Chaos-Based Optimization - A Review

VOLUME: 1 | ISSUE: 1 | 2017 | June

Chaos-Based Optimization - A Review

Roman SENKERIK 1,2,*, Ivan ZELINKA 1,3, Michal PLUHACEK 2

1Modeling Evolutionary Algorithms Simulation and Arti�cial Intelligence, Faculty of Electricaland Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

2Faculty of Applied Informatics, Tomas Bata University in Zlin, Nam T.G. Masaryka 5555, 76001 Zlin, Czech Republic

3Technical University of Ostrava, Faculty of Electrical Engineering and Computer Science, 17.listopadu 15, 708 33 Ostrava-Poruba, Czech Republic

*[email protected]; [email protected](Received: 11-February-2017; accepted: 3-May-2017; published: 8-June-2017)

Abstract. This paper discusses the utilization of

the complex chaotic dynamics given by the se-

lected time-continuous chaotic systems as well

as by the discrete chaotic maps, as the chaotic

pseudo-random number generators and driving

maps for the chaos based optimization. Such

an optimization concept is utilizing direct out-

put iterations of chaotic system transferred into

the required numerical range as the replacement

of traditional and default pseudo-random num-

ber generators, or this concept uses the chaotic

dynamics for mapping the search space mostly

within the smart hybrid local search techniques.

This paper shows totally three groups of complex

chaotic dynamics given by chaotic �ows, oscil-

lators and discrete maps. Simulations of exam-

ples of chaotic dynamics as unconventional gen-

erators or mapped to the search space were per-

formed, and related issues like parametric plots,

distributions of such a systems, periodicity, and

dependency on internally available parameters

are brie�y discussed in this paper.

Keywords

Deterministic chaos, Heuristic, Chaotic

Optimization, Chaotic Pseudo Random

Number Generators.

1. Introduction

Generally speaking, the term "chaos" can de-note anything that cannot be predicted deter-ministically. In the case that the word "chaos"is combined with an attribute such as "determin-istic," then a speci�c type of chaotic phenomenais involved, having their speci�c laws, mathe-matical apparatus and a physical origin. Thedeterministic chaos is a phenomenon that - asits name suggests - is not based on the presenceof a random or any stochastic e�ects. It is evi-dent from the structure of the equations (see thesection: Chaotic Optimization) that no mathe-matical term expressing randomness is present.The seeming randomness in deterministic chaosis related to the extreme sensitivity to the initialconditions [1].

In the past, the chaos has been observed inmany of various systems (including evolutionaryone). Systems exhibiting deterministic chaos in-clude, for instance, weather, biological systems,many electronic circuits (Chua's circuit), me-chanical systems, such as a double pendulum,magnetic pendulum, or so-called billiard prob-lem. The idea of using chaotic systems insteadof random processes (pseudo-number generators- PRNGs) has been presented in several research�elds and many applications with promising re-sults [2], [3].

68 c© 2017 Journal of Advanced Engineering and Computation (JAEC)

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Another research joining deterministic chaosand pseudorandom number generator has beendone for example in [4]. The possibility of gen-eration of random or pseudorandom numbersby use of the ultra-weak multidimensional cou-pling of p 1-dimensional dynamical systems isdiscussed there.

Another paper [5], deeply investigate logis-tic map as a possible pseudorandom numbergenerator and is compared with contemporarypseudo-random number generators. A compar-ison of logistic map results is made with con-ventional methods of generating pseudorandomnumbers. The approach used to determine thenumber, delay, and period of the orbits of thelogistic map at varying degrees of precision (3to 23 bits). Another paper [6] proposed an al-gorithm for generating pseudorandom numbergenerator, which is called (coupled map latticebased on discrete chaotic iteration) and combinethe coupled map lattice and chaotic iteration.Authors also tested this algorithm in NIST 800-22 statistical test suits and for future utilizationin image encryption. In (Narendra et al. 2010)authors exploit interesting properties of chaoticsystems to design a random bit generator, calledCCCBG, in which two chaotic systems are cross-coupled with each other. A new binary streamcipher algorithm based on dual one-dimensionalchaotic maps is proposed in [7] with statistic pro-prieties showing that the sequence is of high ran-domness. Similar studies are also done in [8].

2. Motivation

This paper represents the extension of prelimi-nary suggestions described in [9]. Recently thedeterministic chaos has frequently been usedeither as a replacement of (mostly uniformdistribution based) pseudo-number generators(PRGNs) in metaheuristic algorithms or for sim-ple mapping of solutions/iterations within orig-inal local search engines. The metaheuristicchaotic approach uses the chaotic system in theplace of a pseudo-random number generator [10].This method causes the heuristic to map searchregions based on unique sequencing and period-icity of transferred chaotic dynamics, thus simu-lating the dynamical alternations of several sub-

populations. The task is then to select an ap-propriate chaotic system (either discrete or time-continuous) as the chaotic pseudo random num-ber generator (CPRNG) [11].

Recently, the concept of embedding of chaoticdynamics into the evolutionary algorithms hasbeen studied intensively. The self-adaptivechaos di�erential evolution (SACDE) [12]wasfollowed by the implementation of chaos into thesimple not-adaptive di�erential evolution [13],[14]; the chaotic searching algorithm for the verysame metaheuristic was introduced in [15]. Also,the PSO (Particle Swarm Optimization) algo-rithm with elements of chaos was introduced asCPSO [16]. Many other works focusing on thehybridization of the swarm and chaotic move-ment have been published afterward [17], [18].Later on, the utilization of chaotic sequences be-came to be popular in many interdisciplinary ap-plications and techniques. The question of im-pact and importance of di�erent randomizationwithin heuristic search was intensively studiedin [19].

The primary aim of this work is to try,test, analyze and compare the implementa-tion of di�erent natural chaotic dynamic ei-ther as the mapping procedure for the optimiza-tion/searching process or as the CPRNG. Thispaper presents the discussion about the usabilityof such systems, periodicity, and dependency onaccessible internal parameters; thus the usabilityas CPRNG, or for local search and metaheuristicbased optimization techniques.

3. Chaotic Optimization

There exist three possible utilizations of chaoticdynamics in optimization tasks.

Firstly, as aforementioned in the previous sec-tion, the direct output simulation iterations ofthe chaotic system are transferred into the re-quired numerical range (as simple CPRNG).The idea of CPRNG is to replace the defaultsystem PRNG with the chaotic system. As thechaotic system is a set of equations with a staticstart position (see the next section), we createda random start position of the system, to havedi�erent start position for various experiments.

c© 2017 Journal of Advanced Engineering and Computation (JAEC) 69


Once the origin position of the chaotic systemhas been obtained, the system generates thenext sequence using its current position. Sub-sequently, simple techniques as to how to dealwith the negative numbers as well as with thescaling of the wide range of the numbers givenby the chaotic systems into the typical range 0-1.

Secondly, the complexity of chaotic systemsand its movement in the space is used for dy-namical mapping of the search space mostlywithin the local search techniques [20]. Finally,the hybridization of searching/optimization pro-cess and chaotic systems is represented by chaosbased random walk technique.

4. Chaotic Optimization

This section contains the description of threedi�erent groups of chaotic dynamics: time-continuous chaotic systems (�ows and oscilla-tors), and the discrete chaotic maps.

4.1. Time Continuous systems

as CPRNG

In this research, following chaotic systems wereused: Lorenz system (1) and Rossler system (2)as two examples of chaotic �ows; further unmod-i�ed UEDA oscillator (3); and Driven Van derPol Oscillator (4) as chaotic oscillators [21].

The Lorenz system (1) is a 3-dimensional dy-namical �ow, which exhibits chaotic behavior.It was introduced in 1963 by Edward Lorenz,who derived it from the simpli�ed equations ofconvection rolls arising in the equations of theatmosphere.

The Rossler system (2) exhibits chaotic dy-namics associated with the fractal properties ofthe attractor. It was originally introduced as anexample of very simple chaotic �ow containingchaos similarly to the Lorenz attractor. Thisattractor has some similarities to the Lorenz at-tractor but is simpler.

UEDA oscillator (3) is the simple example ofdriven pendulums, which represent some of the

most signi�cant examples of chaos and regular-ity.

UEDA system can be simply considered as aspecial case of intensively studied Du�ng oscil-lator that has both a linear and cubic restoringforce. Ueda oscillator represents the both biolog-ically and physically relevant dynamical modelexhibiting chaotic motion.

Finally, The Van der Pol oscillator (4) is thesimple example of the limit cycles and chaoticbehavior in electrical circuits employing vacuumtubes. Similarly to the UEDA oscillator, it canbe used to explore physical (unstable) behaviorin biological sciences. [22].

The equations, which describe the chaotic sys-tems, have parameter settings for Lorenz sys-tem: a = 3.0, b = 26.5; Rossler system: a = 0.2,b = 0.2, and c = 5.7; UEDA oscillator: a = 1.0,b = 0.05, c = 7.5 and ω = 1.0; and Van derPol oscillator : µ = 0.2, γ = 8.0, a = 0.35 andω = 1.02.

dx

dt= −a(x− y)

dy

dt= x(b− z)− y.

dz

dt= xy − z (1)

dx

dt= −x− y

dy

dt= x+ ay.

dz

dt= b+ z(x− c) (2)

dx

dt= y

dy

dt= −ax3 − by + csinωt. (3)

dx

dt= y

dy

dt= µ(1− γx2)y − x3 + asinωt. (4)



The parametric plots of the chaotic systemsare depicted in Fig. 1. The Fig. 2 show the ex-ample of dynamical sequencing during the gen-erating of pseudo random numbers transferredinto the range < 0−1 >using particular studiedCPRNGs and with the sampling rate of 0.5 s.The dependency of sequencing and periodicityon the sampling frequency is discussed in detailsin [23]. Finally, the comparison of CPRNGs dis-tribution is depicted in Fig. 3.

4.2. Discrete Chaotic Maps as

CPRNG

The examples of chaotic maps are following:Arnold Cat map (5), Dissipative Standard map(6), Ikeda map (7), and Lozi map (8). Mapequations and parameters values are given in Ta-ble 1, as suggested in [21]. Parametric plots aredepicted in Fig. 4, whereas the examples of dy-namical sequencing are given in Fig. 4 and distri-bution histograms for 5000 samples of CPRNGsequences are in Fig. 6.

4.3. Discrete Chaotic Maps for

Mapping of Search Space

Finally, this last subsection discusses the utiliza-tion of discrete chaotic maps for e�ective dynam-ical mapping of the search space suitable for ei-ther random/local search or random walk basedengines. The reason, as to why the discrete mapswere chosen for the mapping experiment is givenin the previous subsection, where the selecteddiscrete chaotic maps are depicted in graphicsgrid (Fig. 3) and sorted by the highest densityof coverage to the least. The graphical data inFig. 4 lends weight to the argument that somechaotic maps support the basic claim and fea-ture of deterministic chaos. This feature is calleddensity of periodic orbits, assuming that chaoticattractor (system) will visit most of the pointsin the space.

Furthermore presented simple discrete chaoticsystems have additional accessible parameters(See Tab.1.), which can by tuned. This issueopens up the possibility of examining the impactof these parameters on the generation of ran-

dom numbers and thus in�uence on chaos-basedoptimization (including adaptive switching be-tween chaotic systems or sampling rates). Theimpact of parameter changing for the chaoticmap is demonstrated in Fig. 7, Fig. 8, Fig. 9,Fig. 10, Fig. 11, Fig. 12, Fig. 13. Slight changeof original settings for Dissipative map resultedin increasing of the search space mapping den-sity (See Fig. 9 compared to the Fig. 4). Thechange of chaotic dynamics is also transferred todi�erent search trajectories (Figs 7 and 8 com-pared to the Fig. 10 and Fig. 11). Such a changein density of search space coverage and changein dynamical steps is also demonstrated in twohistograms of density: Fig. 12 and Fig. 13.

4.4. Simulation Results

Discussion

The main aim of this research is to try, test,analyze the usability and compare the im-plementation of di�erent natural chaotic dy-namic either as the replacement of tradi-tional/default PRNGs or as the mapping pro-cedure for the optimization/searching process.Eight di�erent complex chaotic time-continuous�ows/oscillators and discrete chaotic maps havebeen simulated, and the output behaviors havebeen transferred into the pseudo-random num-ber sequences. Findings can be summarized inTab. 2 and expanded as follows:

• In many research works, it was proventhat chaos based optimization is very sen-sitive to the hidden chaotic dynamics driv-ing the CPRNG/mapping the search space.Such a chaotic dynamics can be signi�-cantly changed by the selection of samplingtime in the case of the time-continuous sys-tems (both �ows and oscillators). A smallsampling rate of approx. 0.1 s - 0.5s keepsthe information about the chaotic dynam-ics inside the generated pseudo-random se-quence. By changing or simple learningadaptation of such sampling frequency, wecan fully keep, partially suppress or evenfully remove the chaotic information fromCPRNG sequence. There is no such a con-trol possibility for discrete maps.



Fig. 1: Parametric plots of time-continuous chaotic systems; upper left - Lorenz system, upper right - Rosslersystem, bottom left - UEDA oscillator, bottom right - Van der Pol Oscillator.

Fig. 2: Comparison of the dynamical sequencing of pseudo-random real numbers transferred into the range <0−1 >; upper left ? Lorenz system, upper right Rossler system, bottom left UEDA oscillator, bottom rightVan der Pol Oscillator.



Fig. 3: Comparison of distributions of pseudo-random real numbers transferred into the range < 0 − 1 > (5000samples); upper left ? Lorenz system, upper right Rossler system, bottom left UEDA oscillator, bottomright Van der Pol Oscillator.

Tab. 1: Used discrete chaotic systems as CPRNG and parameters set up.



Fig. 4: Parametric plots of discrete chaotic maps; upper left - Arnold Cat map, upper right Dissipative standardmap, bottom left Ikeda map, bottom right Lozi map.

Fig. 5: Comparison of the dynamical sequencing of pseudo-random real numbers transferred into the range <0 − 1 >; upper left ? Arnold Cat map, upper right Dissipative standard map, bottom left Ikeda map,bottom right Lozi map.



Fig. 6: Comparison of distributions of pseudo-random real numbers transferred into the range < 0 − 1 > (5000samples); upper left - Arnold Cat map, upper right Dissipative standard map, bottom left Ikeda map,bottom right Lozi map.

Fig. 7: Mapping of chaotic dynamics to the searchspace: Dissipative standard map - with defaultparameters b = 0.6 and k = 8.8. The coloring ofpath - from yellow to purple. A Small set of 50iterations.

• Oscillators are giving more dynamicalpseudo-random sequences with uniquequasi-periodical sequencing in comparisonwith chaotic �ows (See Fig. 1 and Fig. 2).Therefore chaotic oscillators are more suit-able for CPRNG.

Fig. 8: Mapping of chaotic dynamics to the searchspace: Dissipative standard map - with defaultparameters b = 0.6 and k = 8.8. The coloringof path - from yellow to purple. A larger set of200 iterations.

• Distributions of CPRNGs driven by time-continuous systems are very similar.

• When comparing the time continuous sys-tems and discrete maps - the �rst group issuitable only for direct CPRNG purposes



Fig. 9: Parametric plots of Dissipative standard map -with changed parameters b = 0.9 and k = 8.6.

Fig. 10: Mapping of chaotic dynamics to the searchspace: Dissipative standard map - changed pa-rameters b = 0.9 and k = 8.6. The coloring ofpath - from yellow to purple. A small set of 50iterations.

(and preferably oscillators as stated in theprevious point). Mapping of those systemsto the optimization search space will leadonly to covering a limited area, where thechaotic attractor is moving (See Fig. 1).Again, in the case of chaotic �ows, these

Fig. 11: Mapping of chaotic dynamics to the searchspace: Dissipative standard map - changed pa-rameters b = 0.9 and k = 8.6. The coloring ofpath - from yellow to purple. A larger set of200 iterations.

Fig. 12: Density histogram of mapping of chaotic dy-namics to the search space; Dissipative stan-dard map - with original parameters b = 0.6and k = 8.8.

areas will be restricted only close to the cy-cles of the attractor. Oscillators are show-ing better coverage of the space. Contraryto chaotic �ows and oscillators, the dis-crete maps show high potential for mapping



Fig. 13: Density histogram of mapping of chaotic dy-namics to the search space; Dissipative stan-dard map - with changed parameters b = 0.9and k = 8.6.

Tab. 2: A brief summarization of experiment results.

or local search techniques. This ability isdemonstrated in the previous section.

• Even though the coverage of search spaceis lower (Ikeda map) or very limited (Lozimap), these two chaotic maps can be com-bined within some hybrid multidimensionalmapping.

• Furthermore, presented discrete chaoticmaps generate di�erent distributions ofCPRNGs. Moreover, these systems haveadditional accessible parameters, which canby tuned and by changing of them, wecan obtain even more CPRNG distributionclasses.

5. Conclusion

The novelty of this research represents in-vestigation on the utilization of the complexchaotic dynamics given by the several selectedtime-continuous chaotic systems, as the chaoticpseudo-random number generators and drivingmaps for the chaos based optimization (mappingof chaotic movement to the optimization/searchspace). This paper showed three groups of com-plex chaotic dynamics given by chaotic �ows, os-cillators and discrete chaotic maps.

Plans are including the testing of combinationof the wider set of chaotic systems as well asthe adaptive switching between systems, adap-tive or chaos-like sequencing sampling rates andobtaining a large number of results to performstatistical tests.

Acknowledgment

This work was supported by Grant Agency ofthe Czech Republic - GACR P103/15/06700S,further by the Ministry of Education, Youthand Sports of the Czech Republic within theNational Sustainability Programme projectNo. LO1303 (MSMT-7778/2014) and alsoby the European Regional DevelopmentFund under the project CEBIA-Tech No.CZ.1.05/2.1.00/03.0089, and by Internal GrantAgency of Tomas Bata University under theproject No. IGA/CebiaTech/2016/007.

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About Authors

Roman SENKERIK was born in the CzechRepublic and went to the Tomas Bata Uni-versity in Zlin, where he studied TechnicalCybernetics and obtained his MSc degree in2004, the Ph.D. degree in Technical Cyberneticsin 2008 and Assoc. prof. in 2013 (Informatics).He is now an Assoc. prof. at the same uni-versity (research and courses in EvolutionaryComputation, Applied Informatics, Cryptology,Arti�cial Intelligence, Mathematical Informat-ics).

Ivan ZELINKA was born in the CzechRepublic and went to the Technical Universityof Brno, where he studied Technical Cyberneticsand obtained his degree in 1995. He obtainedthe Ph.D. degree in Technical Cybernetics in2001 at Tomas Bata University in Zlin. Now heis a full professor at the Technical University ofOstrava (research topics: Arti�cial Intelligence,Theory of Information, Computer security andviruses).

Michal PLUHACEK was born in theCzech Republic and went to the Faculty ofApplied Informatics at Tomas Bata Universityin Zlin, where he studied Information Tech-nologies and received his MSc degree in 2011and the Ph.D. in 2016 with the dissertationtopic: Modern method of development andmodi�cations of evolutionary computationaltechniques. He now works as a researcher atthe same university.


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Chaos-Based Optimization - A Review