Correlation Dimensions for different sets of parameters.
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.
- Lyapunov exponents
- chaos indicator
- topological entropy
- correlation dimension
Henri Poincaré, (1892–1908), , 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, , 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].
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, , 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 . Stability occur in the form of stability and bistability, .
Fixed Points and Bifurcations:
For fixed, , and , only one stable steady state solution exits and stable two cycle starts when exceeds this value. Thus, approximately, , is the bifurcation point. At value , the stable steady state solution is x* = 0.720533.
Keeping and varying parameter , bifurcation diagrams are drawn, Figure 1, for four different ranges of values of . Similarly, keeping fixed, and varying 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) = (5.3, 2.76), due to which an unstable fixed point obtained as , and parameters (b) ) = (5.4, 2.9), due to which an unstable fixed point obtained as . For both cases, initial point taken is which lies nearby these points and so, also, unstable.
Calculations of Lyapunov Exponents, (LCEs):
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.
Consider an orbit = , of a map , where s an open bounded set in . To compute correlation dimension of , for a given positive real number , we form the correlation integral,
is the unit-step function, (Heaviside function). The summation indicates number of pairs of vectors closer to when, and . measures the density of pair of distinct vectors and that are closer to .
The correlation dimension of is then defined as
To obtain , is plotted against log , 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:
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 . Therefore the correlation dimension of the attractor in this case is .
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, (, p: 35), presented here.
Let representing quantities of cells per unit volume (in millions) at time and , respectively and are, respectively, the number of cells produced and destroyed during the nth generation then
Then, assuming that
where all positive parameters. With these our one-dimensional discrete model for blood cells populations comes as
The case a , means that during the time interval under consideration all cells that were alive at time are destroyed. In such a case, above models simply comes as
For , , and , three fixed points obtained for system (6) of which only is stable and other two are unstable. Chaotic motion observed for values of parameter , as shown in the time series plot, Figure 8, with initial condition .
Interesting bifurcations observed for this map: For = 1.1 × 106, = 8, two bifurcation diagrams are drawn; (a) in one for =16 and , and (b) in another for and 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, , 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 , and = 1.1 × 106 and . Fluctuations in increase of topological entropies appear, approximately, in the region indicate existence of complexity.
The correlation dimension of its chaotic attractor for values when , and = 1.1 × 106 is obtained as .
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 , , and , 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 and when , and , 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 and when of evolutions showing irregularities due to presence of chaos and complexity.
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 , bifurcation diagrams are drawn along the directions and , by varying for cases = 3, 4, 5 and certain fixed values of other parameters as shown in Figure 13. Then, plots of attractors have been obtained for parameters and (i) for regular case and (ii) for chaotic case and shown in Figure 14. In each case when = 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 , 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 = 3, 4, 5 and for different ranges of parameter Analysis of these plots, gives an impression that for the case = 3, system shows enough complexity in the range 0.05 ≤ ≤ 0.23. For the case = 4, the system shows high complexity in the range 0 ≤ ≤ 0.22 and in case = 5, high complexity appears in 0 ≤ ≤ 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 = 25 and = 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.
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 :
For case = 3 and = 25, = 0.1, = 0.28, = 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, = 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)/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|
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 and of both firms are evolving in discrete time steps.
where () are the time rates at which the sales of both firm decays in the absence of investments. Parameters describe the investment effectiveness of both the firms. Parameter is an “elasticity” measure of the investment strategies. For parameter values = 0.46¸ = 0.7, = 0.16, = 0.9, = 105, we have observed the chaotic attractor of this model.
Bifurcation diagrams for system (9) obtained for = 0.46¸ = 0.7, = 0.16, = 0.9 and by varying parameter , 8 ≤ ≤ 160 and in close range, 6 ≤ ≤ 8, Figure 21. Then, again it obtained for values = 0.46¸ = 0.7, = 0.16, = 0.6, = 110 and 0 ≤ ≤ 0.4, Figure 22. Appearance of period doubling followed by chaos visible from these figures.
Time series plots and a plot of chaotic attractor obtained for values = 0.16, = 0.9, = 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 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.
Following steps used for map (8), correlation dimension of chaotic the attractor for values α = 0.46, β = 0.7, = 0.16, = 0.9, = 105, obtained as = 0.064
2.2.3 Continuous Volterra-Petzoldt Model
Bifurcation diagram for predator z while varying prey parameter b shown there, Petzoldt , is interesting. Periodic bifurcations and chaotic attractor of this model for different parameter space are presented in the figure, Figure 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 + 1 and that of the desired map Yn + 1 can be explained by following mapping:
Also, the neighborhood dynamics of Xn + 1 and Yn + 1 can be represented by the relation:
Matrices AR, AD, BR, BD can be obtained from the following:
Let , are two parameters of the system and (, ) be any unstable fixed point of above system for given values of and . Then, our objective is to obtain two new values for and 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
And en= Xn-Yn.
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 n → ∞, then equation (14) implies
The necessary and sufficient condition for en→0 as n→is
From these, one can obtain matrices CM, CD, CR and then control parameter matrix P* from (13).
A necessary and sufficient condition for the existence of matrices CM, CD, CR, given by:
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, , written as
For = 3.7, = 3.5, = 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 = 3.7, = 3.5, = 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 = 3.7, = 3.5, = 0.2.
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 . When above-mentioned method applied, one obtains matrices:
For the case when = 0.2; new values of and ; = 3.91525, = 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.
3.2.2 Food chain model
Next, we have considered three dimensional food chain model, , written as
For values = 4.1, = 3.7, = 3, = 3.5, = 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.
In the process of stabilizing the desired point (0.5, 0.3, 0.2), calculations performed to replace parameters = 4.1, = 3.5 and = 3.8 to earlier case of map (18). After obtaining all concerned matrices, replacement matrix obtained as
3.2.3 Controlling Chaos in 2-D Burger’s Map
where and are non-zero parameters . This map evolve chaotically when = 0.9, =0.856. To control chaotic motion we have used pulsive feedback control technique, Litak et al.  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.
3.2.4 Controlling Chaos in Volterra-Petzoldt Map
Evolution of Volterra-Petzoldt map already discussed in Section 2, Eq. (10). For parameters = 1, = 1, = 9.7, = 0.205, = 1, = 0.05, = 0 , = 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.
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), . 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, , both the techniques applied here are perfect and very effective in controlling chaos, especially in real systems.
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.