Chaos enhanced differential evolution in the task of evolutionary control of selected set of discrete chaotic systems

Research Article Chaos Enhanced Differential Evolution in the Task of Evolutionary Control of Selected Set of Discrete Chaotic Systems Roman Senkerik,1Ivan Zelinka,2Michal Pluhacek,1 Donald Davendra,2and Zuzana Oplatková Kominkova1 1Faculty of Applied Informatics,Tomas Bata University in Zlin, N´ amˇ est´ ı T.G. Masaryka 5555, 760 01 Zlin, Czech Republic 2Faculty of Electrical Engineering and Computer Science, Vˇ SB-Technical University of Ostrava, 17 Listopadu 15, Poruba, 708 33 Ostrava, Czech Republic Correspondence should be addressed to Roman Senkerik; senkeri[email protected]tb.cz Received 6 March 2014; Accepted 7 July 2014; Published 26 August 2014 Academic Editor: Gerhard-Wilhelm Weber Copyright © 2014 Roman Senkerik et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Evolutionary technique differential evolution (DE) is used for the evolutionary tuning of controller parameters for the stabilization of set of different chaotic systems. The novelty of the approach is that the selected controlled discrete dissipative chaotic system is used also as the chaotic pseudorandom number generator to drive the mutation and crossover process in the DE. The idea was to utilize the hidden chaotic dynamics in pseudorandom sequences given by chaotic map to help differential evolution algorithm search for the best controller settings for the very same chaotic system. The optimizations were performed for three different chaotic systems, two types of case studies and developed cost functions. 1. Introduction In many engineering applications, one of the most innate tasksisthecontrollingofhighlynonlineardynamicalsystems in order to either eliminate or synchronize the chaos. The first pioneering approach to control chaotic dynamics by means of a simple analytical linearization method was introduced in 1990s by Ott et al. (i.e., OGY method) [1]. Subsequently, the rapid development of methods for stabilizing of nonlinear chaotic dynamics has arisen and many advanced techniques have been applied for chaos control and chaos synchronization including methods from the artificial intelligence field. During recent years, usage of new intelligent systems in engineering, technology, modeling, computing, and simulations has attracted the attention of researchers worldwide. The most current methods are mostly based on soft computing, which is a discipline tightly bound to computers, representing a set of methods of special algorithms, belonging to the artificial intelligence paradigm. The most popular of these methods are neural networks, evolutionary algorithms (EAs), fuzzy logic, and tools for symbolic regression like genetic programming. Currently, EAs are known as a powerful set of tools for almost any difficult and complex optimization problem. The interest about the interconnection between evolutionary techniques and control of chaotic systems is rapidly spreading. The initial research was conducted in [2], whereas [3,4] was more concerned with the tuning of parameters inside the chaos control technique based on the Pyragas method: extended delay feedback control (ETDAS) [5]. When compared to the aforementioned research, later works [6,7]showapossibilityastohowtogeneratetheentirecontrol law (not only how to optimize several parameters) for the purpose of stabilization of a chaotic system. Such approach also may overcome the possible sensitivity to initial conditions which may lead to stability issues. The synthesis of control is inspired by the Pyragas’ delayed feedback control technique [8,9].Thismethodisveryadvantageousforthe evolutionary computation, due to its amount of easy accessible control parameters, which can be easily tuned by means of EAs. Other approaches utilizing the EAs for stabilizing chaotic dynamics have mostly applied the particle swarm optimization (PSO) algorithm [10] and multi-interval gradient Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 836484, 12 pages http://dx.doi.org/10.1155/2014/836484 2The Scientific World Journal method [11] or minimum entropy control technique [12]. EAs have been also frequently used in the task of synchronization of chaos [13–15]. In [16], an EA for optimizing local control of chaos based on a Lyapunov approach is presented. Evolutionary techniques were also used for the synthesis of new complex discrete type structures with chaotic behavior [17] as well as the synthesis (identification) of a mathematical model of chaotic system based on the measured data [18]. Another example of interconnection between deterministic chaos and EAs represents the research focused on the embedding of chaotic dynamics into the EAs. Recent research in chaos driven heuristics has been fueled with the predisposition that, unlike stochastic approaches, a chaotic approach is able to bypass local optima stagnation. This one clause is of deep importance to EAs. A chaotic approach generally uses the chaotic map in the place of a pseudorandom number generator [19]. This causes the heuristic to map unique regions, since the chaotic map iterates to new regions. The task is then to select a very good chaotic map as the pseudorandom number generator. The initial concept of embedding chaotic dynamics into EAsisgivenin[20]. Later, the initial study [21]wasfocusedon thesimpleembeddingofchaoticsystemsintheformofchaos pseudorandom number generator (CPRNG) for differential evolution (DE) [22] and self-organizing migrating algorithm (SOMA) [23] in the task of optimal PID tuning. Also, the PSO algorithm with elements of chaos was introduced as the CPSO [24]. This field of research was later extended with the successful experiments with chaos driven DE [25]inreal domain as well as in combinatorial problems domain [26,27]. At the same time, the chaos embedded PSO with inertia weighstrategywascloselyinvestigated[28], followed by the introduction of a PSO strategy driven alternately by two chaotic systems [29] and novel chaotic multiple choice PSO (chaos MC-PSO) strategy [30]. Recently, the chaotic firefly algorithm was also introduced [31]. Finally, the last example represents the research focusing on the EAs and the edge of chaos. An unconventional approach of the edge of chaos and its application to discrete systems and evolutionary algorithms in terms of stagnation avoidance is presented in [32]. The organization of this paper is as follows: firstly, used evolutionary technique, which is DE, is described and is followed by the description of the ChaosDE concept. Thereafter, the problem design and appropriate corresponding cost functions are investigated and proposed. Results and conclusion follow afterwards. 2. Motivation This paper extends the research of evolutionary chaos control optimization by means of ChaosDE algorithm [33–35]. Recent studies have shown that differential evolution [22] is one of the most potent heuristics and it has been used for a number of optimization tasks; [36–38] have explored DE for combinatorial problems; [39,40]havehybridizedDE whereas [41–43] have developed self-adaptive DE variants. In this paper, the DE/rand/1/bin strategy driven by different chaotic maps (systems) was utilized to solve the issue of evolutionary optimization of chaos control for the very same chaotic system used as a CPRNG in the particular case study. Thus, the idea was to utilize the hidden chaotic dynamics in pseudorandom sequences given by chaotic map to help differential evolution algorithm search for the best controller settingsfortheverysamechaoticsystem.Sincetheverypositive contribution of the chaotic dynamics to the performance of DE in the task of evolutionary chaos control optimization was proven in comparison with original canonical DE within the initial study [44], this paper is not primarily focused on the performance comparisons with different heuristic. But this research extends the initial work with the aforementioned idea and with the several case studies combining different chaotic systems and different utilized cost functions. 3. Used Heuristic: Differential Evolution (DE) DE is a simple and powerful population-based optimization method that works either on real-number-coded individuals or with small modifications on discrete type individuals [22, 45,46]. DE is quite robust, fast, and effective, with global optimization ability. This global optimization ability has been proven in many interdisciplinary researches. It works well even with noisy and time-dependent objective functions. The canonical basic principle is as follows. For each individual  𝑥𝑖,𝐺 in the current generation G, DE generates a new trial individual  𝑥󸀠 𝑖,𝐺 by adding the weighted difference between two randomly selected individuals  𝑥𝑟1,𝐺 and  𝑥𝑟2,𝐺 to a randomly selected third individual  𝑥𝑟3,𝐺.The resulting individual  𝑥󸀠 𝑖,𝐺 is crossed over with the original individual  𝑥𝑖,𝐺. The fitness of the resulting individual, referred to as a perturbed vector  𝑢𝑖,𝐺+1, is then compared with the fitness of  𝑥𝑖,𝐺.Ifthefitnessof  𝑢𝑖,𝐺+1 is greater than the fitness of  𝑥𝑖,𝐺,then  𝑥𝑖,𝐺 is replaced with  𝑢𝑖,𝐺+1;otherwise,  𝑥𝑖,𝐺 remains in the population as  𝑥𝑖,𝐺+1. Please refer to (1) for notation of crossover and to [22]for the detailed description of used DERand1Bin strategy and all other DE strategies: 𝑢𝑖,𝐺+1 =𝑥 𝑟1,𝐺 +𝐹⋅(𝑥 𝑟2,𝐺 −𝑥 𝑟3,𝐺). (1) 4. Concept of ChaosDE This section contains the description of discrete dissipative chaotic maps, which can be used as the chaotic pseudorandom generators for DE as well as the main principle of the ChaosDE concept. In this research, direct output iterations of the chaotic maps were used for the generation of real numbers in the process of crossover based on the user defined CR value and for the generation of the integer values used for selection of individuals. 4.1. Chaotic Pseudorandom Number Generator. The general idea of ChaosDE and CPRNG is to replace the default PRNG with the discrete chaotic map. As the discrete chaotic map is a set of equations with a static start position, we created a random start position of the map, in order to have different start position for different experiments (runs of EAs). This The Scientific World Journal 3 random position is initialized with the default PRNG, as a one-off randomizer. Once the start position of the chaotic map has been obtained, the map generates the next sequence using its current position. The first possible way is to generate and store a long data sequence (approximately 50–500 thousands numbers) during the evolutionary process initialization and keep the pointer to theactualusedvalueinthememory.Incaseoftheusingup of the whole sequence, the new one will be generated with the lastknownvalueasthenewinitialone. The second approach is that the chaotic map is not reinitialized during the experiment and any long data series is not stored; thus it is imperative to keep the current state of the map in memory to obtain the new output values. As two different types of numbers are required in ChaosDE, real and integers, the use of modulo operators is used to obtain values between the specified ranges, as given in the following equations (2): rndreal =mod (abs (rndChaos),1.0) rndint =mod (abs (rndChaos),1.0)×Range +1, (2) whereabsreferstotheabsoluteportionofthechaoticmap generated number rndChaos, mod is the modulo operator, and Range specifies the value (inclusive) till where the number is to be scaled. 4.2. Selected Chaotic Systems. This subsection contains the mathematical and graphical description of the three selected discrete dissipative systems, which served both as for CPRNGs and also as the examples of systems to be evolutionary controlled. 4.2.1. Burgers Map. The Burgers mapping is a discretization of a pair of coupled differential equations which were used by Burgerstoillustratetherelevanceoftheconceptofbifurcation to the study of hydrodynamics flows. The map equations are given in (3) with control parameters 𝑎 = 0.75and 𝑏 = 1.75 as suggested in [47]: 𝑋𝑛+1 =𝑎𝑋 𝑛−𝑌2 𝑛, 𝑌𝑛+1 =𝑏𝑌 𝑛+𝑋 𝑛𝑌𝑛.(3) 4.2.2. Delayed Logistic. The delayed logistic is a simple twodimensional discrete system similar to the one-dimensional logistic equation. The map equations are given in (4). The parameterusedinthisworkis𝐴 = 2.27 as also suggested in [47]: 𝑋𝑛+1 =𝐴𝑋 𝑛(1−𝑌𝑛), 𝑌𝑛+1 =𝑋 𝑛.(4) 4.2.3. Lozi Map. The Lozi map is a simple discrete twodimensional chaotic map. The map equations are given in (5). The parameters used in this work are as follows: 𝑎 = 1.7 and 𝑏 = 0.5 as suggested in [47]. For these values, the system exhibits typical chaotic behavior and, with this parameter 1.0 0.5 0.0 −0.5 −1.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 y x Figure 1: 𝑥,𝑦plot of the Burgers map. setting, it is used in most research papers and other literature sources [48]: 𝑋𝑛+1 =1−𝑎󵄨 󵄨 󵄨 󵄨𝑋𝑛󵄨 󵄨 󵄨 󵄨+𝑏𝑌 𝑛, 𝑌𝑛+1 =𝑋 𝑛.(5) 4.3. Graphical Examples. Threechaoticmapswereselected for the CPRNG concept. The 𝑥,𝑦plots of the selected maps are depicted in Figure 1 (Burgers map), Figure 4 (delayed logistic map), and Figure 7 (Lozi map). The chaotic behavior ofthechaoticmaps,representedbytheexamplesofdirect output iterations, is depicted in Figures 2,5,and8.Finally, the illustrative histograms of the distribution of real numbers transferred into the range ⟨0-1⟩ generated by means of chaotic maps are in Figures 3,6,and9. 5. Design of Cost Functions for Chaotic System Stabilization The proposal of the basic cost function (CFSimple) is in general basedonthesimplestCF,whichcouldbeusedproblemfree only for the stabilization of p-1 orbit. The idea was to minimize the area created by the difference between the requiredstateandtherealsystemoutputonthewhole simulation interval, 𝜏𝑖(6). This CF design is very convenient for the evolutionary searching process due to the relatively favorable CF surface. Nevertheless, this simple approach has one big disadvantage, which is the including of initial chaotic transient behavior of not stabilized system into the costfunctionvalue.Asaresultofthis,theverytinychange of control method setting for extremely sensitive chaotic system (given by the very small change of CF value) can be 4The Scientific World Journal 0 50 100 150 200 0.0 Iteration −0.5 −1.0 −1.5 −2.0 x (a) 0 50 100 150 200 0.0 Iteration −0.5 −1.0 −1.5 −2.0 x (b) Figure 2: Iterations of the Burgers map (variable 𝑥) (a), iterations of the Burgers map (variable 𝑥), point-plot (b). Value 500 1000 1500 Frequency 0.2 0.4 0.6 0.8 1.0 Figure 3: Histogram of the distribution of real numbers transferred into the range ⟨0-1⟩generated by means of the chaotic Burgers map, 5000 samples. 0.2 0.0 0.4 0.6 0.8 1.0 0.2 0.0 0.4 0.6 0.8 1.0 y x Figure 4: 𝑥,𝑦plot of the delayed logistic map. suppressed by the aforementioned inclusion of initial chaotic transient behavior. Consider CFSimple =𝜏𝑖 ∑ 𝑡=0 󵄨 󵄨 󵄨 󵄨TS𝑡−AS𝑡󵄨 󵄨 󵄨 󵄨,(6) where TS is target state and AS is actual state. Different type of universal cost function without any selection rules is purely based on searching for the desired stabilized periodic orbit and thereafter calculation of the difference between desired and found actual periodic orbit on the short time interval, 𝜏𝑠(20 iterations), from the point where the first minimal value of difference between desired and actual system output is found (i.e., floating window for minimization; see Figure 10). Such a design of universal CF should secure the successful stabilizationofeitherp-1orbit(stablestate)oranyhigher periodic orbit anywise phase shifted. Furthermore, due to CF values converging towards zero, this CF also allows the use of decision rules, avoiding very time demanding simulations. This rule stops EA immediately, when the first individual with good parameter structure is reached; thus the value of CF is lower than the acceptable (CFacc)one.Basedonthenumerous experiments, typically CFacc = 0.001 at time interval 𝜏𝑠=20 iterations; thus the difference between desired and actual output has the value of 0.0005 per iteration, that is, successful stabilization for the used control technique. The CFUNI has the following form: CFUNI =𝑝𝑒𝑛 1+𝜏2 ∑ 𝑡=𝜏1 󵄨 󵄨 󵄨 󵄨TS𝑡−AS𝑡󵄨 󵄨 󵄨 󵄨,(7) where 𝜏1is the first min value of the difference between TS and AS and 𝜏2is the end of optimization interval (𝜏1+ 𝜏𝑠),𝑝𝑒𝑛1=0if 𝜏𝑖−𝜏 2≥𝜏 𝑠;𝑝𝑒𝑛1=10∗(𝜏 𝑖−𝜏 2)if 𝜏𝑖−𝜏 2<𝜏 𝑠 (i.e., late stabilization). The issue of pure searching for periodic orbits causes very chaotic, erratic, and discrete type CF surfaces. The Scientific World Journal 5 0 50 100 150 200 0.0 0.2 0.4 0.6 0.8 1.0 Iteration x (a) 0 50 100 150 200 0.0 0.2 0.4 0.6 0.8 1.0 Iteration x (b) Figure 5: Iterations of the delayed logistic (variable 𝑥) (a), iterations of the delayed logistic (variable 𝑥), point-plot (b). 200 400 600 800 1000 1200 1400 Value Frequency 0.2 0.4 0.6 0.8 1.0 Figure6:Histogramofthedistributionofrealnumberstransferred into the range ⟨0-1⟩ generated by means of the chaotic delayed logistic map, 5000 samples. 6. Experimental Design This research encompasses six case studies. Six different sets of discrete chaotic systems as CPRNGs/to be controlled and two different cost functions were combined in the following form: (i) case study 1: Burgers map as CPRNG/controlled system with CFSimple, (ii) case study 2: Burgers map as CPRNG/controlled system with CFUNI, (iii) case study 3: delayed logistic map as CPRNG/controlled system with CFSimple, (iv) case study 4: delayed logistic map as CPRNG/controlled system with CFUNI, (v) case study 5: Lozi map as CPRNG/controlled system with CFSimple, (vi) case study 6: Lozi map as CPRNG/controlled system with CFUNI. This work is focused on the utilization of the chaos driven DE for tuning of parameters for ETDAS control method 1.0 0.50.0−0.5−1.0 y x 1.0 0.5 0.0 −0.5 −1.0 x Figure 7: 𝑥,𝑦plot of the Lozi map. to stabilize desired unstable periodic orbits (UPO). In the described research, desired UPO was p-1 (stable state). The original control method, ETDAS, in the discrete form suitable for discrete chaotic maps has the following form: 𝐹𝑛=𝐾[ (1−𝑅)𝑆𝑛−𝑚 −𝑥 𝑛], 𝑆𝑛=𝑥 𝑛+𝑅𝑆 𝑛−𝑚,(8) where 𝐾and 𝑅are adjustable constants, 𝐹is the perturbation, 𝑆isgivenbyadelayequationutilizingpreviousstatesof the system, and 𝑚istheperiodof𝑚-periodic orbit to be stabilized. The perturbation 𝐹𝑛in (8) may have arbitrarily largevalue,whichcancausethedivergenceofthesystem. Therefore, 𝐹𝑛should have a value between—𝐹max and 𝐹max. The ranges of all evolutionary estimated parameters are given in Table 1. 6The Scientific World Journal 0 50 100 150 200 Iteration 1.0 0.5 0.0 −0.5 −1.0 x (a) 0 50 100 150 200 Iteration 1.0 0.5 0.0 −0.5 −1.0 x (b) Figure 8: Iterations of the Lozi map (variable 𝑥) (a), iterations of the Lozi map (variable 𝑥), point-plot (b). 100 200 300 400 Value Frequency 0.2 0.4 0.6 0.8 1.0 Figure 9: Histogram of the distribution of real numbers transferred into the range ⟨0-1⟩ generated by means of the chaotic Lozi map, 5000 samples. x 𝜏1𝜏2=𝜏 1+𝜏 s Iteration 1.5 1.0 0.5 0.0 −0.5 −1.0 0 50 100 150 200 Figure 10: “Floating window” for minimization. Within the research, a total number of 50 simulations with ChaosDE were carried out for each case study. The parameter settings for ChaosDE were obtained analytically based on numerous experiments and simulations (see Table 1: Estimated parameters. Parameter Min Max 𝐾−22 𝑅0 0.99 𝐹max 0 0.9 Table 2: DE settings. Parameter Value Population size 25 𝐹0.8 Cr 0.8 Generations 250 Max cost function evaluations (CFE) 6250 Table 2). Experiments were performed in an environment of Wolfram Mathematica; PRNG operations therefore used the built-in Mathematica software pseudorandom number generator. All experiments used different initialization; that is, different initial population was generated in each run of chaos driven DE. 7. Results All simulations were successful and have given new optimal settings for ETDAS control method securing the fast stabilization of the chaotic system at required behaviour (p-1 orbit). Tables 4,5,6,7,8,and9contain the simple statistical overview of optimization/simulation results as well as the best founded individual solutions of parameters setup for ETDAS control method, corresponding final CF value, also, the Istab. Value representing the number of iterations required for stabilization on desired UPO, and further the average error between desired output value and real system output from the last 20 iterations. Graphical simulation outputs of the best individual solutions for particular case studies are depicted in Figures 11,13, and 15, whereas Figures 12,14,and16 show the simulation The Scientific World Journal 7 0 20 40 60 80 100 Iteration 0.5 0.0 −0.5 −1.0 x Burgers map-CFSimple-best solution (a) 020 40 60 80 100 0.5 0.0 −0.5 −1.0 −1.5 −2.0 x Iteration Burgers map-CFUNI-best solution (b) Figure 11: Simulation of the best individual solution, ChaosDE and Burgers map: case study 1,CFSimple (a); case study 2, CFUNI (b). 020 40 60 80 100 0.5 0.0 −0.5 −1.0 x Iteration Burgers map-CFSimple-all solution (a) 0 20406080100 0.5 0.0 −0.5 −1.0 −1.5 −2.0 x Iteration Burgers map-CFUNI-all solution (b) Figure 12: Simulation of all the 50 independent runs of EA, ChaosDE and Burgers map: case study 2, CFSimple (a); case study 1, CFUNI (b). 010 20 30 40 50 Iteration x 0.8 0.7 0.6 0.5 0.4 Delayed logistic-CFSimple-best solution (a) 010 20 30 40 50 Iteration x 0.8 0.7 0.6 0.5 0.4 Delayed logistic-CFUNI-best solution (b) Figure 13: Simulation of the best individual solution, ChaosDE and delayed logistic: case study 3, CFSimple (a); case study 4, CFUNI (b). 8The Scientific World Journal 010 20 30 40 50 Iteration x 0.8 0.7 0.6 0.5 0.4 Delayed logistic-CFSimple-all solution (a) 0102030 40 50 Iteration x 0.8 0.7 0.6 0.5 0.4 Delayed logistic-CFUNI-all solution (b) Figure 14: Simulation of all the 50 independent runs of EA, ChaosDE and Delayed logistic: case study 3, CFSimple (a); case study 4, CFUNI (b). 0 1020304050 Iteration x 0.8 0.6 0.4 0.2 Lozi map-CFSimple-best solution (a) 0 10203040 50 Iteration x 0.8 0.6 0.4 0.2 Lozi map-CFUNI-best solution (b) Figure 15: Simulation of the best individual solution, ChaosDE and Lozi map: case study 5, CFSimple (a); case study 6, CFUNI (b). 010 20 30 40 50 Iteration x 0.8 0.6 0.4 0.2 Lozi map-CFSimple-all solution (a) 010 20 30 40 50 Iteration x 0.8 0.6 0.4 0.2 Lozi map-CFUNI-all solution (b) Figure 16: Simulation of the all 50 independent runs of EA, ChaosDE and Lozi map: case study 5, CFSimple (a); case study 6, CFUNI (b). The Scientific World Journal 9 Table 3: The values for p-1 UPO. Chaotic system Values of p-1 UPO of unperturbed system Burgers map 𝑥𝐹=−0.7499 Delayed logistic map 𝑥𝐹= 0.559471 Lozi map 𝑥𝐹= 0.454545 Table 4: Simple CF statistics: joined case studies 1 and 2, Burgers map as CPRNG/controlled system. Statistical data Case study 1 CFsimple Case study 2 CFUNI CF value CF value Min 2.16199 1.05 ⋅10−6 Max 2.16199 0.0103 Average 2.16199 6.67 ⋅10−4 Median 2.16199 5.32 ⋅10−7 Std. dev. 5.58 ⋅10−11 1.89 ⋅10−3 Average full stab. (iteration) 45 35 output of all 50 runs of ChaosDE, thus confirming the robustness of this approach. For the illustrative purposes, all graphical simulations outputs are depicted only for the variable 𝑥of the chaotic systems. The values for desired p-1 UPOs (fixed points) of unperturbed chaotic systems based on the mathematical analysis of thesystemsaregiveninTable 3. From the results presented in the Tables 4–9,itfollows that the CF-simple is very convenient for evolutionary process, which means that repeated runs of EA are giving identical optimal results (i.e., very close to the possible global extreme). This is graphically confirmed in Figures 12,14,and 16, which show all 50 simulations. All the runs are basically merged into one line. On the other hand, the disadvantage of including of initial chaotic transient behavior of not stabilized system into the cost function value and subsequent resulting very tiny change of control method setting for extremely sensitive chaotic system is causing suppression of stabilization speed and numerical precision. ResultsobtainedinthecasesutilizingtheCF UNI lend weight to the argument that the technique of pure searching for periodic orbits is advantageous for faster and more precise stabilization of chaotic system. 8. Conclusions Based on obtained results, it may be claimed that the presented ChaosDE driven by selected discrete dissipative chaotic systems has given satisfactory results in the chaos control optimization issue. The results show that the embedding of the chaotic dynamics in the form of chaotic pseudorandom number generator into the differential evolution algorithm may help to improve the performance and robustness of the DE. ChaosDE is able to obtain optimal solutions securing the very fast and precise stabilization for both convenient CF surface, in case of Table 5: Characteristics of the best solution: joined case studies 1 and2,BurgersmapasCPRNG/controlledsystem. Parameter Case study 1 CFsimple Case study 2 CFUNI Value Value 𝐾1.22847 0.732498 𝐹max 0.9 0.48495 𝑅0.574997 0.811742 CF value 2.16199 1.05 ⋅10−6 Istab. Value 45 25 Average error per iteration 5.86 ⋅10−5 1.21 ⋅10−8 Table 6: Simple CF statistics: joined case studies 3 and 4, delayed logistic as CPRNG/controlled system. Statistical data Case study 3 CFsimple Case study 4 CFUNI CF value CF value Min 0.199798 2.3 ⋅10−15 Max 0.199798 2.7 ⋅10−15 Average 0.199798 2.44222 ⋅10−15 Median 0.199798 2.40551 ⋅10−15 Std. dev. 5.11 ⋅10−16 9.04629 ⋅10−17 Average full stab. (iteration) 10 7.5 Table 7: Characteristics of the best solution: joined case studies 3 and 4, delayed logistic as CPRNG/controlled system. Parameter Case study 3 CFsimple Case study 4 CFUNI Value Value 𝐾1.29837 1.31355 𝐹max 0.394579 0.336294 𝑅0.01 0.010219 CF value 0.199798 2.3 ⋅10−15 Istab. Value 10 8 Average error per iteration 2.22 ⋅10−17 0 Table 8: Simple CF statistics: joined case studies 5 and 6, Lozi map as CPRNG/controlled system. Statistical data Case study 5 CFsimple Case study 6 CFUNI CF value CF value Min 0.520639 3.5331 ⋅10−15 Max 0.520639 4.0551 ⋅10−15 Average 0.520639 3.8063 ⋅10−15 Median 0.520639 3.6352 ⋅10−16 Std. dev. 2.41 ⋅10−15 1.19 ⋅10−16 Average full stab. (iteration) 32 11 theCF-simple,aswellasfortheverychaoticandnonlinear CF surface in case of the CF-universal. When comparing both the CF designs, the CF-simple is very convenient for evolutionary process (i.e., repeated