**3.2. Simulations and results**

12 Name of the Book

and simple chaotic systems based on some elements contained in a pre-selected existing chaotic system and a properly defined cost function. The investigation consists of two case studies based on the aforementioned evolutionary algorithms in various versions. For all algorithms, 100 simulations of chaos synthesis were repeated and then averaged to guarantee the reliability and robustness of the proposed method. The most significant results were

Deterministic chaos, discovered by [21] is a fairly active area of research in the last few decades. The Lorenz system produces a well-known chaotic attractor in a simple three-dimensional autonomous system of ordinary differential equations, [21], [22]. For discrete chaos, there is another famous chaotic system, called the logistic equation [23], which was found based on a predator-prey model showing complex dynamical behaviors. These simple models are widely used in the study of chaos today, while other similar models exist (e.g., canonical logistic equation [24] and 1D or 2D coupled map lattices, [25]). To date, a large set of nonlinear systems that can produce chaotic behaviors have been observed and analyzed. Chaotic systems thus have become a vitally important part of science and engineering at the theoretical as well as practical levels of research. The most interesting and applicable notions are, for example, chaos control and chaos synchronization related to secure communications,

The aim of the present investigation is to show that EA-based symbolic regression (i.e., handling with symbolic objects to create more complex structures) is capable of synthesizing chaotic behavior in the sense that mathematical descriptions of chaotic systems are synthesized symbolically by means of evolutionary algorithms. The ability of EAs to successfully solve this kind of black-box problems has been proven many times before (see,

The cost function was in fact a little bit complex decision function. The cost function used for chaos synthesis, comparing with other problems is quite a complex structure which cannot be easily described by a few simple mathematical equations. Instead, it is described by the

1. Take a synthesized function and evaluate it for 500 iterations with and a sampling step of

2. Check if each value of *A* for all 500 iterations is unique or if some data are repeated in the series (the first check for chaos, indirectly). If the data are not unique, then go to step 5 else

4. Check the Lyapunov exponent: If the Lyapunov exponent is positive, write all important data (synthesized functions, number of cost function evaluations, etc.) into a file. Then,

5. If the data are not unique, i.e., if the Lyapunov exponent is not positive, return an individual fitness, and sum all values whose occurrences in the dataset from step 1 are more than 1 (simply, it returns the occurrences of periodicity, quasi-periodicity – higher

3. Take the last 200 values, and for each value of *A*, calculate its Lyapunov exponent.

repeat the simulation for another synthesized system by going to step 1.

penalization of an individual in the evolution).

carefully selected, visualized and commented on in this chapter.

for example, [26], [27]), and is proven once again here in this paper.

amongst others.

**3.1. Cost function**

Δ*A* = 0.1.

go to step 3.

following algorithm procedure:

SA algorithms have been applied 100 times in order to find artificially synthesized functions that can produce chaos. All of these experiments were done using the *Mathematica* software. The primary aim of this comparative study is not to show which algorithm is better or worse, but to show that symbolic regression is able to synthesize some new (at least in the sense of mathematical description and behavior) chaotic systems.

Based on the results from experiments, two different sets of figures were created. The first set shows the performances of different algorithms from different points of view (see [13]), the second set (Figure 11 - Figure 14) shows behaviors of the selected synthesized programs, i.e., bifurcation diagrams. The synthesized programs are also appended to each figure in the form of the mathematical formula. Figure 15 shows an example of the so-called program length histogram, generated from 100 simulations. Program length (in *Mathematica* command: LeafCount, denoted as LC) means a number of elements that create a mathematical formula. As a summary, the following statements are presented:

As a summary, the following statements are presented:


**Figure 12.** ... and its tree representation.

**Figure 14.** ... and its tree representation.

0

**Figure 15.** Histogram of LC for DERand1Bin

evolutionary algorithm can be used.

5

10

15

tiH

20

25

10 20 30 40 50 60 LC

Simulated Annealing in Research and Applications 83

• **Exemplary system synthesis.** Based on the fact that the logistic equation (its elements and range) is used for chaos synthesis, it is logical to expect that during evolution (if repeated many times) the original system should also be synthesized. That event was also observed a few times, exactly in the mathematical form Equation (6). Some selected bifurcation

• **Mutual comparison.** When comparing all algorithms, it is obvious that these algorithms produced good results. Parameter setting for the algorithms was based on a heuristic approach and thus there is a possibility that better settings can be found there. Based on these results, it is clear that for symbolic synthesis via analytic programming any

• **Engineering design.** It is quite clear that evolutionary synthesis of chaos can be applied to engineering design of devices based on chaos (signal transmission via chaos, chaos-based encryption, and so on). Based on principles and results reported in this paper, it should be possible to synthesize systems with some precisely defined chaotic features and attributes.

diagrams of synthesized systems are depicted in Figures 16 - 19.

whose elements were used in the evolution. Despite the a priori known information, a few chaotic systems were located also outside of this interval. That was due to the fact that a part of the chaotic behavior was inside the interval [0, 4] and thus EA was able to identify it. From these facts, it is clear that EA are able to locate chaos in a wider range than those expected from some textbook exemplary systems.

**Figure 14.** ... and its tree representation.

**Figure 11.** Bifurcation diagram of 2

82 Simulated Annealing – Single and Multiple Objective Problems

**Figure 12.** ... and its tree representation.

**Figure 13.** Bifurcation diagram of

expected from some textbook exemplary systems.

2 (2 1) *A x A x* "

2 2 2 ( 2) *Ax A x A x* "

whose elements were used in the evolution. Despite the a priori known information, a few chaotic systems were located also outside of this interval. That was due to the fact that a part of the chaotic behavior was inside the interval [0, 4] and thus EA was able to identify it. From these facts, it is clear that EA are able to locate chaos in a wider range than those

**Figure 15.** Histogram of LC for DERand1Bin


84 Simulated Annealing – Single and Multiple Objective Problems

**Figure 16.** Another synthesized system.

**Figure 19.** Another synthesized system.

For all algorithms each simulation was repeated 100 times to show and check the robustness of the methods used. All data were processed and used in order to obtain summarizing results and graphs. Many methods were adapted for the so-called spatiotemporal chaos represented by coupled map lattices (CML). Control laws derived for CML are usually based on existing system structures, Schuster H. G. (1999), or using an external observer, Chen G. (2000). Evolutionary approach for control was also successfully developed in, for example, Richter H. and Reinschke K. J. (2000), Richter H. and Reinschke Kurt J. (2000a), Richter H. (2002), Zelinka I. (2006). Many published methods of deterministic chaos control (DCC) were (originally developed for classic DCC) adapted for so-called spatiotemporal chaos represented by CML, given by (10). Models of this kind are based on a set of spatiotemporal (for 1D, Figure 20) or spatial cells which represents the appropriate state of system elements. A typical example is CML based on the so-called logistic equation, [23], [5], [29], which is used to simulate the behavior of system which consists of n mutually joined cells via onlinear coupling, usually noted as *#*. A mathematical description of the CML system is given by Equation (10). The function which is represented by *f* (*xn*(*i*)) is arbitraryɆI discrete

system in this case study logistic equations have been selected to substitute *f* (*xn*(*i*)).

H

**Figure 20.** 1D CML with stabilized pattern T1S2

<sup>1</sup>( ) (1 ) ( ( )) ( ( ( 1)) ( ( 1))) <sup>2</sup> *n nn n x i fx i fx i fx i* H

The main aim of this part is to show that evolutionary algorithms (EA) are capable of controlling (as was also shown for temporal DCC in [31], [32] CML as well as deterministic methods without internal system knowledge operating with CML as with a black box. The

(10)

Simulated Annealing in Research and Applications 85

**Figure 17.** Another synthesized system.

**Figure 18.** Another synthesized system.
