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Chaos-Based Optimization - a Review VOLUME: 1 j ISSUE: 1 j 2017 j June Chaos-Based Optimization - A Review Roman SENKERIK 1;2;*, Ivan ZELINKA 1;3, Michal PLUHACEK 2 1Modeling Evolutionary Algorithms Simulation and Articial Intelligence, Faculty of Electrical and 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, 760 01 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 1. Introduction 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 Generally speaking, the term "chaos" can de- maps for the chaos based optimization. Such note anything that cannot be predicted deter- an optimization concept is utilizing direct out- ministically. In the case that the word "chaos" put iterations of chaotic system transferred into is combined with an attribute such as "determin- the required numerical range as the replacement istic," then a specic type of chaotic phenomena of traditional and default pseudo-random num- is involved, having their specic laws, mathe- ber generators, or this concept uses the chaotic matical apparatus and a physical origin. The dynamics for mapping the search space mostly deterministic chaos is a phenomenon that - as within the smart hybrid local search techniques. its name suggests - is not based on the presence This paper shows totally three groups of complex of a random or any stochastic eects. It is evi- chaotic dynamics given by chaotic ows, oscil- dent from the structure of the equations (see the lators and discrete maps. Simulations of exam- section: Chaotic Optimization) that no mathe- ples of chaotic dynamics as unconventional gen- matical term expressing randomness is present. erators or mapped to the search space were per- The seeming randomness in deterministic chaos formed, and related issues like parametric plots, is related to the extreme sensitivity to the initial distributions of such a systems, periodicity, and conditions [1]. dependency on internally available parameters In the past, the chaos has been observed in are briey discussed in this paper. many of various systems (including evolutionary one). 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, Keywords magnetic pendulum, or so-called billiard prob- lem. The idea of using chaotic systems instead of random processes (pseudo-number generators Deterministic chaos, Heuristic, Chaotic - PRNGs) has been presented in several research Optimization, Chaotic Pseudo Random elds and many applications with promising re- Number Generators. sults [2], [3]. 68 c 2017 Journal of Advanced Engineering and Computation (JAEC) VOLUME: 1 j ISSUE: 1 j 2017 j June Another research joining deterministic chaos populations. The task is then to select an ap- and pseudorandom number generator has been propriate chaotic system (either discrete or time- done for example in [4]. The possibility of gen- continuous) as the chaotic pseudo random num- eration of random or pseudorandom numbers ber generator (CPRNG) [11]. by use of the ultra-weak multidimensional cou- Recently, the concept of embedding of chaotic pling of p 1-dimensional dynamical systems is dynamics into the evolutionary algorithms has discussed there. been studied intensively. The self-adaptive Another paper [5], deeply investigate logis- chaos dierential evolution (SACDE) [12]was tic map as a possible pseudorandom number followed by the implementation of chaos into the generator and is compared with contemporary simple not-adaptive dierential evolution [13], pseudo-random number generators. A compar- [14]; the chaotic searching algorithm for the very ison of logistic map results is made with con- same metaheuristic was introduced in [15]. Also, ventional methods of generating pseudorandom the PSO (Particle Swarm Optimization) algo- numbers. The approach used to determine the rithm with elements of chaos was introduced as number, delay, and period of the orbits of the CPSO [16]. Many other works focusing on the logistic map at varying degrees of precision (3 hybridization of the swarm and chaotic move- to 23 bits). Another paper [6] proposed an al- ment have been published afterward [17], [18]. gorithm for generating pseudorandom number Later on, the utilization of chaotic sequences be- generator, which is called (coupled map lattice came to be popular in many interdisciplinary ap- based on discrete chaotic iteration) and combine plications and techniques. The question of im- the coupled map lattice and chaotic iteration. pact and importance of dierent randomization Authors also tested this algorithm in NIST 800- within heuristic search was intensively studied 22 statistical test suits and for future utilization in [19]. in image encryption. In (Narendra et al. 2010) The primary aim of this work is to try, authors exploit interesting properties of chaotic test, analyze and compare the implementa- systems to design a random bit generator, called tion of dierent natural chaotic dynamic ei- CCCBG, in which two chaotic systems are cross- ther as the mapping procedure for the optimiza- coupled with each other. A new binary stream tion/searching process or as the CPRNG. This cipher algorithm based on dual one-dimensional paper presents the discussion about the usability chaotic maps is proposed in [7] with statistic pro- of such systems, periodicity, and dependency on prieties showing that the sequence is of high ran- accessible internal parameters; thus the usability domness. Similar studies are also done in [8]. as CPRNG, or for local search and metaheuristic based optimization techniques. 2. Motivation 3. Chaotic Optimization This paper represents the extension of prelimi- nary suggestions described in [9]. Recently the There exist three possible utilizations of chaotic deterministic chaos has frequently been used dynamics in optimization tasks. either as a replacement of (mostly uniform distribution based) pseudo-number generators Firstly, as aforementioned in the previous sec- (PRGNs) in metaheuristic algorithms or for sim- tion, the direct output simulation iterations of ple mapping of solutions/iterations within orig- the chaotic system are transferred into the re- inal local search engines. The metaheuristic quired numerical range (as simple CPRNG). chaotic approach uses the chaotic system in the The idea of CPRNG is to replace the default place of a pseudo-random number generator [10]. system PRNG with the chaotic system. As the This method causes the heuristic to map search chaotic system is a set of equations with a static regions based on unique sequencing and period- start position (see the next section), we created icity of transferred chaotic dynamics, thus simu- a random start position of the system, to have lating the dynamical alternations of several sub- dierent start position for various experiments. c 2017 Journal of Advanced Engineering and Computation (JAEC) 69 VOLUME: 1 j ISSUE: 1 j 2017 j June Once the origin position of the chaotic system most signicant examples of chaos and regular- has been obtained, the system generates the ity. next sequence using its current position. Sub- UEDA system can be simply considered as a sequently, simple techniques as to how to deal special case of intensively studied Dung oscil- with the negative numbers as well as with the lator that has both a linear and cubic restoring scaling of the wide range of the numbers given force. Ueda oscillator represents the both biolog- by the chaotic systems into the typical range 0-1. ically and physically relevant dynamical model Secondly, the complexity of chaotic systems exhibiting chaotic motion. and its movement in the space is used for dy- Finally, The Van der Pol oscillator (4) is the namical mapping of the search space mostly simple example of the limit cycles and chaotic within the local search techniques [20]. Finally, behavior in electrical circuits employing vacuum the hybridization of searching/optimization pro- tubes. Similarly to the UEDA oscillator, it can cess and chaotic systems is represented by chaos be used to explore physical (unstable) behavior based random walk technique. in biological sciences. [22]. The equations, which describe the chaotic sys- tems, have parameter settings for Lorenz sys- 4. Chaotic Optimization tem: a = 3:0, b = 26:5; Rossler system: a = 0:2, b = 0:2, and c = 5:7; UEDA oscillator: a = 1:0, , and ; and Van der This section contains the description of three b = 0:05 c = 7:5 ! = 1:0 Pol oscillator : , , and dierent groups of chaotic dynamics: time- µ = 0:2 γ = 8:0 a = 0:35 . continuous chaotic systems (ows and oscilla- ! = 1:02 tors), and the discrete chaotic maps. dx = −a(x − y) dt 4.1. Time Continuous systems dy = x(b − z) − y: as CPRNG dt dz = xy − z (1) In this research, following chaotic systems were dt used: Lorenz system (1) and Rossler system (2) as two examples of chaotic ows; further unmod- ied UEDA oscillator (3); and Driven Van der dx = −x − y Pol Oscillator (4) as chaotic oscillators [21]. dt dy The Lorenz system (1) is a 3-dimensional dy- = x + ay: namical ow, which exhibits chaotic behavior. dt dz It was introduced in 1963 by Edward Lorenz, = b + z(x − c) (2) who derived it from the simplied equations of dt convection rolls arising in the equations of the atmosphere. dx The Rossler system (2) exhibits chaotic dy- = y namics associated with the fractal properties of dt dy the attractor. It was originally introduced as an = −ax3 − by + csin!t: (3) example of very simple chaotic ow containing dt chaos similarly to the Lorenz attractor.
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