unstable system.

Then, applying asymptotic stability discussed above for the map (18). For fixed value c = 0.2, unstable fixed point obtained as (0.25712, 0.49961). Nearby this point take initial point (0.3, 0.5) and p=ab=3.73.5. When above-mentioned method applied, one obtains matrices:


For the case when c= 0.2; new values of aand b; a= 3.91525, b= 2.99538 along with initial point (0.3, 0.5)a phase plot and a plot of Lyapunov exponents (LEC), are given in Figure 30 .

Figure 30.

Phase plot and LCE plot of controlled system whenc= 0.2,a= 3.91525,b= 2.99538.

3.2.2 Food chain model

Next,we have considered three dimensional food chain model, [23], written as


For values a= 4.1, b= 3.7, c= 3, d= 3.5, r= 3.8 five fixed points exist for system (19) given by: P0(0, 0, 0), P1(0, 0.2632, 0.2857), P2(0.518614, 0.263158, 0.158812), P3(0.7561, 0, 0) and P4(0.3333, 0.4685, 0). Then, by stability analysis it has obtained that the fixed points P2(0.518614, 0.263158, 0.158812) and P4(0.3333, 0.4685, 0) are unstable. Then, taking nearby P2, a desired initial point P*(0.5, 0.3, 0.2), chaotic attractors drawn, Figure 31 .

Figure 31.

Time series and attractors of unstable system.

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, dand 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.

Figure 32.

Phase plot and LCE plot of map(19)showing regular motion and chaos is controlled. Pulsive Feedback Technique to Chaos Control

Pulsive chaos control technique is discussed in detail in recent articles, [86, 87, 88]. As an application of this technique let us consider a simple 2 – dimension discrete time Burger’s map

3.2.3 Controlling Chaos in 2-D Burger’s Map


where aand bare 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 (−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 behavior after re-writing equations (1) as follows:


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 controlled.

Figure 33.

Chaos in Burgerger’s map for a = 1, b = 0.9.

3.2.4 Controlling Chaos in Volterra-Petzoldt Map

Evolution of Volterra-Petzoldt map already discussed in Section 2, Eq. (10). For parameters a= 1, b= 1, c= 9.7, α1= 0.205, α2= 1, k1= 0.05, k2= 0 , w= 0.006, this map shows chaotic motion. An unstable equilibrium solution P* (19.5374, 9.64328, 1.02602) exists in this case.

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 = 0.45.


4. 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 non-integers 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.

Figure 34.

Plot of regular attractor fora= 1,b= 0.9 and ε = 0.435.

Figure 35.

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



The author wishes to present his sincere gratitude to Professor M.K. Das of Institute of Informatics & Communication, University of Delhi South Campus, for his all support and help in preparation of this article.


  1. 1. Poincaré, H. (1957): Les Methodes Nouvelles de la Mecanique Celeste. Dover, New York (1957) Paris, 1899; reprint.
  2. 2. Lorenz EN. Deterministic non-periodic flow. Journal of the Atmospheric Sciences.1963;20(2):130-141
  3. 3. Sharkovskii, A. N. (1964).: Co-existence of cycles of a continuous mapping of the line into itself.Ukrainian Math. J.16: 61–71.
  4. 4. Smale, S.: Differentiable dynamical systems. Bull. Amer. Math. with Soc., 1967; 73 (6), 747--817.
  5. 5. Hènon M. Numerical study of quadratic area-preserving maoings. Quart. J. Math. 1969;27:291-311
  6. 6. Ruelle D, Takens F. On the nature of turbulence. Commun. Math. Phys. 1971;20:167-192
  7. 7. Guckenheimer, J., Holmes, P. (1971): Nonlinear Oscillations Dynamical Systems, and Bifurcations of Vector Field. Applied Mathematical Sciences, Book Series, Springer.
  8. 8. May RM. Stability and Complexity in Model Ecosystem. Princeton N. J: Princeton University Press; 1974
  9. 9. Li T-Y, Yorke JA. Period Three Implies Chaos.The American Mathematical Monthly. 1975. DOI:
  10. 10. May RM. Simple mathematical models with very complicated dynamics. Nature. 1976;261:459-467
  11. 11. Rössler OE. An equation for hyperchaos. Physics Letters A. 1979;71:155-157
  12. 12. K. Ikeda, H. Daido and O. Akimoto: Chaotic behavior of transmitted light from a ring cavity. Phys. Rev. Lett., 1980; 45 (9), 709 – 712.
  13. 13. Feigenbaum MJ. Universal behavior in nonlinear systems. Physica. 1983;7D:16-39
  14. 14. Gleick, J. (1987). “Chaos: Making a New Science”.
  15. 15. F. C. Moon (1987): Chaotic Vibrations., John Wiley & Sons New York 1987
  16. 16. Devaney RL. An Introduction to Chaotic Dynamical System. Reading: Addison-Wesley; 1989
  17. 17. Ueda Y. Randomly transitional phenomena in the system governed by Duffing's equation.J. Stat. Phys.1979;20:181-196
  18. 18. Stewart I. Does God Play Dice? Penguin Books; 1989
  19. 19. Tanaka Y and Saha LM. (2012): Nonlinear Behaviors of Pulsating Stars with Convective Zones. PASJ:Publ. Astron. Soc. Japan 2012;64, L8-1- 4.
  20. 20. Lotka AJ.Elements of Physical Biology. Baltimore MD: Williams and Wilkins; 1925
  21. 21. Volterra, V. (1931): .Lecons sur la Thorie Mathmatique de la Lutte pour la Vie, Gauthiers-Viallars, Paris.
  22. 22. Allee WC. Animal aggregations.The Quarterly Review of Biology. 1927;2:367-398
  23. 23. Allee WC, Bowen E. Studies in animal aggregations: mass protection against colloidal silver among goldfishes. Journal of Experimental Zoology. 1932;61(2):185-207
  24. 24. Elsadany AA. Dynamical complexities in a discrete-time food chain.Computational Ecology and Software. 2012;2(2):124-139
  25. 25. Smith M. Mathematical Ideas in Biology. Cambridge: Cambridge University Press; 1968
  26. 26. Freedman HI. Deterministic Mathematical Models in Population Ecology. Marcel Dekker; 1980
  27. 27. J.R. Beddington, C.A. Free and J.H. Lawton. Dynamic complexity in predator–prey models framed in difference equations, Nature. 1975;255:58-60.
  28. 28. Abrams PA, Ginzburg LR. The nature of predation: prey dependent, ratio dependent or neither? Trends in Ecology & Evolution. 2000;15(8):337-341
  29. 29. Grafton, R.Q., Silva-Echenique J. (1994). Predator–Prey Models: Implications for Management.Atlantic Canada Economics Association Papers 23, pp. 61–71.
  30. 30. Kaitala V, Heino M. Complex non-unique dynamics in ecological interactions. Proc R Soc London B. 1996;263:1011-1015
  31. 31. Quentin Grafton R, Silva-Echenique J. How to manage nature? Strategies, predator-prey models, and chaos. Marine Resource Economics. 1997;12(2):127-143
  32. 32. Yakubu A-A. Prey dominance in discrete predator-prey systems with a prey refuge. Mathematical Biosciences. 1997;144:155-178
  33. 33. Xiao Y, Cheng D, Tang S. Dynamic complexities in predator–prey ecosystem models with age-structure for predator.Chaos, Solitons and Fractals. 2002;14:1403-1411
  34. 34. Liu X, Xiao D. Complex dynamic behaviors of a discrete-time predator-prey system. Chaos, Solitons & Fractals. 2007;32:80-94
  35. 35. Andrecut, M. and Kauffman, S. A. Chaos in a Discrete Model of a Two-Gene System. Physics Letters A. 2007;367, 281-287.
  36. 36. Canan C¸ elik, Oktay Duman. Allee effect in a discrete-time predator–prey system. Chaos, Solitons & Fractals, 2009; 40, 1956–1962.
  37. 37. Hadeler KP, Freedman HI. Predator-prey populations with parasitic infection. Journal of Mathematical Biology. 1989;27(6):609-631
  38. 38. Danca M, Codreanu S, Bako B. Detailed analysis of a nonlinear predator-prey model. Journal of Biological Physics. 1997;23(1):11-20
  39. 39. Wan-Xiong, Yan-Bo-Zhang and Chang-zhong Liu. Analysis of a discrete-time predator-prey system with Allee effect. Ecological Complexity, 2011, 8: 81 – 85.
  40. 40. Zhao M, Yunfei D. Stability of a discrete-time predator-prey system with Allee effect. Nonlinear Analysis and Differential Equations. 2016;4(5):225-233
  41. 41. Xian-wei X-l F, Jing Z-j. Dynamics in a discrete-time predator-prey system with Allee effect. Acta Mathematicae Applicatae Sinica. 2013;20:143-164
  42. 42. Stephens PA, Sutherland WJ, Freckleton RP. What is the Allee effect? Oikos. 1999;87:185-190
  43. 43. Tang S, Chen LA discrete predator–prey system with age-structure for predator and natural barriers for prey.Mat. Model. Numer. Anal.2001;35:675-690
  44. 44. Tang S, Chen L. Density-dependent birth rate, birth pulses and their population dynamic consequences. J. Math. Biol. 2001;44(2):185-199
  45. 45. W. Weaver,(1948): “Science and complexity,” American Scientist, vol. 36, no. 4, p. 536.
  46. 46. Gribbin, J (2004) Deep Simplicity: Chaos, Complexity and the Emergence of Life. Penguin Press Science.
  47. 47. Simon HA. The architecture of complexity. Proceedings of the American Philosophical Society. 1962;106(6):467-482
  48. 48. Gribble, S. 1995, Topological Entropy as a Practical Tool for Identification and Characterization of Chaotic System. Physics 449 Thesis.
  49. 49. Iwai K. Continuity of topological entropy of one dimensional map with degenerate critical points.J. Math. Sci. Univ. Tokyo. 1998;5:19-40
  50. 50. Hefferman DM. Multistability, intermittency and remerging Feigenbaum trees in an externally pumped ring cavity laser system.Phys. Lett.A. 1985;108:413-422
  51. 51. Walby S. Complexity theory, systems theory, and multiple intersecting social inequalities. Philosophy of the Social Sciences. 2007;37(4):449-470
  52. 52. Benettin G, Galgani L, Giorgilli A, Strelcyn JM. Lyapunov Characteristic Exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1 & 2: Theory.Meccanica. 1980;15:9-30
  53. 53. Katok A. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms.Publ. Math. IHES. 1980;51:137-174
  54. 54. P. Grassberger and Itamar Procaccia. "Measuring the Strangeness of Strange Attractors". Physica D: Nonlinear Phenomena, 1983;9 (1–2): 189–208.
  55. 55. Grassberger P, Procaccia I. Characterization of Strange Attractors.Physical Review Letters. 1983;50(5):346-349
  56. 56. Bryant P, Brown R, Abarbanel H. Lyapunov exponents from observed time series. Physical Review Letters. 1990;65(13):1523-1526
  57. 57. Brown R, Bryant P, Abarbanel H. Computing the Lyapunov spectrum of a dynamical system from an observed time series.Phys. Rev. A. 1991;43:2787-2806
  58. 58. Abarbanel HDI, Brown R, Kennel MB. Local Lyapunov exponents computed from observed data.Journal of Nonlinear Science. 1992;2(3):343-365
  59. 59. Skokos C. The Lyapunov characteristic exponents and their computation.Lect. Notes. Phys.2009;790:63-135
  60. 60. Syta A, Litak G, Budhraja M, Saha LM. Detection of the chaotic behavior of a bouncing ball by 0 – 1 test.Chaos, Solitons & Fractals. 2009;42:1511-1517
  61. 61. Adler RL, Konheim AG, McAndrew MH. Topological entropy. Trans. Amer. Math. Soc. 1965;114:309-319
  62. 62. Nagashima H, Baba Y. Introduction to Chaos: Physics and Mathematics of Chaotic Phenomena. Overseas Press India Private Limited; 2005
  63. 63. Bowen R. Topological entropy for noncompact sets. Trans. Amer. Math. Soc.1973;184:125-136
  64. 64. Holmes P. ‘Strange’ phenomena in dynamical systems and their physical implications.App. Math. Modelling. 1977;7(1):362-366
  65. 65. P. Holmes (1979) A nonlinear oscillator with a strange attractor,Phil. Trans. Roy. Soc.Lond.A 292(1394): 419 – 448.
  66. 66. Balmforth NJ, Spiegel EA, Tresser C. Topological entropy of one dimenonal maps: approximations and bounds.Phys. Rev. Lett.1994;72:80-83
  67. 67. Stewart L, Edward ES. Calculating topological entropy. J. Stat. Phys. 1997;89:1017-1033
  68. 68. Yuasa M, Saha LM. Indicators of chaos. Science and Technology, Kinki University. Japan. No. 2008;20:1-12
  69. 69. Saha LM, Prasad S and Yuasa M Measuring Chaos: Topological Entropy and Correlation Dimension in Discrete Maps, Science and Technology, Vol. 24, 2012, pg. 10 – 23.
  70. 70. DeCoster GP, Mitchell DW. The efficacy of the correlation dimension technique in detecting determinism in small samples.Journal of Statistical Computation and Simulation. 1991;39:221-229
  71. 71. Saha LM, Sharma R. Dynamics of Two-Gene Andrecut-Kauffman System: Chaos and Complexity. Accepted.Italian Journal of Pure and Applied Mathematica (IJPAM). 2018;41:405-413
  72. 72. Saha LM, Das MK. Complexities in Micro-Economic Behrens Feichtinger Model.Indian Journal of Industrial and Applied Mathematics. 2016;7(2):127-135
  73. 73. M. Martelli (1999)Introduction to Discrete Dynamical Systems and Chaos, Wiley- Interscience
  74. 74. Guanrong Chen and Xiaoning Dong (1998): From Chaos to Order: Methodologies, Perspectives and Applications. World Scientific, Singapore, New Jersey, London, Hong Kong.
  75. 75. Auerbach D, Grebogi C, Ott E, Yorke JA. Controlling chaos in high dimensional systems. Phys. Rev. Lett. 1992;69:3479-3482
  76. 76. Erjaee GH, Atabakzade MH, Saha LM. Interesting synchronization-like behavior.Int. Jour. Bifur.. Chaos. 2004;14(4):1447-1453
  77. 77. Carroll TL, Pecora LM. Cascading synchronized chaotic systems. Physica D. 1993;67:126-140
  78. 78. Chen G. Optimal control of chaotic systems. Int’l J. of Bifur. Chaos. 1994;4:461-463
  79. 79. Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys. Rev. Lett. 1990;64:1196-1199
  80. 80. Pan S, Yin F. Using chaos to control chaotic systems. Phys. Lett. A. 1997;231:173
  81. 81. Pyragas K. Continuous control of chaos by self-controlling feedback. Phys. Lett. A. 1992;170:421-428
  82. 82. Shinbrot T, Grebogi C, Ott E, Yorke JA. Using small perturbations to control chaos. Nature. 1993;363:411-417
  83. 83. Erjaee GH. On the asymptotic stability of a dynamical system.IJST, Transaction A. 2002;26(A1):131-135
  84. 84. Saha LM, Erjaee GH and Budhraja M. Controlling chaos in 2-dimensional systems,Iranian Jour. Sci. Tech.,Trans. A, 2004;28, No.A2, 219 – 226.
  85. 85. Saha LM, Das MK, Bhardwaj R. Asymptotic stability analysis applied to price dynamics.Ind. J, Industrial and Appl, Math. (IJIAM). 2018;9(2):186-195
  86. 86. Litak G, Ali M, Saha LM. Pulsating feedback control for stabilizing unstable periodic orbits in a nonlinear oscillator with a non-symmetric potential.Int. J. Bifur. Chaos. 2007;17:2797-2803
  87. 87. Litak G, Borowiec M, Ali M, Saha LM, Friswell MI. Pulsive feedback control of a quarter car forced by a road profile.Chaos Soliton and Fractals. 2007;33:1672-1676
  88. 88. G. Litak, L. M. Saha and M. Ali (2010): Continuous and Pulsive Feedback Control of Chaos,Recent Progress in Controlling Chaos, by Miguel A. F. Sanjuan and Celso Grebogi, World Scientific (eBooks), p. 337 – 369.
  89. 89. O’Cairbre F, O’Farrell AG, O’Reilly A. Bistability, bifurcation and chaos in a laser system.Int. Jour. Bifurcation and Chaos. 1995;5(4):1021-1031
  90. 90. Bonifacio R, Lugiato LA. Bistable absorption in a ring cavity, Lett.l. Nuovo Cimento. 1978;21(15):505-510
  91. 91. Benefacio R, Lugato LA. Theory of optical bistability. In:Dissipative Systems in Quantum Optics. Ed. R. Benefacio: Springer-Verlag; 1982. pp. 61-92
  92. 92. de Souza SLT, Lima AA, Caldas IL, Medrano-T RO, Guimarães-Filho ZO. Self-similarities of periodic structures for a discrete model of a two-gene system. Physics Letter A. 2012;376:1290-1294
  93. 93. Holyst JA, Hagel T, Haag G, Weidlich W. How to control chaotic economy? J. Evol. Econ. 1996;6:31-42
  94. 94. Behrens DA. Feichtinger G., Prskawetz A. Complexity dynamics and control of arms race. European Journal of Operational Research, 1997;100:192-215
  95. 95. Perc M. Microeconomic uncertainties cooperative allince and social welfare. Economics Letters. 2007;95:104-109
  96. 96. Thomas Petzoldt (2003): R as a simulation platform in ecological modelling. R. News, Vol. 3/3, 8 – 16.
  97. 97. B. Blasius, A. Huppert and L. Stone. Complex dynamics and phase synchronization in spatially extended ecological systems. Nature, 1999;399:354 – 359.
  98. 98. Blasius B, Stone L. Chaos and phase synchronization in ecological systems. Int. J. Bifur. And chaos. 2000;10:2361-2380

Written By

Lal Mohan Saha

Submitted: July 27th, 2020 Reviewed: October 2nd, 2020 Published: December 2nd, 2020