Impact of chaotic dynamics on the performance of metaheuristic optimization algorithms: An experimental analysis

Impact of chaotic dynamics on the performance of metaheuristic optimization algorithms: An experimental analysis Ivan Zelinka a, ⇑ , Quoc Bao Diep a , Václav Snášel a , Swagatam Das b , Giacomo Innocenti c , Alberto Tesi c , Fabio Schoen c , Nikolay V. Kuznetsov d,e,f a Department of Computer Science, Faculty of Electrical Engineering and Computer Science, VŠB-TUO, 17.listopadu 2172/15, 708 00 Ostrava-Poruba, Czech Republic b Electronics and Communication Sciences Unit, Indian Statistical Institute, 203 B T Road, Kolkata 700108, India c Dept. Information Engineering (DINFO) – University of Florence, via di Santa Marta 3, Firenze, Italy d Faculty of Mathematics and Mechanics, St. Petersburg State University, 198504 Peterhof, St. Petersburg, Russia e Faculty of Information Technology, University of Jyväskylä 40014 Jyväskylä, Finland f Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, Bolshoj pr. 61, St. Petersburg, 199178, Russia article info Article history: Received 21 June 2021 Received in revised form 25 October 2021 Accepted 30 October 2021 Available online 10 November 2021 Keywords: Deterministic chaos Swarm intelligence Evolutionary algorithms Algorithm dynamics Algorithm performance abstract Random mechanisms including mutations are an internal part of evolutionary algorithms, which are based on the fundamental ideas of Darwin’s theory of evolution as well as Mendel’s theory of genetic heritage. In this paper, we debate whether pseudo-random processes are needed for evolutionary algorithms or whether deterministic chaos, which is not a random process, can be suitably used instead. Specifically, we compare the performance of 10 evolutionary algorithms driven by chaotic dynamics and pseudo-random number generators using chaotic processes as a comparative study. In this study, the logistic equation is employed for generating periodical sequences of different lengths, which are used in evolutionary algorithms instead of randomness. We suggest that, instead of pseudorandom number generators, a specific class of deterministic processes (based on deterministic chaos) can be used to improve the performance of evolutionary algorithms. Finally, based on our findings, we propose new research questions. Ó2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). 1. Introduction Chaos (deterministic chaos) is a broad term that refers to dynamical phenomena showing random-like behaviors at a first glance, even if they are generated by deterministic systems. This kind of behavior is typical of processes that are strictly stochastic in nature, such as the motion of molecules in a vessel filled with gas. However since chaotic systems are inherently nonlinear, traditional statistical methods, which are often linear, are inadequate for their study. Chaotic system behaviors share several features with random-like mechanisms, thus making them easily mistaken for random noises. Deterministic chaos and its applications can be observed in control theory, computer science, physics, biology, and many other fields. Deterministic chaos research can be discussed on two levels. On the first level, it is an object of research where https://doi.org/10.1016/j.ins.2021.10.076 0020-0255/Ó2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). ⇑ Corresponding author. E-mail addresses: [email protected] (I. Zelinka), [email protected] (V. Snášel), [email protected] (S. Das), [email protected] (G. Innocenti), [email protected] (A. Tesi), [email protected] (F. Schoen), [email protected] (N.V. Kuznetsov). Information Sciences 587 (2022) 692–719 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins the properties of cause and effect are examined, and on the second, it is the use of chaos as a tool. This article focuses on the use of modified chaos in heuristic algorithms to investigate their performance. However, for a better understanding of the issue, we will mention both areas now - chaos as an object of research and chaos as a tool that can be used in other disciplines. Firstly, we will discuss chaos relating to the phenomenon and object of the research. Chaos was the subject of research in the field of dynamical nonlinear systems as early as the nineteenth century when the well-known French mathematician Henri Poincaré solved the problem of three bodies [1]. The introduction of chaos on the scientific scene did not occur until the 1960s when E.N. Lorenz discovered deterministic chaos in mathematical models modeling weather behavior [2]. From that moment on, chaos was found practically everywhere. Some examples of places chaos can be found are; fluid flow [3], plasma flow and turbulence [4], heart rhythm, ECG and EEG recordings [5–7], biological systems, chemical systems, and economic systems. Around deterministic chaos, a whole mathematical theory has been developed which allows us to grasp this extraordinary phenomenon and work with it. Thanks to this, we can now solve problems that were previously unsolvable and thus develop better technologies. One example of many is the use of wingtip winglets, which divert turbulent flow from the wingtip of away [8]. Due to this and regarding the operation of the aircraft, wings with a smaller area are therefore sufficient, even with less resistance at the same lifting force, thus leading to significant fuel savings. It must also be said that chaos has been observed not only in natural and economic systems but also in man-made technological systems. Most of the time, however, the chaos observed in technological systems was induced by the interaction of human technology with natural systems like atmospheric ones. Surprisingly, chaos has also been discovered in systems such as computer systems and algorithms. For example, chaos is observed in the dynamics of evolutionary algorithms [9,10], i.e. chaos within the algorithmic system. It can be said that chaos is a phenomenon that is not intended to occur only in certain types of systems but a phenomenon that can occur virtually anywhere. Therefore, in modern engineering, it is necessary to consider its presence when designing new technologies. Many such examples can be found in today’s scientific literature. On the second level, chaos is used as a tool in, usually interdisciplinary, research. If we look at deterministic chaos as a tool that can be used in research in other scientific fields, then we also come across a vibrant set of possible applications of deterministic chaos. For example, it can be used in the design of various engineering devices, data encryption, and information transmission, but also in improving the functionality of some types of algorithms. Here we will be focusing on the mutual fusion of chaos and computer sciences with attention to the heuristic algorithm. Concerning chaos and computer science in general, a lot of interesting applications have been published. In [11], for example, researchers combined deterministic chaos and a pseudo-random number generator. The use of an ultra-weak multidimensional coupling of p 1-dimensional dynamical systems to generate random or pseudo-random numbers is discussed there. In another article, [12] examines the logistic map as a potential pseudo-random number generator and compares it to current pseudo-random number generators. The results of logistic maps are compared to traditional pseudo-random number generation methods. The method is used to calculate the number, delay, and period of the logistic map’s orbits with varying degrees of precision. The paper [13] suggests a pseudo-random number generator algorithm, which combines coupled map lattice and chaotic iteration. This algorithm was also put to the test in NIST 800–22 statistical test suits, which are often used in image encryption. The authors of [14] use properties of chaotic systems to construct the CCCBG, a pseudo-random bit generator in which two chaotic systems are cross-coupled with each other. The four basic tests are used to evaluate the bitstreams developed by the CCCBG: mono bit test, serial test, auto-correlation, and Poker test. The NIST suit checks, which are the most rigorous randomness tests, were also used. Paper [15] proposes a binary stream-cipher algorithm using dual one-dimensional chaotic maps, with statistic properties indicating that the sequence is of strong randomness. Similar research is carried out in [16–19]. If we stay focused on chaos and heuristic algorithms, which is the core of this paper, then chaos has previously been found in a variety of processes, including evolutionary systems [9] (the first report on observed chaos inside genetic algorithm dynamics) or [10] which is a book discussing the fusion of chaos and evolutionary algorithms (EAs). Chaos has also been used to substitute pseudo-random number generators (PRNGs) in evolutionary algorithms in recent years. Specifically, the use of chaos inside EAs is reported in [17] (discussing whether or not pure chaotic sequences can improve the performance of evolutionary algorithms). Alteratively, papers [20,21] discuss the use of deterministic chaos inside the particle swarm algorithm instead of PRNGs, [11–15] investigate relations between chaos and randomness or the others like [22] (applies chaotic genetic algorithm on cyclic electric load forecasting), [23] (uses chaos driven differential evolution on optimization of the batch reactor) and [24] which uses chaos driven evolutionary algorithms for PID control. Other papers reporting the use of chaos in algorithms are [25] (use of the chaos-based evolutionary algorithm in nonlinear programming) and [26] which discusses adaptive differential evolution algorithms powered by multi-chaotic framework used for parent selection and tested on CEC2014 1 benchmark set. Furthermore recent algorithms like a Whale algorithm [27] with chaos was reported in [28,29] (using chaos game optimization). Another recent algorithm, the Invasive Weed Optimization algorithm (IWO) [30] based on chaos theory, was used for the optimal design of PID controller is principally similar to [24,31]. Of course, our list of classical and modern algorithms using chaos to increase their performance does not end here. For example, a paper using multiple chaoses embedded gravitational search algorithm, paper [32], is another representative of a new generation of algorithm testing chaos hybridization and the optimization algorithm itself. The list of all research papers that contain a fusion of 1 http://www5.zzu.edu.cn/cilab/info/1005/1013.htm I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 693 chaos and a heuristic algorithm would be for the article itself is not the goal of this work, and therefore we refer only to this brief list. Based on those and other research papers, it can be stated that several interesting contributions have been made dealing with the use of chaos within optimization algorithms. It is worth mentioning work [33], which also deals with the chaos use in a newly designed algorithm, which has been tested on a large number of test functions. Additionally, [34] investigates the performance, scalability and convergence and robustness of chaos-enhanced evolutionary algorithms with boundary constraints while papers like [35–37] or [38] deal with chaos application inside evolutionary algorithms. Therefore, it is clear that recently the interest of scientists has become to be attracted by the hybridization of modern optimization algorithms and chaotic dynamics. The mentioned publications uses chaos application within algorithms from different points of view, but primarily only as an improvement of one specific algorithm, tested for significant test functions. Some of them compared the impact of a chaotic version on a CEC (Congress of Evolutionary Algorithms) test benchmark function (however older and typically one set) to a non-chaotic version of the same algorithm. We have agreed to broaden the scope of this research to include a newer CEC collection of benchmark test functions, as well as more algorithms of various types (evolutionary vs swarm). We also included one critical aspect of the experiment: we looked at the effect of modified chaos on algorithm performance. This paper focuses on using deterministic chaos to produce Nperiodic sequences (forced by low calculation precision), which are used in evolutionary algorithms instead of pseudorandom number generators or just pure deterministic chaos series. Various EAs of different kinds are used here, see Table 3. Algorithms itself did not performed any analysis on how and if used pseudo-random numbers are random, what is its distribution etc. Pseudo-random, chaotic, and periodic (chaos-based) series are only simply used. A research study, especially a theoretic one, dealing with how much the performance of different algorithms depends on different levels of chaotic dynamics has not yet been satisfactorily developed. For these reasons, this work was created, which aims to point out how much the performance of the evolutionary algorithm depends on the specifically modified chaotic dynamics, which is used inside the algorithm instead of a pseudo-random number generator. This research also raises interesting research questions that suggest an interpretation of evolutionary algorithms on the level of discrete dynamic systems with feedback and thus allow the use of the theoretical mathematical apparatus of cybernetics to analyze and describe it. This could explain why chaos has a positive effect on the performance of evolutionary algorithms. Applying this existing mathematical apparatus to evolutionary and swarm intelligence algorithms shall help shed light on the many unanswered questions in the algorithm community today while providing exciting and powerful mathematical tools for researching these algorithms. However, answering this question belongs to future research. EAs with deterministic chaos systems (DCHS), as demonstrated here, produce mostly the same or better results, as seen in the section Results. The article’s structure is as follows. We will clarify the relevance of research in this direction and point out the unanswered questions in the use of chaos in evolutionary algorithms in the Motivation and Novelty section. The main idea of our paper is then repeatedly formulated in the Hypothesis section, where we repeat the basic ideas from the field of various heuristic algorithms using deterministic chaos and explain why we chose the procedure used in this paper. The Experiment design section follows, which explains the conditions under which our experiments were conducted including a link to GitHub with all source codes from our experimentation. We also explore and discuss the impact of accuracy on the chaotic course’s periodicity, as well as the nature of our own experiments in the section Results. The CEC test functions were used to test ten selected evolutionary algorithms. Whole idea is captured in Fig. 9. The results are discussed and summarized in the section Conclusions. 2. Motivation and novelty The experiments described in this article were motivated by the current state, which was discussed in the previous section, as well as the widespread perception that operations like mutations and others that need random operations cannot be performed without randomness. From the description of evolutionary algorithms, it is evident, that they are discrete dynamic systems. That means they are feedback systems processing their output (in this case the population with fitness) as the input for the next generation procedures. Which individual will be used in the following generation (iteration, migration, etc.) is determined by an objective function (e.g. fitness). If we consider evolutionary or swarm algorithms to be dynamic systems, the assumption that successful control of these systems requires (pseudo) random processes is a bit odd in regards to what the control engineering community normally expects (remember that in control theory randomness usually represents a signal, that has to be eliminated). As a result, our tests were meant to see if the pseudo-random numbers created by the traditional type of pseudo-random number generators (Mersenne - Twister like) are actually necessary for the performance of evolutionary algorithms, or if alternative processes, such as those generated by deterministic chaos, can be used instead. It is also possible to create periodic series (from chaotic ones) of various lengths. Chaos as well as pure randomness, does not exist in computers, only pseudo-chaotic or pseudo-random processes are present. Chaos, unlike pseudo-random processes generated by other algorithms, may alter the length of its period by altering the precision of its calculation. For example, if we employ a logistic Eq. 1 with the parameter A¼4 and let this equation loop in a computer, we have a theoretically quasi-random series (for the external observer) that never repeats and, in fact, belongs to the world of deterministic chaos due to the high accuracy in which today’s computers work. When the precision of the calculation is reduced, however, rounding happens, resulting in I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 694 the effect that this otherwise potentially endlessly long series of numbers begins to be shorter and recurrent. As shown in Fig. 2–8, the occurrence of repetitions is dependent on the extent of the computation accuracy. This study’s tests are not only based on determining the impact of chaotic dynamics as pseudo-random numbers on the algorithm’s performance, but also on determining if modifying the computation accuracy or shortening the chaotic period will alter the EAs’ performance quality. That means no pure chaotic series were used in EAs instead of randomness. We employed periodic series of various lengths Nbased on the chaotic generator ”truncating”, which has never been done before. It is also debated whether pseudo-random processes are required or whether short pseudo-chaotic processes generated using a deterministic method are sufficient. This raises several new research problems, such as whether EAs require pseudo-random processes to work, and whether there is a link between the type of test function (which typically contains nonlinear elements and it can be seen as a nonlinear system interacting with a specific algorithm) and algorithm performance. All of this is demonstrated by our findings. In the conclusion section, further possible questions and research ideas are presented in the section Open Questions and Future Research. Therefore, if we can summarize the previous information, it can be stated that the novelty of our contribution lies, in comparison with others, in the following facts. Our experiments consider not only one or two algorithms but a total of ten which reasonably represent the whole spectrum of evolutionary algorithms and swarm intelligence. Furthermore, these algorithms are applied to recognized benchmark test functions and evaluated by a widely accepted statistical validity test. Another novelty of this paper is that in addition to the influence of pure chaos on the performance of the algorithm, we also study the influence of periodic behavior generated by a given chaotic system. These series are no longer random-like or chaotic but fully deterministic. As our experiments have found that, from a certain length of the Nperiodic behavior, the algorithms usually start to show better performance concerning randomness or pure chaos. This raises many interesting questions about the dynamics and performance of evolutionary algorithms as such. Regarding the used chaotic system, we chose the logistic equation, which is a simple representative of chaos from the area of chaotic dynamics. It is very well known, well studied, and sufficiently rich for our experiments. Of course, other chaotic systems can be used, of which there are many today [39]. However, given our experience with these systems and evolutionary techniques, we believe that the logistic equation is quite suitable for the set of experiments and confirmation of our hypotheses presented in this article. Especially if we force the chaos generator to produce Nperiodic behavior. The use of other chaotic systems and other evolutionary algorithms is certainly an issue to be investigated in future researches. 3. Hypothesis As mentioned in previous sections, a large number of research papers have been produced in the last ten to fifteen years discussing the presence of chaos in evolutionary algorithms, the use of these algorithms for synthesis and chaos control and also the use of chaos as a random generator in the algorithms themselves. If we think about this development, then it is clear that algorithms using chaos more or less benefit in terms of increasing the performance of evolutionary algorithms, which is evident in already published works. If the first generation of researchers used EAs with pseudo-random generators, these EAs were improved by chaos generators; then the question arises as to whether this research could be further extended in the same direction. High-quality random generators produce time series that really repeat after a very large number of iterations, similarly to chaos. Oppositely, chaos as a parametrically adjustable system can change the degree of chaos, which turns into deterministic windows, which can be called Nperiodic orbit-series. Our first idea was to use these deterministic windows, Fig. 1, to have Nperiodic orbits and thus replace the random series produced by pseudo-random number generators or chaotic generators. The problem is that the number of deterministic windows in each chaotic system is limited and usually of very low periodicity (e.g. N=4,8,...), see Fig. 1. We decided to work around this with a simple trick, i.e. the low accuracy of the calculation, which results in the respective chaotic generator producing periodic series. In other words, the limited accuracy of the calculation degrades the chaos generator to a generator of periodic events (e.g. Fig. 2–8). Thus a question of whether a chaotic generator operating in forced deterministic regime mode has the same or better benefit of an evolutionary algorithm performance exists. To avoid using only a few existing deterministic windows (existing in virtually every chaotic system), we decided to force the generator to create periodic series by changing the accuracy of the calculation. In other words, based on the accuracy of the calculation of the individual iterations, which then enter back into the chaotic generator, we are able to force the generator to generate periodic series. Furthermore, these are then used as a source instead of a random number generator. Thus, we got into an area that has not yet been explored and deals with the question of whether short or long periodic series can be used in evolutionary algorithms instead of a random number generator. Of course, we could do without chaotic number generators and immediately generate some pre-selected Nperiodic series. However, it makes sense to use already proven chaotic systems that can be relatively easily forced to generate periodic series. There is such a logical continuity: chaos replaced randomness, periodicity replaced chaos. The whole principle is shown in Table 1. We, therefore, formulated a hypothesis: What is the effect of using periodic series instead of a random number generator in evolutionary algorithms? Is there any impact, positive or negative? The findings and our answers then open up many other interesting research questions, which seem to connect these algorithms with the theory of discrete dynamical systems and thus allow new insights into evolutionary dynamics. I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 695 4. Experiment design The experiments that have been performed can be divided into two categories. The first category examines how the existence of periodicity generated by deterministic chaos systems is influenced by computation precision, while the second one employs periodical time series generated by chaotic systems inside EAs and compared with the same EAs powered by PRNGs. Our experiments were designed according to the following facts. To verify/demonstrate our hypothesis about chaos usability it must be firstly remembered that there are a large number of algorithms, test problems, and repeated simulations with different initial conditions (necessary to demonstrate the robustness of the results). All this must then be statistically evaluated, and it is clear that it is not possible to cover all possible scenarios. Therefore, we focused on a narrower selection of more well-known and less well-known algorithms, both in the field of classical evolutionary algorithms and in the field of swarm intelligence. These are algorithms such as differential evolution (DE) of particles swarm (PSO), grey wolf algorithm (GWA), SOMA algorithm, and many others. These algorithms were tested regularly on selected test functions used at CEC. Regular statistical tests and evaluations were then performed on a huge amount of results, which in the end allowed us to evaluate the impact of using chaos with different degrees of accuracy on the performance of these algorithms. The experiments were performed on a standard PC with a classic i7 processor and 8 GB of memory, as well as partly on a supercomputer of the national supercomputer center IT4 (https://www.it4i.cz/en) to speed up the calculations and shorten the time. There was no other reason to use a supercomputer - these experiments can be repeated on any fast enough computer. Only the logistic Eq. 1was used to generate chaos because it is a well-known basic system for generating chaos. Our study can be extended by a larger class of chaotic systems. However, it is clear from the results we obtained that extending a larger class of chaotic systems would only yield more simulations, but the results would probably not be fundamentally different. 5. Generators of chaotic or periodic series? The so-called logistic equation (Eq. 1) was used to demonstrate the reliability of pseudo-chaos periodicity on numerical precision. Several experiments were carried out, and the results are displayed in Fig. 2–8. Other chaotic generators of various mathematical descriptions, such as Lozi, Henon, Ikeda, or others, like those experimentally manufactured and published in [39], can also be employed. On the other hand, we recall that the basic idea of this paper is the use of Nperiodic series generated by a chaotic generator due to the reduced accuracy of the calculation. Thus, our work differs from previous ones, where only the influence of purely chaotic generators was investigated (see, e.g. [53–57] amongst the others). Fig. 1. Chaotic system behavior with deterministic windows containing Nperiodic behavior. The grainy area is chaos, the deterministic windows are for example at A¼1:4;2:1;2:55fg, see [10]. Table 1 Hypothesis and selected literature representing individual periods of development and use of evolutionary algorithms. Classic era of evolutionary computation using different pseudorandom number generators. Evolutionary computation using deterministic chaos instead of pseudorandom number generators. Can periodic series generated by deterministic chaos systems, used instead of pseudo-random number generators, improve evolutionary algorithms performance? [40–52],...,[53–56] [57],... [58,59], this paper I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 696 In chaotic dynamics it can be also observed the existence of pseudo-patterns, that is, the situation where a time series does not repeat itself exactly, but it periodically displays a very similar behavior. This is a feature of true chaos, which has an inner structure made of a ”basic recurring shape”, which is persistently approached by the system trajectory. Each pseudo-pattern has its unique features, which makes a number of the sequences to be somewhat correlated with the previous ones. As a consequence, the probability density function is no longer evenly distributed among different sequences of numbers, which is a strong difference from white noises. Therefore, using chaos for random-like numbers can make the generator ”biased”, as well as the algorithms which exploit it. However, to observe this, shorter parts of ‘‘clear” chaotic series shall be taken into consideration where these pseudo-patterns are dominant (i.e. take a major part of chaotic series). In our experimentation, we have used max 100 000 cost function evaluations, which means more chaotic numbers have been used for one algorithm run (remember, a ”randomness” is generally needed more than once, to build new offspring for one cost function evaluation). Thus, the eventual occurrence of pseudo-patterns should have a negligible impact on algorithm performance. Due to the divergence of nearby trajectories, such patterns shall shortly end if appears in chaotic series. x nþ1 ¼Ax n 1x n ðÞ ð1Þ The logistic equation has been employed in several stages of research here as well as in another papers, i.e. Identification of a Nperiodic orbit with numerical precision as a criterion, see Fig. 2–8. A global picture of the behavior of a logistic equation with a fixed parameter A= 4, see Fig. 3 and 4. A broad view of the behavior of logistic equations considering a variety of parameters A2½3:4;4and numerical precision 2½1;20, logistic equation has been iterated for 1 000, 10 000, 100 000 iterations, see Table 2. Investigate and visualize why and when Nperiodic series occur, see Fig. 3–4. The analysis of the dynamics of logistic equation has been done. For parameter A¼4, numerical precision 2½1;13, initial conditions x start 2½0:01;0:99(xwas incremented periodically by = 0.01) and 1 000 000 iterations for each combination of this parameters, see Table 2. see Fig. 5–7,. The impact of the precision on the dynamics of logistic equation is depicted on Fig. 3 (compare with Fig. 4).The original mapping function is changed to a step-wise mapping function with low accuracy. This is the origin of many of the periodic series, that we later used in our studies. 5.1. Determinism or randomness If we are talking about using random or, more precisely, pseudo-random generators, it is necessary to realize the difference between a purely random process and a pseudo-random process generated by an algorithm, i.e. the difference between true random and pseudo-random numbers. In principle, there are two ways to generate random numbers. The first one is to generate so-called pseudo-random numbers (PRNG), which consists of producing seemingly random numbers via a suitable algorithm. The second way is to generate random numbers using the true random number generator (TRNG), which uses randomness obtained from physical phenomena. Recognition of true and pseudo-random numbers can be very problematic. Statistical analyses are used to verify the trustworthiness of algorithms. Fig. 2. The period 36 (precision = 4) based on Eq. 1for A¼4, see Table 2. I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 697 TRNG generation methods use physical/hardware units. A physical random number generator can be based on many principles, one of which may be the unpredictability of the random behavior of atomic and subatomic phenomena, which can be observed using the laws of quantum mechanics. One of them is the use of a radioactive source. The time intervals in which the radioactive source decays are entirely unpredictable. Another interesting feature was the Lavarand generator, built by Silicon Graphics and using lava lamp images to generate truly random numbers. It no longer works today. Closer to common options are devices that use network peaks, measuring noise or user input, such as mouse movement, keystroke delay, and more. Fig. 3. Cobweb diagram: a low numerical precision = 1 (one decimal behind zero) impact on the mapping function shape. Four periodic orbit is observable (blue line). Fig. 4. Cobweb diagram: full chaotic dynamics with of logistic equation. Compare with Fig. 3, the same initial conditions are used. Table 2 Periodicity dependence of Eq. 1on various numerical precision (up to precision 13). Numerical Precision Minimal Period Maximal Period 144 2210 31029 41536 5 67 170 6 143 481 7 421 758 8 1030 4514 9 2277 11227 10 2948 35200 11 9668 57639 12 65837 489154 13 518694 518694 I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 698 However, computer/software methods are important for our needs. Pseudo-random number generators are algorithms that produce long strings of numbers having a seemingly good random distribution, but later these sequences are repeated, and the quality of the distribution decreases. One of the most used algorithms is a linear congruent generator operating based on the recurrent relationship, Eq. 2. The maximum number of numbers it can create is modulo bmodm. Slightly different values of the multiplication factor are used to avoid some undesirable properties of the linear congruent generator. Most programming languages have special libraries or functions for creating pseudo-random numbers integrated. Nowadays, of course, they are used in much more modern algorithms. Their output are number series, which for common Fig. 5. Periodicity of logistic equation under precision 1 and x start ¼0:45. Fig. 6. Periodicity of logistic equation under precision 2 and x start ¼0:8. Fig. 7. Periodicity of logistic equation under precision 3 and x start ¼0:6. I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 699 Table 3 The detailed control parameters of the algorithms. Algorithms The control parameter values DE pop ¼100;F min ¼0:4;F max ¼1;Cr ¼0:5 SOMA pop ¼50;Step ¼0:11;PRT ¼0:1;PathLength ¼3:0 PSO pop ¼100;w i ¼1;w damp ¼0:99;c 1 ¼1:5;c 2 ¼2:0; v max ¼0:1½100;100 D ; v min ¼ v max ABC pop ¼50;n onlooker ¼pop;l¼0:6D:pop;a¼1 ACO pop ¼50;n sample ¼40;q¼0:5;f¼1 FA pop ¼25;a¼0:2;b 0 ¼1;c¼1;a damp ¼0:98;d¼0:05:½100;100 D CA pop ¼50;a¼0:3;b¼0:5;p accept ¼0:35;n accept ¼p accept :pop GWO pop ¼30;~ A¼2~ a:~ r 1 ~ a;~ C¼2:~ r 2 WOA pop ¼30;b¼1;l¼ða 2 1Þrnd þ1 HFPSO pop ¼30;a¼0:2;b 0 ¼2;c¼1;c 1 ¼c 2 ¼1:49445; v max ¼0:1½100;100 D ; v min ¼ v max ;w i ¼0:9;w f ¼0:5 Fig. 9. Experiment idea. Fig. 8. Periodicity of logistic equation under precision 4 and x start ¼0:02. I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 700 Table 5 Comparison of algorithms (SOMA ATO, HFPSO, and ABC original vs chaos version) on the CEC 2017 benchmark suite (10 dimensions, 51 runs, full precision, WRT at 5%). CEC 2017 SOMA ATO HFPSO ABC Function Original Chaos Original Chaos Original Chaos Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) F 1 1:66eþ03ð2:12eþ03Þ1:14eþ01ð1:61eþ01Þ+2:57eþ03ð2:94eþ03Þ3:62eþ03ð4:10eþ03Þ 5:91eþ02ð6:18eþ02Þ1:29eþ04ð2:30eþ04Þ F 2 1:43eþ00ð7:09eþ00Þ0:00eþ00ð0:00eþ00Þ+7:37eþ00ð3:46eþ01Þ6:28eþ02ð4:44eþ03Þ 2:49eþ06ð3:09eþ06Þ1:78eþ06ð7:67eþ06Þ+ F 3 1:67eþ03ð9:72eþ02Þ7:57e02ð1:21e01Þ+0:00eþ00ð0:00eþ00Þ0:00eþ00ð0:00eþ00Þ 1:39eþ04ð3:89eþ03Þ1:21eþ03ð2:90eþ03Þ+ F 4 4:02eþ00ð1:73eþ00Þ3:96eþ00ð1:28eþ00Þ 1:31eþ00ð7:76eþ00Þ1:47eþ00ð5:45e01Þ 3:89eþ00ð1:40e01Þ7:04eþ00ð1:38eþ01Þ F 5 7:15eþ00ð1:60eþ00Þ4:55eþ00ð1:68eþ00Þ+1:64eþ01ð6:01eþ00Þ1:09eþ01ð5:18eþ00Þ+2:73eþ01ð3:74eþ00Þ1:92eþ01ð8:85eþ00Þ+ F 6 2:13e05ð6:60e06Þ2:54e07ð1:62e06Þ+9:78e02ð4:08e01Þ1:07e02ð7:61e02Þ 3:12e06ð6:35e06Þ5:76e02ð2:13e01Þ F 7 1:94eþ01ð2:39eþ00Þ1:59eþ01ð3:56eþ00Þ+1:79eþ01ð3:87eþ00Þ1:90eþ01ð3:18eþ00Þ 3:82eþ01ð4:43eþ00Þ2:78eþ01ð8:51eþ00Þ+ F 8 7:54eþ00ð2:01eþ00Þ4:19eþ00ð1:49eþ00Þ+1:27eþ01ð6:14eþ00Þ8:74eþ00ð2:87eþ00Þ+2:64eþ01ð3:85eþ00Þ1:77eþ01ð7:48eþ00Þ+ F 9 7:37e07ð1:10e06Þ0:00eþ00ð0:00eþ00Þ+0:00eþ00ð0:00eþ00Þ0:00eþ00ð0:00eþ00Þ 0:00eþ00ð0:00eþ00Þ2:35e01ð6:46e01Þ F 10 3:23eþ02ð1:00eþ02Þ1:48eþ02ð1:17eþ02Þ+5:42eþ02ð2:10eþ02Þ4:07eþ02ð2:36eþ02Þ+1:45eþ03ð1:18eþ02Þ8:40eþ02ð3:78eþ02Þ+ F 11 3:29eþ00ð1:26eþ00Þ2:56eþ00ð1:81eþ00Þ+9:48eþ00ð9:19eþ00Þ5:91eþ00ð3:78eþ00Þ+6:87eþ00ð1:11eþ00Þ9:19eþ00ð5:24eþ00Þ F 12 2:88eþ04ð4:39eþ04Þ1:35eþ04ð1:33eþ04Þ+1:56eþ04ð1:36eþ04Þ1:43eþ04ð1:28eþ04Þ 2:52eþ06ð1:29eþ06Þ2:58eþ04ð3:06eþ04Þ+ F 13 2:41eþ02ð3:52eþ02Þ8:37eþ00ð3:51eþ00Þ+5:84eþ03ð6:44eþ03Þ5:94eþ03ð6:08eþ03Þ 1:12eþ04ð3:03eþ03Þ1:07eþ04ð1:62eþ04Þ+ F 14 1:63eþ00ð9:68e01Þ1:61eþ00ð1:18eþ00Þ 6:43eþ01ð3:50eþ01Þ5:14eþ01ð2:14eþ01Þ 7:21eþ02ð4:77eþ02Þ4:58eþ01ð2:66eþ01Þ+ F 15 1:14eþ00ð6:25e01Þ7:76e01ð7:50e01Þ+4:87eþ01ð5:87eþ01Þ3:49eþ01ð3:12eþ01Þ 3:62eþ03ð1:90eþ03Þ6:33eþ02ð9:63eþ02Þ+ F 16 2:02eþ00ð3:13eþ00Þ2:08eþ00ð3:07eþ00Þ 2:44eþ02ð1:28eþ02Þ2:32eþ02ð1:44eþ02Þ 3:97eþ01ð1:83eþ01Þ7:09eþ01ð6:97eþ01Þ F 17 2:01eþ00ð3:28eþ00Þ1:43eþ00ð2:51eþ00Þ+5:24eþ01ð3:43eþ01Þ4:24eþ01ð3:27eþ01Þ 4:61eþ01ð6:56eþ00Þ4:56eþ01ð2:00eþ01Þ+ F 18 4:34eþ00ð3:03eþ00Þ5:46eþ00ð1:04eþ01Þ 1:11eþ04ð1:06eþ04Þ1:06eþ04ð1:04eþ04Þ 6:69eþ04ð4:69eþ04Þ6:70eþ03ð1:54eþ04Þ+ F 19 2:97e01ð3:63e01Þ1:13eþ01ð1:55eþ01Þ 1:16eþ02ð2:53eþ02Þ3:55eþ02ð1:82eþ03Þ 9:93eþ02ð5:90eþ02Þ6:61eþ02ð1:22eþ03Þ+ F 20 9:94e02ð2:20e01Þ2:09e01ð3:85e01Þ 7:02eþ01ð5:92eþ01Þ8:26eþ01ð6:57eþ01Þ 2:41eþ01ð2:07eþ00Þ2:60eþ01ð7:60eþ00Þ F 21 1:07eþ02ð8:82eþ00Þ9:90eþ01ð2:50eþ01Þ+1:97eþ02ð4:53eþ01Þ1:85eþ02ð5:00eþ01Þ+1:65eþ02ð2:11eþ01Þ2:07eþ02ð4:15eþ01Þ F 22 6:56eþ01ð3:49eþ01Þ7:24eþ01ð4:16eþ01Þ 1:02eþ02ð1:87eþ01Þ1:15eþ02ð9:84eþ01Þ 1:03eþ02ð1:45eþ00Þ2:03eþ02ð3:10eþ02Þ F 23 3:11eþ02ð2:92eþ00Þ3:02eþ02ð4:30eþ01Þ+3:26eþ02ð1:31eþ01Þ3:18eþ02ð7:45eþ00Þ+3:30eþ02ð4:13eþ00Þ3:25eþ02ð1:09eþ01Þ+ F 24 1:20eþ02ð4:35eþ01Þ1:26eþ02ð7:37eþ01Þ 3:03eþ02ð1:02eþ02Þ3:21eþ02ð8:19eþ01Þ 3:53eþ02ð2:08eþ01Þ3:57eþ02ð9:11eþ00Þ F 25 4:01eþ02ð8:14eþ00Þ4:19eþ02ð2:30eþ01Þ 4:22eþ02ð5:28eþ01Þ4:05eþ02ð6:63eþ01Þ 4:35eþ02ð1:18eþ01Þ4:28eþ02ð2:50eþ01Þ F 26 2:28eþ02ð1:17eþ02Þ2:75eþ02ð8:49eþ01Þ 4:60eþ02ð3:60eþ02Þ3:22eþ02ð2:39eþ02Þ+4:89eþ02ð8:49eþ01Þ5:40eþ02ð1:51eþ02Þ F 27 3:92eþ02ð2:37eþ00Þ3:93eþ02ð2:77eþ00Þ 4:10eþ02ð2:63eþ01Þ4:02eþ02ð2:50eþ01Þ+4:57eþ02ð3:24eþ01Þ4:77eþ02ð3:40eþ01Þ F 28 2:66eþ02ð9:05eþ01Þ2:62eþ02ð9:80eþ01Þ+4:80eþ02ð1:43eþ02Þ4:64eþ02ð1:44eþ02Þ 4:92eþ02ð6:59eþ00Þ4:87eþ02ð1:27eþ01Þ F 29 2:63eþ02ð8:87eþ00Þ2:51eþ02ð7:32eþ00Þ+3:02eþ02ð5:65eþ01Þ2:84eþ02ð3:54eþ01Þ 3:42eþ02ð2:20eþ01Þ3:32eþ02ð5:85eþ01Þ F 30 3:57eþ04ð4:78eþ04Þ3:83eþ03ð4:29eþ03Þ+5:06eþ05ð8:52eþ05Þ2:81eþ05ð5:13eþ05Þ 7:15eþ02ð3:50eþ02Þ1:95eþ03ð2:50eþ03Þ Chaos wins + 19 + 8 + 14 Original wins 627 Similar 520 9 I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 707 Table 6 Comparison of algorithms (GWO, WOA, and ACOR original vs chaos version) on the CEC 2017 benchmark suite (10 dimensions, 51 runs, full precision, WRT at 5%). CEC 2017 GWO WOA ACOR Function Original Chaos Original Chaos Original Chaos Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) F 1 3:08eþ08ð3:98eþ08Þ9:42eþ07ð1:68eþ08Þ+3:93eþ05ð8:14eþ05Þ1:88eþ05ð2:73eþ05Þ 1:90eþ03ð2:31eþ03Þ7:85eþ02ð1:18eþ03Þ+ F 2 1:52eþ09ð3:01eþ09Þ3:75eþ07ð2:31eþ08Þ+2:10eþ04ð3:19eþ04Þ1:37eþ04ð3:25eþ04Þ 7:96eþ05ð5:03eþ06Þ0:00eþ00ð0:00eþ00Þ+ F 3 1:78eþ04ð9:02eþ03Þ1:14eþ03ð1:87eþ03Þ+5:14eþ02ð7:52eþ02Þ2:67eþ02ð3:37eþ02Þ 8:02eþ02ð1:34eþ03Þ0:00eþ00ð0:00eþ00Þ+ F 4 3:67eþ01ð2:39eþ01Þ1:47eþ01ð1:79eþ01Þ+1:78eþ01ð2:80eþ01Þ2:08eþ01ð3:35eþ01Þ 2:16eþ00ð1:38e01Þ2:88eþ00ð3:95e01Þ F 5 3:63eþ01ð1:45eþ01Þ1:65eþ01ð8:30eþ00Þ+5:34eþ01ð1:78eþ01Þ4:52eþ01ð1:67eþ01Þ+2:86eþ01ð4:64eþ00Þ2:01eþ01ð6:40eþ00Þ+ F 6 7:76eþ00ð7:55eþ00Þ1:14eþ00ð1:25eþ00Þ+3:05eþ01ð1:30eþ01Þ2:93eþ01ð1:28eþ01Þ 0:00eþ00ð0:00eþ00Þ2:22e07ð1:59e06Þ F 7 5:68eþ01ð1:62eþ01Þ3:31eþ01ð1:02eþ01Þ+7:78eþ01ð2:21eþ01Þ8:02eþ01ð2:47eþ01Þ 3:81eþ01ð4:78eþ00Þ3:10eþ01ð3:31eþ00Þ+ F 8 3:14eþ01ð1:56eþ01Þ1:40eþ01ð6:49eþ00Þ+3:98eþ01ð1:24eþ01Þ3:93eþ01ð1:60eþ01Þ 2:88eþ01ð4:62eþ00Þ2:02eþ01ð4:74eþ00Þ+ F 9 1:18eþ02ð2:36eþ02Þ1:62eþ01ð3:90eþ01Þ+4:13eþ02ð2:81eþ02Þ4:71eþ02ð3:46eþ02Þ 0:00eþ00ð0:00eþ00Þ8:91e03ð6:36e02Þ F 10 1:46eþ03ð5:83eþ02Þ4:97eþ02ð2:82eþ02Þ+1:02eþ03ð3:31eþ02Þ9:35eþ02ð3:55eþ02Þ 1:70eþ03ð1:18eþ02Þ1:29eþ03ð1:70eþ02Þ+ F 11 3:90eþ02ð1:05eþ03Þ3:08eþ01ð2:11eþ01Þ+1:09eþ02ð9:81eþ01Þ9:69eþ01ð7:19eþ01Þ 6:55eþ00ð1:51eþ00Þ7:61e01ð7:11e01Þ+ F 12 3:05eþ06ð3:51eþ06Þ4:21eþ05ð6:65eþ05Þ+3:83eþ06ð5:03eþ06Þ3:88eþ06ð5:00eþ06Þ 2:79eþ04ð8:38eþ04Þ7:38eþ03ð3:64eþ03Þ+ F 13 2:01eþ04ð1:42eþ04Þ1:14eþ04ð8:40eþ03Þ+1:43eþ04ð1:26eþ04Þ1:44eþ04ð1:17eþ04Þ 8:92eþ03ð7:91eþ03Þ1:10eþ04ð5:52eþ03Þ F 14 6:26eþ03ð6:96eþ03Þ5:27eþ02ð1:11eþ03Þ+4:55eþ02ð9:68eþ02Þ2:25eþ02ð4:04eþ02Þ+8:92eþ02ð8:66eþ02Þ3:26eþ03ð2:67eþ03Þ F 15 1:91eþ04ð1:74eþ04Þ1:30eþ03ð1:43eþ03Þ+2:39eþ03ð2:08eþ03Þ2:59eþ03ð2:86eþ03Þ 6:19eþ03ð3:33eþ03Þ1:54eþ03ð1:78eþ03Þ+ F 16 2:47eþ02ð1:39eþ02Þ1:41eþ02ð1:29eþ02Þ+2:65eþ02ð1:36eþ02Þ2:18eþ02ð1:28eþ02Þ 3:93eþ01ð2:33eþ01Þ3:57eþ00ð8:41eþ00Þ+ F 17 1:34eþ02ð6:42eþ01Þ4:84eþ01ð1:58eþ01Þ+8:46eþ01ð3:69eþ01Þ9:54eþ01ð4:75eþ01Þ 4:24eþ01ð1:07eþ01Þ3:89eþ01ð1:14eþ01Þ+ F 18 7:03eþ04ð4:89eþ04Þ2:97eþ04ð1:42eþ04Þ+1:69eþ04ð1:21eþ04Þ1:27eþ04ð1:23eþ04Þ+1:03eþ05ð7:94eþ04Þ6:94eþ03ð6:31eþ03Þ+ F 19 6:53eþ04ð1:33eþ05Þ3:01eþ03ð4:75eþ03Þ+1:75eþ04ð3:92eþ04Þ1:60eþ04ð2:31eþ04Þ 2:60eþ03ð2:64eþ03Þ4:17eþ03ð3:27eþ03Þ F 20 1:87eþ02ð9:43eþ01Þ6:59eþ01ð4:30eþ01Þ+1:55eþ02ð7:46eþ01Þ1:43eþ02ð6:95eþ01Þ 2:04eþ01ð2:11e01Þ2:29eþ01ð5:39eþ00Þ F 21 2:35eþ02ð2:72eþ01Þ2:08eþ02ð3:17eþ01Þ+2:24eþ02ð5:63eþ01Þ2:06eþ02ð6:51eþ01Þ 2:29eþ02ð8:25eþ00Þ2:23eþ02ð5:58eþ00Þ+ F 22 1:53eþ02ð2:13eþ02Þ1:17eþ02ð2:54eþ01Þ+1:31eþ02ð1:24eþ02Þ1:33eþ02ð1:34eþ02Þ 2:06eþ02ð3:70eþ02Þ1:00eþ02ð2:52e01Þ+ F 23 3:49eþ02ð1:81eþ01Þ3:20eþ02ð9:43eþ00Þ+3:49eþ02ð2:00eþ01Þ3:38eþ02ð1:91eþ01Þ+3:33eþ02ð4:79eþ00Þ3:17eþ02ð8:42eþ00Þ+ F 24 3:71eþ02ð3:72eþ01Þ3:45eþ02ð1:27eþ01Þ+3:54eþ02ð8:24eþ01Þ3:64eþ02ð4:05eþ01Þ 3:60eþ02ð4:26eþ00Þ3:50eþ02ð6:18eþ00Þ+ F 25 4:51eþ02ð1:81eþ01Þ4:29eþ02ð1:46eþ01Þ+4:36eþ02ð5:26eþ01Þ4:41eþ02ð2:96eþ01Þ 4:42eþ02ð1:31eþ01Þ4:46eþ02ð7:01eþ00Þ F 26 7:88eþ02ð4:17eþ02Þ4:81eþ02ð3:43eþ02Þ+8:87eþ02ð5:71eþ02Þ6:88eþ02ð5:28eþ02Þ+6:92eþ02ð6:70eþ01Þ5:03eþ02ð1:46eþ02Þ+ F 27 4:20eþ02ð2:81eþ01Þ3:94eþ02ð1:06eþ01Þ+4:34eþ02ð3:74eþ01Þ4:10eþ02ð2:32eþ01Þ+4:81eþ02ð2:15eþ01Þ4:09eþ02ð4:83eþ01Þ+ F 28 6:40eþ02ð1:23eþ02Þ6:00eþ02ð4:58eþ01Þ+6:36eþ02ð1:68eþ02Þ5:93eþ02ð6:37eþ01Þ+4:96eþ02ð8:08eþ00Þ4:73eþ02ð3:75eþ00Þ+ F 29 3:75eþ02ð6:51eþ01Þ2:88eþ02ð3:91eþ01Þ+4:27eþ02ð7:80eþ01Þ3:97eþ02ð8:64eþ01Þ 3:45eþ02ð4:14eþ01Þ2:73eþ02ð9:72eþ00Þ+ F 30 4:07eþ06ð5:31eþ06Þ7:01eþ05ð7:75eþ05Þ+5:34eþ05ð9:98eþ05Þ2:54eþ05ð4:20eþ05Þ 1:96eþ03ð1:57eþ03Þ2:47eþ02ð2:29eþ01Þ+ Chaos wins + 30 + 7 + 22 Original wins 005 Similar 023 3 I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 708 Table 7 Comparison of algorithms (FA, PSO, and CA original vs chaos version) on the CEC 2017 benchmark suite (10 dimensions, 51 runs, full precision, WRT at 5%). CEC 2017 FA PSO CA Function Original Chaos Original Chaos Original Chaos Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) F 1 3:85eþ03ð4:27eþ03Þ4:19eþ03ð3:63eþ03Þ 1:79eþ03ð1:66eþ03Þ2:41eþ03ð2:59eþ03Þ 4:36eþ03ð6:32eþ03Þ1:45eþ07ð2:48eþ07Þ F 2 3:92e02ð1:96e01Þ0:00eþ00ð0:00eþ00Þ 0:00eþ00ð0:00eþ00Þ0:00eþ00ð0:00eþ00Þ 9:71eþ83ð8:71eþ68Þ4:13eþ63ð2:95eþ64Þ+ F 3 1:24e05ð3:27e06Þ9:16e04ð2:57e04Þ 0:00eþ00ð0:00eþ00Þ0:00eþ00ð0:00eþ00Þ 4:85eþ04ð2:38eþ04Þ3:29eþ04ð1:03eþ04Þ+ F 4 2:78eþ00ð1:12eþ00Þ1:92eþ00ð3:35e01Þ+2:64eþ00ð9:39eþ00Þ2:84eþ00ð7:52e01Þ 5:93eþ00ð2:06e01Þ8:53eþ00ð8:19e01Þ F 5 1:74eþ01ð8:18eþ00Þ8:99eþ00ð3:90eþ00Þ+1:69eþ01ð7:95eþ00Þ1:49eþ01ð7:50eþ00Þ 4:23eþ01ð5:02eþ00Þ1:27eþ01ð6:22eþ00Þ+ F 6 1:58e03ð2:38e04Þ1:40e02ð1:89e03Þ 3:83e01ð8:88e01Þ1:83e01ð4:12e01Þ+6:86eþ01ð8:81eþ00Þ1:74e06ð7:56e06Þ+ F 7 1:73eþ01ð3:01eþ00Þ1:71eþ01ð2:96eþ00Þ 1:91eþ01ð4:29eþ00Þ1:72eþ01ð4:05eþ00Þ+5:21eþ01ð5:23eþ00Þ2:41eþ01ð6:61eþ00Þ+ F 8 1:32eþ01ð5:13eþ00Þ7:24eþ00ð3:01eþ00Þ+1:29eþ01ð5:18eþ00Þ1:03eþ01ð3:77eþ00Þ+4:32eþ01ð6:31eþ00Þ1:16eþ01ð6:01eþ00Þ+ F 9 2:67e06ð7:27e07Þ2:07e04ð5:38e05Þ 8:91e03ð6:36e02Þ0:00eþ00ð0:00eþ00Þ 9:31eþ00ð1:93eþ01Þ1:92e02ð4:83e02Þ+ F 10 6:07eþ02ð3:30eþ02Þ2:36eþ02ð1:67eþ02Þ+6:55eþ02ð2:53eþ02Þ5:48eþ02ð2:30eþ02Þ+2:09eþ03ð1:93eþ02Þ6:73eþ02ð3:63eþ02Þ+ F 11 5:95eþ00ð3:28eþ00Þ4:61eþ00ð2:77eþ00Þ+1:76eþ01ð1:09eþ01Þ1:09eþ01ð6:78eþ00Þ+3:33eþ03ð2:33eþ03Þ7:96eþ02ð1:09eþ03Þ+ F 12 1:34eþ04ð1:18eþ04Þ1:65eþ04ð1:62eþ04Þ 9:25eþ03ð7:31eþ03Þ1:40eþ04ð1:01eþ04Þ 3:23eþ08ð1:67eþ08Þ5:72eþ05ð9:68eþ05Þ+ F 13 4:20eþ03ð5:32eþ03Þ4:06eþ03ð5:03eþ03Þ 6:56eþ03ð5:15eþ03Þ5:55eþ03ð4:62eþ03Þ 1:28eþ06ð1:38eþ06Þ1:59eþ04ð5:38eþ03Þ+ F 14 4:95eþ01ð8:45eþ01Þ2:52eþ01ð1:15eþ01Þ+5:48eþ01ð2:17eþ01Þ5:55eþ01ð2:27eþ01Þ 8:22eþ03ð8:10eþ03Þ2:43eþ03ð1:50eþ03Þ+ F 15 8:59eþ01ð3:31eþ02Þ2:76eþ01ð2:37eþ01Þ 6:18eþ01ð4:84eþ01Þ4:06eþ01ð2:43eþ01Þ+5:27eþ04ð4:49eþ04Þ3:75eþ03ð3:19eþ03Þ+ F 16 7:22eþ01ð7:74eþ01Þ1:37eþ01ð1:81eþ01Þ+2:09eþ02ð1:30eþ02Þ1:69eþ02ð1:40eþ02Þ+2:14eþ02ð5:76eþ01Þ1:36eþ01ð1:72eþ01Þ+ F 17 2:71eþ01ð1:88eþ01Þ2:61eþ01ð1:15eþ01Þ 5:27eþ01ð3:07eþ01Þ4:73eþ01ð3:25eþ01Þ 9:29eþ01ð2:01eþ01Þ4:38eþ01ð1:47eþ01Þ+ F 18 7:66eþ03ð7:19eþ03Þ1:07eþ04ð9:04eþ03Þ 4:18eþ03ð6:40eþ03Þ4:01eþ03ð6:06eþ03Þ 8:54eþ06ð1:06eþ07Þ1:33eþ04ð1:40eþ04Þ+ F 19 2:45eþ02ð7:24eþ02Þ1:65eþ01ð1:75eþ01Þ+2:13eþ02ð5:35eþ02Þ1:01eþ02ð1:90eþ02Þ 5:76eþ04ð9:30eþ04Þ4:91eþ03ð3:56eþ03Þ+ F 20 1:62eþ01ð1:57eþ01Þ1:11eþ01ð2:14eþ01Þ+6:79eþ01ð5:54eþ01Þ6:28eþ01ð5:62eþ01Þ 1:88eþ02ð3:46eþ01Þ5:89eþ01ð1:27eþ01Þ+ F 21 1:72eþ02ð5:55eþ01Þ1:65eþ02ð5:85eþ01Þ 1:77eþ02ð5:51eþ01Þ1:70eþ02ð5:65eþ01Þ 2:38eþ02ð1:99eþ01Þ2:09eþ02ð1:79eþ01Þ+ F 22 9:79eþ01ð1:59eþ01Þ9:00eþ01ð3:08eþ01Þ 1:02eþ02ð8:75e01Þ1:12eþ02ð9:49eþ01Þ 1:41eþ03ð4:13eþ02Þ1:14eþ02ð1:02eþ02Þ+ F 23 3:18eþ02ð6:82eþ00Þ3:10eþ02ð3:26eþ00Þ+3:20eþ02ð8:10eþ00Þ3:17eþ02ð9:27eþ00Þ+3:64eþ02ð9:14eþ00Þ3:16eþ02ð5:58eþ00Þ+ F 24 2:98eþ02ð9:91eþ01Þ3:40eþ02ð3:78eþ00Þ 2:94eþ02ð9:88eþ01Þ2:84eþ02ð1:03eþ02Þ 3:79eþ02ð4:82eþ00Þ3:49eþ02ð7:08eþ00Þ+ F 25 4:22eþ02ð2:32eþ01Þ4:27eþ02ð2:33eþ01Þ 4:21eþ02ð2:33eþ01Þ4:20eþ02ð2:36eþ01Þ 4:58eþ02ð4:97eþ00Þ4:46eþ02ð6:92eþ00Þ+ F 26 3:30eþ02ð6:13eþ01Þ3:00eþ02ð3:83e03Þ 2:87eþ02ð3:95eþ01Þ3:26eþ02ð2:07eþ02Þ 9:11eþ02ð6:73eþ01Þ5:28eþ02ð1:04eþ02Þ+ F 27 3:96eþ02ð3:87eþ00Þ3:90eþ02ð7:33e01Þ+4:07eþ02ð2:78eþ01Þ4:01eþ02ð1:88eþ01Þ 4:74eþ02ð2:32eþ01Þ4:45eþ02ð2:14eþ01Þ+ F 28 4:14eþ02ð1:38eþ02Þ4:26eþ02ð1:52eþ02Þ 4:53eþ02ð1:47eþ02Þ4:71eþ02ð1:48eþ02Þ 4:98eþ02ð2:65eþ00Þ4:98eþ02ð1:66eþ00Þ F 29 2:77eþ02ð3:22eþ01Þ2:42eþ02ð8:48eþ00Þ+3:05eþ02ð3:84eþ01Þ2:87eþ02ð3:84eþ01Þ+6:81eþ02ð1:28eþ02Þ4:65eþ02ð1:51eþ02Þ+ F 30 1:88eþ05ð3:60eþ05Þ2:67eþ05ð3:99eþ05Þ 8:43eþ04ð2:44eþ05Þ1:19eþ05ð3:19eþ05Þ 7:97eþ05ð8:59eþ05Þ2:95eþ03ð2:68eþ03Þ+ Chaos wins + 12 + 9 + 27 Original wins 522 Similar 13 19 1 I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 709 Table 8 Comparison of algorithms (SOMA ATO, HFPSO, and ABC original vs chaos version) on the CEC 2020 benchmark suite (10 dimensions, 51 runs, full precision, WRT at 5%). CEC 2020 SOMA ATO HFPSO ABC Function Original Chaos Original Chaos Original Chaos Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) F 1 1:31eþ03ð1:83eþ03Þ2:76eþ01ð4:58eþ01Þ+1:01eþ03ð1:12eþ03Þ2:49eþ03ð3:51eþ03Þ 9:33eþ02ð1:33eþ03Þ1:06eþ04ð1:71eþ04Þ F 2 1:77eþ02ð8:09eþ01Þ6:93eþ01ð6:87eþ01Þ+3:06eþ02ð2:68eþ02Þ2:18eþ02ð1:24eþ02Þ 1:37eþ03ð1:59eþ02Þ7:29eþ02ð2:96eþ02Þ+ F 3 1:97eþ01ð2:53eþ00Þ1:67eþ01ð2:94eþ00Þ+1:46eþ01ð6:65e01Þ1:99eþ01ð4:60eþ00Þ 3:71eþ01ð3:68eþ00Þ2:76eþ01ð7:80eþ00Þ+ F 4 1:50eþ00ð3:16e01Þ9:70e01ð3:27e01Þ+6:79e01ð1:35e01Þ1:56eþ00ð3:67eþ00Þ 1:83eþ00ð3:58e01Þ1:52eþ00ð6:22e01Þ+ F 5 1:90eþ04ð2:40eþ04Þ5:87eþ02ð1:01eþ03Þ+3:39eþ03ð2:44eþ03Þ2:46eþ03ð2:64eþ03Þ 5:13eþ04ð2:65eþ04Þ1:20eþ04ð1:63eþ04Þ+ F 6 4:14eþ00ð1:68eþ01Þ1:72eþ00ð2:98eþ00Þ 1:24eþ02ð1:60eþ02Þ1:65eþ02ð8:01eþ01Þ 7:50eþ01ð2:90eþ01Þ1:12eþ02ð8:06eþ01Þ F 7 7:59eþ02ð1:36eþ03Þ5:48eþ00ð1:10eþ01Þ+2:22eþ02ð2:52eþ02Þ1:39eþ02ð1:48eþ02Þ 1:45eþ04ð7:35eþ03Þ6:00eþ02ð6:49eþ02Þ+ F 8 7:69eþ01ð3:42eþ01Þ7:70eþ01ð3:73eþ01Þ 1:01eþ02ð1:53e01Þ1:23eþ02ð1:27eþ02Þ 1:03eþ02ð1:19eþ00Þ1:74eþ02ð1:80eþ02Þ F 9 1:24eþ02ð3:74eþ01Þ1:31eþ02ð7:19eþ01Þ 3:54eþ02ð9:20eþ00Þ3:29eþ02ð6:83eþ01Þ+3:45eþ02ð2:85eþ01Þ3:58eþ02ð1:48eþ01Þ F 10 4:01eþ02ð1:10eþ01Þ4:14eþ02ð2:19eþ01Þ 4:20eþ02ð2:33eþ01Þ4:26eþ02ð2:36eþ01Þ 4:30eþ02ð1:37eþ01Þ4:21eþ02ð3:94eþ01Þ Chaos wins + 6 + 1 + 5 Original wins 152 Similar 343 I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 710 Table 9 Comparison of algorithms (GWO, WOA, and ACOR original vs chaos version) on the CEC 2020 benchmark suite (10 dimensions, 51 runs, full precision, WRT at 5%). CEC 2020 GWO WOA ACOR Function Original Chaos Original Chaos Original Chaos Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) F 1 2:12eþ07ð9:72eþ07Þ2:70eþ07ð9:83eþ07Þ 3:34eþ05ð7:26eþ05Þ2:14eþ05ð3:40eþ05Þ 8:27eþ02ð9:97eþ02Þ1:16eþ03ð1:91eþ03Þ F 2 3:82eþ02ð2:63eþ02Þ4:48eþ02ð2:68eþ02Þ 9:65eþ02ð2:77eþ02Þ9:69eþ02ð2:56eþ02Þ 1:60eþ03ð1:86eþ02Þ1:26eþ03ð2:01eþ02Þ+ F 3 2:83eþ01ð9:54eþ00Þ2:91eþ01ð8:09eþ00Þ 8:24eþ01ð2:49eþ01Þ7:28eþ01ð2:21eþ01Þ 4:09eþ01ð2:05eþ00Þ2:97eþ01ð3:78eþ00Þ+ F 4 3:29eþ00ð1:07eþ01Þ1:76eþ00ð9:01e01Þ 6:23eþ00ð4:34eþ00Þ5:17eþ00ð2:95eþ00Þ 2:62eþ00ð2:10e01Þ1:74eþ00ð3:52e01Þ+ F 5 4:50eþ04ð1:12eþ05Þ5:59eþ03ð4:72eþ03Þ 1:39eþ05ð2:82eþ05Þ8:35eþ04ð1:33eþ05Þ 2:27eþ04ð1:48eþ04Þ6:60eþ04ð5:87eþ04Þ F 6 1:32eþ02ð9:58eþ01Þ1:38eþ02ð7:97eþ01Þ 2:00eþ02ð1:01eþ02Þ1:74eþ02ð8:80eþ01Þ 5:97eþ01ð3:59eþ01Þ7:33eþ00ð1:79eþ01Þ+ F 7 5:49eþ03ð4:61eþ03Þ4:30eþ03ð3:85eþ03Þ 1:80eþ04ð1:39eþ04Þ1:70eþ04ð1:43eþ04Þ 3:58eþ03ð1:19eþ03Þ1:15eþ04ð8:56eþ03Þ F 8 1:21eþ02ð5:99eþ01Þ1:26eþ02ð7:78eþ01Þ 1:74eþ02ð2:35eþ02Þ1:78eþ02ð2:65eþ02Þ 1:04eþ02ð6:33e01Þ1:00eþ02ð2:83e01Þ+ F 9 3:45eþ02ð1:03eþ01Þ3:47eþ02ð1:19eþ01Þ 3:72eþ02ð4:53eþ01Þ3:66eþ02ð4:19eþ01Þ 3:64eþ02ð5:84eþ00Þ3:51eþ02ð5:62eþ00Þ+ F 10 4:30eþ02ð1:83eþ01Þ4:35eþ02ð1:85eþ01Þ 4:26eþ02ð6:44eþ01Þ4:33eþ02ð6:40eþ01Þ 4:24eþ02ð2:42eþ01Þ4:44eþ02ð1:17eþ01Þ Chaos wins + 0 + 0 + 6 Original wins 203 Similar 810 1 I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 711 Table 10 Comparison of algorithms FA, PSO, and CA original vs chaos version) on the CEC 2020 benchmark suite (10 dimensions, 51 runs, full precision, WRT at 5%). CEC 2020 FA PSO CA Function Original Chaos Original Chaos Original Chaos Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) F 1 3:58eþ03ð3:91eþ03Þ6:80eþ03ð3:93eþ03Þ 2:49eþ03ð3:16eþ03Þ2:35eþ03ð2:87eþ03Þ 3:53eþ03ð2:46eþ03Þ4:40eþ07ð1:47eþ08Þ F 2 5:44eþ02ð2:90eþ02Þ2:40eþ02ð1:45eþ02Þ+4:22eþ02ð2:61eþ02Þ3:94eþ02ð2:05eþ02Þ 2:12eþ03ð1:19eþ02Þ6:01eþ02ð2:67eþ02Þ+ F 3 1:79eþ01ð4:43eþ00Þ1:66eþ01ð3:70eþ00Þ+1:97eþ01ð4:86eþ00Þ1:79eþ01ð3:53eþ00Þ+5:46eþ01ð1:96eþ00Þ2:35eþ01ð5:83eþ00Þ+ F 4 8:36e01ð3:31e01Þ8:38e01ð2:58e01Þ 9:47e01ð3:22e01Þ9:37e01ð3:28e01Þ 3:15eþ00ð5:54e01Þ9:37e01ð5:22e01Þ+ F 5 2:40eþ03ð2:76eþ03Þ4:31eþ03ð4:26eþ03Þ 2:30eþ03ð1:98eþ03Þ2:26eþ03ð2:14eþ03Þ 2:91eþ06ð9:08eþ05Þ4:63eþ05ð3:95eþ05Þ+ F 6 1:09eþ02ð1:01eþ02Þ4:98eþ01ð6:21eþ01Þ+1:48eþ02ð9:08eþ01Þ1:57eþ02ð8:34eþ01Þ 7:32eþ02ð9:80eþ01Þ1:96eþ01ð9:12eþ00Þ+ F 7 1:71eþ02ð1:12eþ02Þ1:60eþ02ð1:39eþ02Þ 2:25eþ02ð1:62eþ02Þ1:50eþ02ð1:20eþ02Þ+3:46eþ05ð2:32eþ05Þ8:16eþ04ð7:08eþ04Þ+ F 8 9:77eþ01ð1:62eþ01Þ9:35eþ01ð2:67eþ01Þ 9:86eþ01ð1:55eþ01Þ9:30eþ01ð2:79eþ01Þ+1:35eþ03ð4:62eþ02Þ1:08eþ02ð5:44eþ01Þ+ F 9 3:23eþ02ð7:45eþ01Þ3:08eþ02ð8:41eþ01Þ+3:06eþ02ð9:02eþ01Þ2:88eþ02ð1:02eþ02Þ 3:81eþ02ð2:63eþ00Þ3:50eþ02ð4:30eþ00Þ+ F 10 4:29eþ02ð2:29eþ01Þ4:25eþ02ð2:32eþ01Þ 4:14eþ02ð2:20eþ01Þ4:29eþ02ð2:22eþ01Þ 4:59eþ02ð4:25eþ00Þ4:47eþ02ð6:81eþ00Þ+ Chaos wins + 4 + 3 + 9 Original wins 111 Similar 560 I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 712 Table 11 Comparison of algorithms (SOMA ATO, HFPSO, and ABC original vs chaos version) on the CEC 2015 benchmark suite (10 dimensions, 51 runs, full precision, WRT at 5%). CEC 2015 SOMA ATO HFPSO ABC Function Original Chaos Original Chaos Original Chaos Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) F 1 1:90eþ05ð1:65eþ05Þ5:97eþ04ð5:64eþ04Þ+1:13eþ04ð8:36eþ03Þ4:83eþ04ð3:95eþ04Þ 7:03eþ06ð2:25eþ06Þ3:26eþ06ð5:16eþ06Þ+ F 2 6:85eþ03ð7:10eþ03Þ4:52eþ02ð8:12eþ02Þ+6:25eþ03ð7:88eþ03Þ6:40eþ03ð7:74eþ03Þ 4:89eþ03ð4:05eþ03Þ1:85eþ04ð2:28eþ04Þ F 3 2:01eþ01ð3:55e02Þ2:01eþ01ð2:75e02Þ 2:00eþ01ð2:05e03Þ2:00eþ01ð1:04e02Þ 2:03eþ01ð6:62e02Þ2:03eþ01ð8:17e02Þ F 4 6:65eþ00ð1:85eþ00Þ5:32eþ00ð1:82eþ00Þ+1:45eþ01ð5:35eþ00Þ8:63eþ00ð3:67eþ00Þ+2:77eþ01ð3:79eþ00Þ1:77eþ01ð7:87eþ00Þ+ F 5 2:34eþ02ð1:08eþ02Þ2:01eþ02ð1:19eþ02Þ 3:68eþ02ð1:93eþ02Þ2:82eþ02ð1:47eþ02Þ 1:41eþ03ð1:46eþ02Þ8:90eþ02ð3:75eþ02Þ+ F 6 1:68eþ03ð1:73eþ03Þ1:95eþ02ð2:21eþ02Þ+2:83eþ03ð2:58eþ03Þ2:39eþ03ð2:81eþ03Þ 2:67eþ04ð1:48eþ04Þ6:33eþ03ð9:32eþ03Þ+ F 7 4:43e01ð2:65e01Þ4:18e01ð3:68e01Þ+1:65eþ00ð1:00eþ00Þ1:46eþ00ð9:74e01Þ 2:52eþ00ð1:88e01Þ2:61eþ00ð6:50e01Þ F 8 2:17eþ03ð3:50eþ03Þ6:93eþ01ð7:73eþ01Þ+1:40eþ03ð1:46eþ03Þ8:89eþ02ð1:02eþ03Þ 9:31eþ03ð5:80eþ03Þ7:12eþ03ð7:79eþ03Þ+ F 9 1:00eþ02ð3:75e02Þ1:00eþ02ð3:94e02Þ+1:03eþ02ð1:67eþ01Þ1:00eþ02ð3:65e01Þ 1:00eþ02ð4:34e02Þ1:00eþ02ð6:12e02Þ F 10 1:26eþ03ð1:40eþ03Þ3:68eþ02ð3:31eþ02Þ+9:87eþ02ð8:14eþ02Þ7:50eþ02ð6:28eþ02Þ+1:51eþ04ð6:93eþ03Þ6:42eþ03ð8:93eþ03Þ+ F 11 6:47eþ01ð1:15eþ02Þ1:32eþ02ð1:48eþ02Þ 2:87eþ02ð7:52eþ01Þ2:78eþ02ð1:12eþ02Þ 3:31eþ02ð2:91eþ01Þ3:93eþ02ð9:57eþ01Þ F 12 1:03eþ02ð4:90e01Þ1:02eþ02ð4:85e01Þ+1:02eþ02ð8:11e01Þ1:02eþ02ð8:46e01Þ 1:04eþ02ð6:61e01Þ1:03eþ02ð1:08eþ00Þ+ F 13 2:95eþ01ð1:56eþ00Þ2:72eþ01ð2:16eþ00Þ+3:98eþ01ð4:43eþ00Þ3:77eþ01ð5:66eþ00Þ 3:90eþ01ð1:38eþ00Þ3:58eþ01ð3:64eþ00Þ+ F 14 2:40eþ03ð1:15eþ03Þ2:63eþ03ð9:34eþ02Þ 4:29eþ03ð3:73eþ03Þ4:42eþ03ð3:51eþ03Þ 3:71eþ02ð1:65eþ01Þ3:54eþ02ð5:64eþ01Þ+ F 15 1:00eþ02ð9:37e06Þ1:00eþ02ð0:00eþ00Þ+1:00eþ02ð0:00eþ00Þ1:00eþ02ð0:00eþ00Þ 1:00eþ02ð0:00eþ00Þ1:00eþ02ð0:00eþ00Þ Chaos wins + 11 + 2 + 9 Original wins 122 Similar 311 4 I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 713 Table 12 Comparison of algorithms (GWO, WOA, and ACOR original vs chaos version) on the CEC 2015 benchmark suite (10 dimensions, 51 runs, full precision, WRT at 5%). CEC 2015 GWO WOA ACOR Function Original Chaos Original Chaos Original Chaos Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) F 1 3:41eþ06ð3:15eþ06Þ2:58eþ06ð2:78eþ06Þ 4:43eþ06ð2:47eþ06Þ4:05eþ06ð2:78eþ06Þ 5:73eþ06ð3:29eþ06Þ2:96eþ05ð9:64eþ04Þ+ F 2 4:61eþ06ð5:40eþ06Þ6:01eþ06ð5:87eþ06Þ 4:86eþ05ð8:11eþ05Þ9:45eþ04ð7:30eþ04Þ+5:21eþ03ð5:60eþ03Þ6:51eþ03ð4:47eþ03Þ F 3 2:04eþ01ð7:64e02Þ2:04eþ01ð7:07e02Þ 2:01eþ01ð1:07e01Þ2:01eþ01ð9:44e02Þ 2:04eþ01ð6:36e02Þ2:04eþ01ð8:54e02Þ F 4 1:47eþ01ð6:48eþ00Þ1:40eþ01ð5:09eþ00Þ 3:98eþ01ð1:65eþ01Þ3:89eþ01ð1:43eþ01Þ 2:85eþ01ð3:64eþ00Þ1:84eþ01ð7:94eþ00Þ+ F 5 4:70eþ02ð2:42eþ02Þ4:72eþ02ð2:91eþ02Þ 9:47eþ02ð2:89eþ02Þ9:04eþ02ð3:05eþ02Þ 1:64eþ03ð1:68eþ02Þ1:30eþ03ð1:57eþ02Þ+ F 6 1:70eþ04ð2:02eþ04Þ1:36eþ04ð1:34eþ04Þ+2:66eþ05ð3:27eþ05Þ2:82eþ05ð3:38eþ05Þ 9:30eþ03ð1:18eþ04Þ5:84eþ02ð8:71eþ02Þ+ F 7 2:43eþ00ð2:18eþ00Þ2:44eþ00ð1:09eþ00Þ 5:61eþ00ð1:38eþ00Þ5:12eþ00ð1:14eþ00Þ 3:19eþ00ð2:81e01Þ2:57eþ00ð3:10e01Þ+ F 8 3:47eþ04ð2:33eþ05Þ2:02eþ03ð1:19eþ03Þ 5:63eþ03ð4:74eþ03Þ6:42eþ03ð5:31eþ03Þ 5:49eþ03ð3:93eþ03Þ6:27eþ02ð8:36eþ02Þ+ F 9 1:00eþ02ð1:50e01Þ1:00eþ02ð1:29e01Þ 1:01eþ02ð2:35e01Þ1:01eþ02ð2:62e01Þ 1:00eþ02ð4:40e02Þ1:00eþ02ð4:43e02Þ+ F 10 3:31eþ03ð4:49eþ03Þ3:03eþ03ð3:30eþ03Þ 8:15eþ03ð1:07eþ04Þ9:21eþ03ð8:12eþ03Þ 2:21eþ04ð1:21eþ04Þ1:45eþ03ð1:31eþ03Þ+ F 11 3:00eþ02ð8:68eþ01Þ2:83eþ02ð1:34eþ02Þ 3:25eþ02ð1:05eþ02Þ3:03eþ02ð3:91eþ01Þ 5:69eþ02ð8:49eþ01Þ3:89eþ02ð1:19eþ02Þ+ F 12 1:02eþ02ð7:92e01Þ1:02eþ02ð8:04e01Þ 1:08eþ02ð2:60eþ00Þ1:08eþ02ð3:29eþ00Þ 1:04eþ02ð8:60eþ00Þ1:02eþ02ð3:17e01Þ+ F 13 3:15eþ01ð4:43eþ00Þ3:09eþ01ð3:93eþ00Þ 4:14eþ01ð3:84eþ00Þ4:00eþ01ð3:22eþ00Þ+4:15eþ01ð3:19eþ00Þ3:21eþ01ð3:51eþ00Þ+ F 14 5:91eþ03ð2:48eþ03Þ5:81eþ03ð2:70eþ03Þ 6:46eþ03ð1:28eþ03Þ6:34eþ03ð1:47eþ03Þ+3:50eþ02ð1:08eþ01Þ7:27eþ02ð1:47eþ03Þ F 15 1:07eþ02ð4:99eþ00Þ1:09eþ02ð5:45eþ00Þ 1:01eþ02ð5:91e01Þ1:00eþ02ð1:06e01Þ+1:00eþ02ð0:00eþ00Þ1:00eþ02ð0:00eþ00Þ Chaos wins + 1 + 4 + 11 Original wins 102 Similar 13 11 2 I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 714 Table 13 Comparison of algorithms (FA, PSO, and CA original vs chaos version) on the CEC 2015 benchmark suite (10 dimensions, 51 runs, full precision, WRT at 5%). CEC 2015 FA PSO CA Function Original Chaos Original Chaos Original Chaos Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) Mean (Std Dev) F 1 9:95eþ04ð1:16eþ05Þ1:61eþ05ð1:97eþ05Þ 1:06eþ04ð9:46eþ03Þ7:25eþ04ð4:59eþ04Þ 1:35eþ08ð7:17eþ07Þ5:87eþ07ð2:32eþ07Þ+ F 2 6:09eþ03ð8:00eþ03Þ1:28eþ04ð1:39eþ04Þ 6:67eþ03ð7:16eþ03Þ6:75eþ03ð7:44eþ03Þ 1:01eþ04ð1:29eþ04Þ2:70eþ07ð9:68eþ07Þ F 3 1:91eþ01ð4:83eþ00Þ1:97eþ01ð2:82eþ00Þ 2:00eþ01ð2:64e02Þ2:02eþ01ð1:35e01Þ 2:03eþ01ð7:46e02Þ2:03eþ01ð7:27e02Þ F 4 1:31eþ01ð4:94eþ00Þ8:25eþ00ð4:14eþ00Þ+1:58eþ01ð7:76eþ00Þ1:18eþ01ð5:16eþ00Þ+4:18eþ01ð5:13eþ00Þ1:05eþ01ð5:89eþ00Þ+ F 5 5:87eþ02ð2:89eþ02Þ2:45eþ02ð1:85eþ02Þ+5:27eþ02ð2:02eþ02Þ3:89eþ02ð2:12eþ02Þ+2:10eþ03ð1:46eþ02Þ6:71eþ02ð3:39eþ02Þ+ F 6 2:59eþ03ð2:34eþ03Þ4:32eþ03ð3:16eþ03Þ 1:73eþ03ð2:06eþ03Þ1:20eþ03ð9:40eþ02Þ 1:74eþ06ð1:44eþ06Þ7:15eþ04ð9:78eþ04Þ+ F 7 1:01eþ00ð4:68e01Þ1:21eþ00ð4:91e01Þ 2:07eþ00ð9:91e01Þ1:19eþ00ð8:60e01Þ+3:23eþ00ð2:44e01Þ2:34eþ00ð2:70e01Þ+ F 8 1:64eþ03ð2:66eþ03Þ7:09eþ02ð8:50eþ02Þ+1:46eþ03ð1:73eþ03Þ1:15eþ03ð1:40eþ03Þ 3:30eþ05ð3:26eþ05Þ5:44eþ04ð8:06eþ04Þ+ F 9 1:00eþ02ð4:98e02Þ1:00eþ02ð4:56e02Þ 1:00eþ02ð5:64e02Þ1:00eþ02ð3:96e02Þ+1:00eþ02ð5:28e02Þ1:00eþ02ð4:30e02Þ F 10 1:21eþ03ð1:54eþ03Þ8:39eþ02ð1:15eþ03Þ 9:37eþ02ð7:30eþ02Þ7:52eþ02ð4:76eþ02Þ 4:51eþ05ð3:54eþ05Þ1:11eþ05ð1:42eþ05Þ+ F 11 2:71eþ02ð8:93eþ01Þ2:83eþ02ð7:08eþ01Þ 2:77eþ02ð8:05eþ01Þ2:71eþ02ð8:92eþ01Þ 4:94eþ02ð9:63eþ01Þ4:09eþ02ð5:23eþ01Þ+ F 12 1:02eþ02ð4:37e01Þ1:02eþ02ð3:67e01Þ 1:02eþ02ð7:13e01Þ1:02eþ02ð5:42e01Þ 1:40eþ02ð2:70eþ01Þ1:03eþ02ð5:54eþ00Þ+ F 13 3:26eþ01ð4:32eþ00Þ2:95eþ01ð2:40eþ00Þ+3:50eþ01ð3:82eþ00Þ3:29eþ01ð4:44eþ00Þ+4:48eþ01ð7:75e01Þ3:83eþ01ð1:80eþ00Þ+ F 14 5:12eþ03ð2:31eþ03Þ5:12eþ03ð2:10eþ03Þ 2:61eþ03ð2:69eþ03Þ2:99eþ03ð2:97eþ03Þ 2:87eþ03ð1:27eþ03Þ5:23eþ02ð6:05eþ01Þ+ F 15 1:00eþ02ð2:88e04Þ1:00eþ02ð3:01e03Þ 1:00eþ02ð0:00eþ00Þ1:00eþ02ð0:00eþ00Þ 1:00eþ02ð0:00eþ00Þ1:01eþ02ð2:77eþ00Þ Chaos wins + 4 + 5 + 11 Original wins 423 Similar 781 I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 715 This raises several interesting questions that deserve further research. Questions such as how complex and how periodic a given numerical series must be for the algorithm to work properly, whether there is a link between the chaotic mode and the nonlinearity of the test function, or how these algorithms process information about their previous activities such as Nsteps back and forth, etc. Another open question, which follows from our results, is how the ability of the algorithm to find the optimal solution depends on the correlation between individual increments of time series-generated sequences (pseudo-random or chaotic). It is clear that a significant difference between randomness and chaos, from our experiment point of view, are the underlying correlations between sections of the sequences. While sections of random sequences are by definition uncorrelated, in the case of chaos short-term correlations between an initial state and subsequent states are present, see, for example, a logistic Eq. 1with A¼4. How these different correlations between sections of sequences are related to algorithm performance is an open question, partly already discussed in [72–74] for example. Based on the repeatability and accuracy of the results obtained in these experiments, it is believed that this area still offers a wide range of interesting questions and issues for further research. We also expect that the evolutionary computing community should begin to understand these algorithms as dynamic systems, for which there already exists productive mathematical apparatus dealing with their stability, control, and many other aspects. We firmly believe that applying this pre-existing mathematical apparatus to evolutionary and swarm intelligence algorithms would help shed light on the many unanswered questions that exist in the algorithm community today. 9. Conclusion In this paper, a well-known logistic equation was used as a chaos generator, which produced deterministic chaos, as it has been verified and proven many times. The behavior of this equation was generated with varying degrees of calculation accuracy/precision. This accuracy affected whether the chaotic series was theoretically infinite and therefore unrepeatable began to repeat after Niterations. The resulting sequences of various lengths, i.e. infinitely long chaotic series but also a series of repeated partially chaotic series (so it was a kind of pseudo-chaotic periodic processes), were then used instead of pseudorandom numbers in all considered evolutionary and swarm intelligence algorithms. In other words, whenever pseudorandom numbers were needed, a chaotic series or periodic series (chaos-based) numbers were used. The results obtained from all the experiments described in section Results in Figs. 12,13, and Table 4–13 clearly show the interesting positive impact of the use of N periodic series on the operation of the algorithm. Results do not need detailed comment - as clearly visible from the attached figures and tables. These results then generate the questions mentioned in section Open Questions and Future Research. In our experiments, a set of algorithms were used, which, in our opinion, is sufficiently representative to demonstrate the basic idea of the paper. Of course, this does not mean that the simulations and results demonstrated in this paper cannot be extended to other algorithms such as [75–80], and many others. This leaves open the possibility for further research in this direction. Thus, our experiments have regularly shown that chaotic series, and even periodic sequences obtained by reducing the accuracy of these series, certainly increases the performance of evolutionary algorithms. This leads to the striking fact that the algorithm performance increases even if random-like processes are not used, which contradicts a generally accepted dogma. We believe that it is an interesting research area that combines computer science in terms of algorithm study, theoretical cybernetics and control theory. CRediT authorship contribution statement Ivan Zelinka: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Visualization, Writing - original draft, Writing - review & editing. Quoc Bao Diep: Conceptualization, Methodology, Software, ValidaFig. 14. The problem of integer number sequences. I. Zelinka, Quoc Bao Diep, Václav Snášel et al. Information Sciences 587 (2022) 692–719 716