SIMULATION OF CHAOTIC DYNAMICS FOR CHAOS BASED OPTIMIZATION – AN EXTENDED STUDY

Roman Senkerik, Michal Pluhacek, Adam Viktorin, Zuzana Kominkova Oplatkova and Tomas Kadavy

Tomas Bata University in Zlin , Faculty of Applied Informatics Nam T.G. Masaryka 5555, 760 01 Zlin, Czech Republic {senkerik, oplatkova , pluhacek , aviktorin , kadavy}@fai.utb.cz

KEYWORDS The idea of using chaotic dynamics as a replacement of Deterministic chaos; Chaotic oscillators; Heuristic; classical pseudo-number generators – (PRNGs) has Chaotic Optimization; Chaotic Pseudo Random Number been presented in several research papers and in many Generators applications with promising results (Lee and Chang 1996; Wu and Wang, 1999). ABSTRACT Another research joining deterministic chaos and pseudorandom number generator has been done for This paper discuss the utilization of the complex chaotic example in (Lozi 2012). Possibility of generation of dynamics given by the selected time-continuous chaotic pure random or pseudorandom numbers by use of the systems as well as by the discrete chaotic maps, as the ultra weak multidimensional coupling of 1-dimensional chaotic pseudo random number generators and driving dynamical systems is discussed there. maps for the chaos based optimization. Such an Another paper (Persohn and Povinelli 2012), deeply optimization concept is utilizing direct output iterations investigate well-known as a possible of chaotic system transferred into the required pseudorandom number generator and is compared with numerical range or it uses the chaotic dynamics for contemporary pseudo-random number generators. A mapping the search space mostly within the local search comparison of logistic map results is made with techniques. This paper shows totally three groups of conventional methods of generating pseudorandom complex chaotic dynamics given by chaotic flows, numbers. The approach used to determine the number, oscillators and discrete maps. Simulations of examples delay, and period of the orbits of the logistic map at of chaotic dynamics mapped to the search space were varying degrees of precision (3 to 23 bits). Another performed and related issues like parametric plots, paper (Wang and Qin 2012) proposed an algorithm of distributions of such a systems, periodicity, and generating pseudorandom number generator, which is dependency on internal accessible parameters are briefly called (couple map lattice based on discrete chaotic discussed in this paper. iteration) and combine the couple map lattice and chaotic iteration. Authors also tested this algorithm in INTRODUCTION NIST 800-22 statistical test suits and for future Generally speaking, the term “chaos” can denote utilization in image encryption. In (Narendra et al. anything that cannot be predicted deterministically. 2010) authors exploit interesting properties of chaotic When the term “chaos” is combined with an attribute systems to design a random bit generator, called such as “deterministic”, a specific type of chaotic CCCBG, in which two chaotic systems are cross- paradigm is involved, having their specific laws, coupled with each other. A new binary stream-cipher mathematical apparatus, and a physical origin. algorithm based on dual one-dimensional chaotic maps Recently, the chaos has been observed in many of is proposed in (Yang and Wang 2012) with statistic various systems (including evolutionary one). Systems proprieties showing that the sequence is of high exhibiting deterministic chaos include, for instance, . Similar studies are also done in (Bucolo et weather, biological systems, many electronic circuits al. 2002). (Chua’s circuit), mechanical systems, such as , magnetic pendulum, or so called billiard MOTIVATION problem. This paper represents the extension of preliminary The deterministic chaos is not based on the presence of suggestions described in (Senkerik et al. 2016). randomness or any stochastic effects. It is clear from the Recently the deterministic chaos has been frequently mathematic definition and structure of the equations used either as a replacement of (mostly uniform (see the section “Chaotic Optimization”), that no distribution based) pseudo-number generators (PRGNs) mathematical term expressing randomness is present. in metaheuristic algorithms or for simple mapping of The seeming randomness in deterministic chaos is solutions/iterations within intelligent local search related to the extreme sensitivity to the initial conditions engines. The metaheuristic chaotic approach generally (Celikovsky and Zelinka 2010). uses the chaotic system in the place of a pseudo random number generator (Aydin et al. 2010). This causes the

Proceedings 31st European Conference on Modelling and Simulation ©ECMS Zita Zoltay Paprika, Péter Horák, Kata Váradi, Péter Tamás Zwierczyk, Ágnes Vidovics-Dancs, János Péter Rádics (Editors) ISBN: 978-0-9932440-4-9/ ISBN: 978-0-9932440-5-6 (CD)

heuristic to map search regions based on unique sequencing and periodicity of transferred chaotic CHAOTIC SYSTEMS dynamics, thus simulating the dynamical alternations of This section contains the description of three different several subpopulations. The task is then to select a very groups of chaotic dynamics: time-continuous chaotic good chaotic system (either discrete or time-continuous) systems (flows and oscillators), and the discrete chaotic as the pseudo random number generator (Caponetto et maps. al. 2003).

Recently, the concept of embedding of chaotic Time-continuous Chaotic Systems dynamics into the evolutionary algorithms has been studied intensively. The self-adaptive chaos differential In this research, following chaotic systems were used: evolution (SACDE) (Zhenyu et al. 2006) was followed (1) and Rossler system (2) as two by the implementation of chaos into the simple not- examples from chaotic flows; further unmodified adaptive differential evolution (Davendra et al. 2010), UEDA oscillator (3); and Driven (Senkerik et al. 2014); chaotic searching algorithm for (4) as chaotic oscillators (Sprott 2003). the very same metaheuristic was introduced in (Liang et The Lorenz system (1) is a 3-dimensional dynamical al. 2011). Also the PSO (Particle Swarm Optimization) flow, which exhibits chaotic behavior. It was introduced algorithm with elements of chaos was introduced as by Edward Lorenz in 1963, who derived it from the CPSO (Coelho and Mariani 2009). Many other works simplified equations of convection rolls arising in the focusing on the hybridization of the swarm and chaotic equations of the atmosphere. movement have been published afterwards (Pluhacek et The Rossler system (2) exhibits chaotic dynamics al. 2013), (Pluhacek et al. 2014). Later on, the associated with the properties of the . It utilization of chaotic sequences became to be popular in was originally introduced as an example of very simple many interdisciplinary applications and techniques. The chaotic flow containing chaos similarly to the Lorenz question of impact and importance of different attractor. This attractor has some similarities to the randomization within heuristic search was intensively Lorenz attractor, but is simpler studied in (Zamuda and Brest 2015) UEDA oscillator (3) is the simple example of driven The primary aim of this work is to try, analyze and pendulums, which represent some of the most compare the implementation of different natural chaotic significant examples of chaos and regularity. dynamic as the mapping procedure for the UEDA system can be simply considered as a special optimization/searching process. This paper presents the case of intensively studied Duffing oscillator that has discussion about the usability of such systems, both a linear and cubic restoring force. Ueda oscillator periodicity, and dependency on internal accessible represents the both biologically and physically parameters; thus the usability for local search or important dynamical model exhibiting chaotic motion. metaheuristic based optimization techniques. Finally, The Van der Pol oscillator (4) is the simple example of the limit cycles and chaotic behavior in CHAOTIC OPTIMIZATION electrical circuits employing vacuum tubes. Similarly to the UEDA oscillator, it can be used to explore physical Generally, there exist three possible utilizations of (unstable) behaviour in biological sciences. (Bharti and chaotic dynamics in optimization tasks. Yuasa 2010). Firstly, as aforementioned in previous section, the direct The equations, which describe the chaotic systems, have output simulation iterations of chaotic system are parameter settings for Lorenz system: a = 3.0, b = 26.5; transferred into the required numerical range (as simple Rössler system: a = 0.2, b = 0.2, and c = 5.7; UEDA CPRNG). The idea of CPRNG is to replace the default oscillator: a = 1.0 b = 0.05, c = 7.5 and ω = 1.0; and system PRNG with the chaotic system. As the chaotic Van der Pol oscillator : μ = 0.2 γ = 8.0, a = 0.35 and system is a set of equations with a static start position ω = 1.02. (See the next section), we created a random start dx position of the system, in order to have different start = −a()x − y position for different experiments. Once the start dt position of the chaotic system has been obtained, the dy = x()b − z − y (1) system generates the next sequence using its current dt position. Subsequently, simple techniques as to how to dz deal with the negative numbers as well as with the = xy − z dt scaling of the wide range of the numbers given by the chaotic systems into the typical range 0 – 1 dx Secondly, the of chaotic systems and its = −y − z movement in the space is used for dynamical mapping dt of the search space mostly within the local search dy = x + ay (2) techniques (Hamaizia and Lozi 2011). dt Finally, the hybridization of searching/optimization dz process and chaotic systems is represented by chaos = b + z()x − c dt based random walk technique.

dx = means of particular studied CPRNGs and with the y sampling rate of 0.5s. The dependency of sequencing dt . (3) dy and periodicity on the sampling rate is discussed in = −ax3 − by + csinωt details in (Senkerik et al. 2015) dt

Discrete Chaotic Maps dx = y dt The examples of chaotic maps are following: Arnold (4) Cat map (5), Dissipative (6), Ikeda map dy 2 3 = μ()1−γx y − x + asinωt (7), and Lozi map (8). Map equations and parameters dt values are given in Table 1. Parametric plots are

depicted in Figure 3, whereas the examples of The parametric plots of the chaotic systems are depicted dynamical sequencing are given in Figure 4. in Figure 1. The Figure 2 show the example of dynamical sequencing during the generating of pseudo number numbers transferred into the range <0 - 1> by

Table 1: Used discrete chaotic systems as CPRNG and parameters set up. Chaotic system Notation Parameters values Arnold Cat Map X = X + Y ()mod1 k = 2.0 n+1 n n = + () Yn+1 X n kYn mod1 Dissipative Standard Map X = X + Y (mod 2π ) b = 0.6 and k = 8.8 n+1 n n+1 = + π Yn+1 bYn k sin X n (mod 2 ) = γ + μ φ + φ Ikeda Map X n+1 (X n cos Yn sin ) α = 6, β = 0.4, γ = 1 and μ = 0.9 = μ φ + φ Yn+1 (X n sin Yn cos ) φ = β −α ()+ 2 + 2 / 1 X n Yn Lozi Map X =1− a X + bY a = 1.7 and b = 0.5 n+1 n n = Yn+1 X n

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

Fig. 2: Comparison of 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 right Van der Pol Oscillator.

Fig. 3: Parametric plots of discrete chaotic maps; upper left – Arnold Cat map, upper right Dissipative standard map, bottom left Ikeda map, bottom right Lozi map.

Fig. 4: Comparison of 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.

as stated in the previous point). Mapping of those EXPERIMENT DISCUSSION systems to the optimization search space will lead The main aim of this research is to try, test, analyze the only to covering a limited area, where the chaotic usability and compare the implementation of different attractor is moving (See Figure 1). Again, in case natural chaotic dynamic as the mapping procedure for of chaotic flows, these areas will be restricted only the optimization/searching process. Eight different close to the cycles of the attractor. Oscillators are complex chaotic time-continuous flows/oscillators and showing better coverage of the space. Selected discrete chaotic maps have been simulated and the discrete chaotic maps are depicted in graphics grid output behaviors have been transferred into the pseudo (Figure 3) and sorted from the highest density of random number sequences. Findings can be summarized coverage to the least. The graphical data in Figure as follows: 3 lends weight to the argument, that some chaotic maps support the basic claim and feature of • In many research works, it was proven that chaos deterministic chaos. This feature is called density based optimization is very sensitive to the hidden of periodic orbits, assuming that chaotic attractor chaotic dynamics driving the CPRNG/mapping the (system) will visit most of the points in the space. search space. Such a chaotic dynamics can be • significantly changed by the selection of sampling Even though the coverage of search space is lower time in the case of the time-continuous systems (Ikeda map) or very limited (Lozi map), these two (both flows and oscillators). Very small sampling chaotic maps can be combined together within rate of approx. 0.1s - 0.5s keeps the information some hybrid multidimensional mapping. • about the chaotic dynamics inside the generated Furthermore presented chaotic systems have pseudo-random sequence. By changing or simple additional accessible parameters, which can by learning adaptation of such sampling rate, we can tuned. This issue opens up the possibility of fully keep, partially suppress or even fully remove examining the impact of these parameters to the chaotic information from CPRNG sequence. generation of random numbers, and thus influence There is no such a control possibility for discrete on chaos based optimization (including adaptive maps. switching between chaotic systems or sampling • Oscillators are giving more dynamical pseudo rates). The impact of parameter changing for random sequences with unique quasi-periodical chaotic map is demonstrated in Figures 5 – 10. sequencing in comparison with chaotic flows (See Very small change of original settings for Figs 1 and 2). Therefore chaotic oscillators are Dissipative map resulted in increasing of the more suitable for CPRNG. search space mapping density (See Figure 7 compared to the Figure 3). The change of chaotic • When comparing the time continuous systems and dynamics is also transferred to different search discrete maps – the first group is suitable only for trajectories (Figures 5, 8 and 8) and distributions direct CPRNG purposes (and preferably oscillators (Figures 6 and 9).

Fig. 8: Mapping of chaotic dynamics to the search space: Fig. 5: Mapping of chaotic dynamics to the search space: Dissipative standard map – with changed parameters b = 0.8 Dissipative standard map – with original parameters b = 0.6 and k = 8.8. Coloring of path – from yellow to purple. and k = 8.8. Coloring of path – from yellow to purple.

Fig. 9: Original chaotic CPRNG and identified distributions Fig. 6: Original chaotic CPRNG and identified distributions for Dissipative standard map – with changed parameters for Dissipative standard map – with original parameters b = b = 0.8 and k = 8.8 (5000 samples). 0.6 and k = 8.8 (5000 samples).

Fig. 10: Mapping of chaotic dynamics to the search space: Fig. 7: Parametric plots of Dissipative standard map – with Dissipative standard map – with changed parameters b = 0.9 changed parameters b = 0.8 and k = 8.8 and k = 8.6. Coloring of path – from yellow to purple.

CONCLUSION application to support vector regression machine', Journal of Software, 6(7), 1297- 1304. The novelty of this research represents investigation on Lozi, R. (2012) 'Emergence of Randomness from Chaos', the utilization of the complex chaotic dynamics given International Journal of Bifurcation and Chaos, 22(02), by the several selected time-continuous chaotic systems, 1250021. as the chaotic pseudo random number generators and Narendra, K. P., Vinod, P. and Krishan, K. S. (2010) 'A driving maps for the chaos based optimization (mapping Random Bit Generator Using Chaotic Maps', International of chaotic movement to the optimization/search space). Journal of Network Security, 10, 32 - 38. This paper showed three groups of complex chaotic Persohn, K. J. and Povinelli, R. J. (2012) 'Analyzing logistic map pseudorandom number generators for periodicity dynamics given by chaotic flows, oscillators and induced by finite precision floating-point representation', discrete chaotic maps. Chaos, Solitons & , 45(3), 238-245. Future plans are including the testing of combination of Pluhacek, M., Senkerik, R. and Zelinka, I. (2014) 'Multiple wider set of chaotic systems as well as the adaptive Choice Strategy Based PSO Algorithm with Chaotic switching between systems, adaptive or chaos-like Decision Making – A Preliminary Study' in Herrero, Á., sequencing sampling rates and obtaining a large number Baruque, B., Klett, F., Abraham, A., Snášel, V., Carvalho, of results to perform statistical tests. A. C. P. L. F., Bringas, P. G., Zelinka, I., Quintián, H. and Corchado, E., eds., International Joint Conference ACKNOWLEDGEMENT SOCO’13-CISIS’13-ICEUTE’13, Springer, 21-30. Pluhacek, M., Senkerik, R., Davendra, D., Kominkova This work was supported by Grant Agency of the Czech Oplatkova, Z. and Zelinka, I. (2013) 'On the behavior and Republic - GACR P103/15/06700S, further by the Ministry of performance of chaos driven PSO algorithm with inertia Education, Youth and Sports of the Czech Republic within the weight', Computers & with Applications, National Sustainability Programme project No. LO1303 66(2), 122-134. (MSMT-7778/2014) and also by the European Regional Senkerik, R., Pluhacek, M., Zelinka, I., Oplatkova, Z., Vala, Development Fund under the project CEBIA-Tech No. R. and Jasek, R. (2014) 'Performance of Chaos Driven CZ.1.05/2.1.00/03.0089, and by Internal Grant Agency of Differential Evolution on Shifted Benchmark Functions Tomas Bata University, No. IGA/CebiaTech/2017/004. Set' in Herrero, Á., Baruque, B., Klett, F., Abraham, A., Snášel, V., Carvalho, A. C. P. L. F., Bringas, P. G., REFERENCES Zelinka, I., Quintián, H. and Corchado, E., eds., International Joint Conference SOCO’13-CISIS’13- Aydin, I., Karakose, M. and Akin, E. (2010) 'Chaotic-based ICEUTE’13, Springer International Publishing, 41-50. hybrid negative selection algorithm and its applications in Senkerik R., Pluhacek M., Davendra D., Zelinka I., fault and anomaly detection', Expert Systems with Kominkova Oplatkova Z. (2015). Simulation Of Time- Applications, 37(7), 5285-5294. Continuous Chaotic UEDA Oscillator As The Generator Bharti, L. and Yuasa, (2010) M. Energy Variability and Chaos Of Random Numbers For Heuristic, ECMS 2015 in Ueda Oscillator. Available: Proceedings edited by: Valeri M. Mladenov, Petia http://www.rist.kindai.ac.jp/no.23/yuasa-EVCUO.pdf Georgieva, Grisha Spasov, Galidiya Petrova European Bucolo, M., Caponetto, R., Fortuna, L., Frasca, M. and Rizzo, Council for Modeling and Simulation. doi:10.7148/2015- A. (2002) 'Does chaos work better than noise?', Circuits 0543 and Systems Magazine, IEEE, 2(3), 4-19. Senkerik R., Pluhacek M., Viktorin A, Kominkova Caponetto, R., Fortuna, L., Fazzino, S. and Xibilia, M. G. OplatkovaZ. (2016). On The Simulation Of Complex (2003) 'Chaotic sequences to improve the performance of Chaotic Dynamics For Chaos Based Optimization, ECMS evolutionary algorithms', IEEE Transactions on 2016 Proceedings edited by: Thorsten Claus, Frank Evolutionary Computation, 7(3), 289-304. Herrmann, Michael Manitz, Oliver Rose European Celikovsky, S. and Zelinka, I. (2010) ' for Council for Modeling and Simulation. doi:10.7148/2016- Evolutionary Algorithms Researchers' in Zelinka, I., 0258. Celikovsky, S., Richter, H. and Chen, G., eds., Sprott, J. C. (2003) Chaos and Time-Series Analysis, Oxford Evolutionary Algorithms and Chaotic Systems, Springer University Press. Berlin Heidelberg, 89-143. Wang, X.-y. and Qin, X. (2012) 'A new pseudo-random Coelho, L. d. S. and Mariani, V. C. (2009) 'A novel chaotic number generator based on CML and chaotic iteration', particle swarm optimization approach using Hénon map Nonlinear Dynamics, 70(2), 1589-1592. and implicit filtering local search for economic load Wu, J., Lu, J. and Wang, J. (2009) 'Application of chaos and dispatch', Chaos, Solitons & Fractals, 39(2), 510-518. fractal models to water quality time series prediction', Davendra, D., Zelinka, I. and Senkerik, R. (2010) 'Chaos Environmental Modelling & Software, 24(5), 632-636. driven evolutionary algorithms for the task of PID control', Yang, L. and Wang, X.-Y. (2012) 'Design of Pseudo-random Computers & Mathematics with Applications, 60(4), 1088- Bit Generator Based on Chaotic Maps', International 1104. Journal of Modern Physics B, 26(32), 1250208. ׳Hamaizia, T. and Lozi, R. (2011) Improving Chaotic Zamuda, A., Brest, J. (2015) Self-adaptive control parameters Optimization Algorithm using a new global locally randomization frequency and propagations in differential averaged strategy, translated by pp. 17-20. evolution. Swarm and Evolutionary Computation 25, 72- Lee, J. S. and Chang, K. S. (1996) 'Applications of chaos and 99. fractals in process systems engineering', Journal of Zhenyu, G., Bo, C., Min, Y. and Binggang, C. (2006) 'Self- Process Control, 6(2–3), 71-87. Adaptive Chaos Differential Evolution' in Jiao, L., Wang, Liang, W., Zhang, L. and Wang, M. (2011) 'The chaos L., Gao, X.-b., Liu, J. and Wu, F., eds., Advances in differential evolution optimization algorithm and its Natural Computation, Springer Berlin Heidelberg, 972- 975.