PID control design for chaotic synchronization using a tribes optimization approach

Chaos, Solitons and Fractals 42 (2009) 634–640

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos

PID control design for chaotic synchronization using a tribes optimization approach Leandro dos Santos Coelho *, Diego Luis de Andrade Bernert Industrial and Systems Engineering Graduate Program, LAS/PPGEPS, Pontifical Catholic University of Paraná, PUCPR, Imaculada Conceição, 1155, 80215-901 Curitiba, Paraná, Brazil

a r t i c l e

i n f o

Article history: Accepted 27 January 2009

a b s t r a c t Recently, the investigation of synchronization and control problems for discrete chaotic systems has stimulated a wide range of research activity including both theoretical studies and practical applications. This paper deals with the tuning of a proportional-integralderivative (PID) controller using a modified Tribes optimization algorithm based on truncated chaotic Zaslavskii map (MTribes) for synchronization of two identical discrete chaotic systems subject the different initial conditions. The Tribes algorithm is inspired by the social behavior of bird flocking and is also an optimization adaptive procedure that does not require sociometric or swarm size parameter tuning. Numerical simulations are given to show the effectiveness of the proposed synchronization method. In addition, some comparisons of the MTribes optimization algorithm with other continuous optimization methods, including classical Tribes algorithm and particle swarm optimization approaches, are presented. Ó 2009 Published by Elsevier Ltd.

1. Introduction Over the past decades, chaos plays a more and more important role in nonlinear science field [1–24]. It is well known that the trajectories of two chaotic systems starting from two different nearby initial conditions separate exponentially in the course of the time, as their evolution sensitively depends on the initial conditions. Since Pecora and Carroll [5] demonstrated that coupled chaotic systems with different initial values, can be synchronized, there has been an increasing interest in the study of chaos synchronization and its applications in various fields ranging from physics, biology, chemistry, mathematics to engineering. Generally the two chaotic systems in synchronization are called drive system and response system, respectively. A basic configuration for chaos synchronization is the master–slave (drive–response) pattern, where the response chaotic system must track the drive chaotic trajectory. Many approaches have been presented for the control and synchronization of chaotic systems such as feedback control [6,7], impulsive control [8–10], fuzzy control [11–15] and adaptive control [16–18]. Moreover, many types of design and optimization methods have been suggested for the synchronization of chaotic systems via proportional-integral-derivative (PID) control such as linear matrix inequality [19], observers [20], evolutionary programming [21,22], particle swarm optimization [23], and harmony search [24]. In this paper, the tuning procedure of a PID using the modified Tribes optimization method based on truncated chaotic Zaslavskii map for synchronization of two Lozi chaotic systems subject the different initial conditions is presented and

* Corresponding author. E-mail addresses: [email protected] (L.d.S. Coelho), [email protected] (D.L.d.A. Bernert). 0960-0779/$ - see front matter Ó 2009 Published by Elsevier Ltd. doi:10.1016/j.chaos.2009.01.032

L.d.S. Coelho, D.L.d.A. Bernert / Chaos, Solitons and Fractals 42 (2009) 634–640

635

evaluated. Simulation results of the modified Tribes optimization method to PID’s gains tuning are compared with other optimization methods, including classical Tribes algorithm and particle swarm optimization. Recently, many research activities have been devoted to particle swarm optimization (PSO) approaches [25–30]. PSO is a population-based heuristic optimization technique proposed by Kennedy and Eberhart [31,32] inspired by social behavior of bird flocking. The fundamental point of developing PSO is a hypothesis in which the exchange of information among creatures of the same species offers some sort of evolutionary advantage. Clerc [33–36] proposed a parameter-free PSO configuration called Tribes, in which details of the topology, including the size of the population, evolve over time in response to performance feedback. The Tribes is an optimization adaptive procedure that does not require sociometric or swarm size parameter tuning. Since then, the Tribes has attracted attention from researchers in applications such as optimization of milling operations [37,38], flow shop scheduling [39], molecular docking [40], and electromagnetics optimization [41]. The outline for the rest of the paper is as follows. Section 2 describes the problem, while Section 3 explains the concepts of Tribes. Section 4 presents the simulation results of PID’s tuning and chaotic synchronization of two Lozi chaotic systems. Finally, Section 5 outlines a brief conclusion about this study. 2. Problem formulation 2.1. PID controller PID control is a term usually used for denoting the control with the Proportional, Integral, and Derivative actions. Despite the huge development in control theory, the majority of industrial processes are controlled by the well-established PID controller [42,43]. The popularity of PID control can be attributed to its simplicity (in terms of design and from the point of view of parameter tuning) and to its good performance in a wide range of operating conditions [44]. As modelled in this paper, the transfer function of PID controller is described by the following equation in the continuous s-domain (Laplace operator)

UðsÞ Ki ¼ Kp þ þ Kd  s EðsÞ s

GPID ðsÞ ¼ P þ I þ D ¼

ð1Þ

or

  1 GPID ðsÞ ¼ K p  1 þ þ Td  s ; Ti  s

ð2Þ

where U(s) and E(s) are the control (controller output) and tracking error signals in s-domain, respectively; Kp is the proportional gain, Ki is the integration gain, and Kd is the derivative gain. Ti is the integral action time or reset time and Td is referred to as the derivation action time or rate time [24]. In this context, the output of the PID controller in time domain is given by

uðtÞ ¼ K p  eðtÞ þ K i

Z

t

eðsÞ ds þ K d 

0

deðtÞ ; dt

ð3Þ

where u(t) and e(t) are the control and tracking error signals in time domain, respectively. Using trapezoidal approximations for Eq. (3) to obtain the discrete control law, we have

uðkÞ ¼ uðk  1Þ þ K p  ½eðkÞ  eðk  1Þ þ K i 

Ts Ts  ½eðkÞ  eðk  1Þ þ K d :  ½eðkÞ  2eðk  1Þ þ eðk  2Þ; 2 2

ð4Þ

where Ts is the sampling period and k is the sample. Via feedback of the system output the PID controller has the ability to eliminate steady-state offsets through integral action and it can also ‘anticipate’ the future through its derivative action. 2.2. Nonlinear discrete chaotic system In this study, two identical discrete chaotic systems are considered to be synchronized using the proposed PID control. The master system is given by a typical discrete chaotic system, Lozi’s model [45] is employed as an example in this paper. The Lozi’s piecewise linear model is a simplification of the Hénon map [4,46] and it admits strange attractors. The only difference between Lozi’s and Hénon’s maps is that the (x1(k  1))2 term is replaced by jx1(k  1)j. This chaotic map involves also non-differentiable functions which difficult the modeling of the associate time series. The Lozi map is given by

x1 ðk þ 1Þ ¼ 1  a  jx1 ðkÞj þ xðkÞ;

ð5Þ

xðk þ 1Þ ¼ b  x1 ðkÞ;

ð6Þ

where a = 1.7, b = 0.5, and x is the master state. On the other hand, the corresponding slave system is described by

636

L.d.S. Coelho, D.L.d.A. Bernert / Chaos, Solitons and Fractals 42 (2009) 634–640

y1 ðk þ 1Þ ¼ 1  a  jy1 ðkÞj þ yðkÞ þ uðkÞ;

ð7Þ

yðk þ 1Þ ¼ b  y1 ðkÞ;

ð8Þ

where y is the slave state and u is the external control force that adopts the PID control of Eq. (4). For two identical discrete chaotic systems (6) and (8) without control u, the state trajectories of these two chaotic systems will separate each other if their initial conditions are not the same. However, the state trajectories can approach synchronization for any initial condition if an appropriate controller is utilized. Hence the purpose of this paper is to apply the Tribes optimization approaches to find out the optimal PID control gains such that chaos synchronization for two Lozi’s chaotic systems is achieved. The objective function F used in the study is defined as

F¼

N X

jxðkÞ  yðkÞj ¼

k¼1

N X

jeðkÞj;

ð9Þ

k¼1

where e(k) is the error signal between the master and slave states and N is the total number of sampling. The optimization problem involves finding g  ¼ ½K p ; K i ; K d  such that the F performance index of the system is minimized. 3. Fundamentals of Tribes algorithm This section describes the Tribes optimization algorithm. First, a brief overview of the Tribes is provided, and finally a modified Tribes algorithm is proposed. 3.1. Tribes algorithm The Tribes algorithm is inspired by the social behavior of bird flocking. Clerc’s Tribes mechanism is auto-parameterising. The principles of Tribes are: (i) the swarm is divided in tribes; (ii) at the beginning, the swarm is composed of only one particle; (iii) according to tribes’ behaviors, particles are added or removed; and (iv) according to the performances of the particles, their strategies of displacement are adapted [47]. In Tribes, the concepts used were named tribes and informer groups (i groups). Clerc [33] used the random uniform hypersphere generation that does not need constriction coefficient nor weighting parameters. Each particle belongs to a single tribe. The concept of informer is analogous to that – described in population topologies – of neighbor: someone who can influence the current particle. The set of informers of a particle, its i-group, contains, but is not limited to, the elements of that particle’s tribe. The adaptation rules describe when a particle is created or removed and when a particle becomes the informer of another [48]. In Tribes algorithm, the population of particles is divided in subpopulations, each maintaining its own order and structure. ‘‘Good” tribes may benefit by removal of their weakest member, as they already possess good problem solutions and thus may afford to reduce their population; ‘‘bad” tribes, on the other hand, may benefit by addition of a new member, increasing the possibility of improvement. New particles are randomly generated. Clerc reevaluates and modifies the population structure every L/2 iterations, where Lis the number of links in the population. Summarizing, the Tribes algorithm is given by steps as follows [33–41]. 3.1.1. Initialization of swarm Set iteration g = 1. Initialize a population of i = 1, . . ., M particles or cells (real-valued n-dimensional solution vectors) with random values generated according to a uniform probability distribution in the n dimensional problem space. In this step, initialize the entire solution vector population in the given upper and lower limits of the search space. 3.1.2. Evaluation of each particle in the swarm Evaluate the fitness value (objective function to be minimized in this work) of each particle. 3.1.3. Swarm moves Determine the promising search areas using hyper-spheres and position update of particles. Unlike the classical PSO [31,32], Tribes algorithm has no explicit velocity. 3.1.4. Adaptation scheme Some concepts are necessary to elaborate the rules for removing particles and creating new tribes. A particle that improves its performance is said to be good; otherwise, a particle is bad. In a given set of particles (either a tribe or igroup) a particle is the best if its performance is better than all others. A particle is the worst if it is bad and its performance is worse than all others. In both cases, ties are broken at random. A tribe is said to be bad if its best particle is bad. A tribe is said to be good if at least half of its particles are good. When at least one tribe is bad, a new empty tribe T is generated.

637

L.d.S. Coelho, D.L.d.A. Bernert / Chaos, Solitons and Fractals 42 (2009) 634–640

From time to time social adaptation is performed, where: (i) Each tribe can have the status: good, neutral or bad. For each tribe, the count of how many particles have improved their best performance (m) is made, such that: if m = 0, the tribe is said to be bad; if (m = size(tribe)/2) then the tribe is said to be good. (ii) If there is at least one bad tribe, a new tribe is generated. (iii) For each bad tribe: (a) the best particle is found and ‘generates’ a new particle in the new tribe using uniform distribution between the boundaries of decision variables in the new tribe; (b) this new particle is added to the i group of this best particle. (iv) Each particle can have the status: good or neutral. For each particle of the new tribe, the i group contains: (a) the particle itself; (b) the best particle of the new tribe; (c) the particle that has generated it (a case of symmetry; it is noted that with the i group concept, non-symmetrical relationships could be built). (v) For each ‘good’ tribe: (a) the worst particle is removed (if it is bad); (b) in all igroups it is replaced by the best one of the tribe (general case) or by the best particle of the i group of the removed particle (mono-particle tribe case); (c) its i group (except of course this removed particle itself) is merged to the one that replaces it.

3.1.5. Stopping criterion Set the generation number for g = g + 1. Proceed to step of Evaluation of each particle in the swarm until a stopping criterion is met, usually a maximum number of iterations or maximum number of evaluations of objective function. 3.2. Modified Tribes algorithm In step iii(a) of classical Tribes algorithm, for each bad tribe, the best particle is found and ‘generates’ a new particle using uniform distribution in the new tribe. In this paper is proposed a modified scheme for the step iii(a), which is outlined below. This scheme uses the step iii(a) of classical Tribes in iteration g = 1. After, this scheme called MTribes uses a truncated chaotic Zaslavskii map, Z, between [0, 1] (space search of Kp, Ki, Kd) in following iterations. In this case, the generation of a new particle is stated by the Zaslavskii map [49] given by:

wðtÞ ¼ mod½wðt  1Þ þ v þ a  zðtÞ; 1;

ð10Þ

zðtÞ ¼ cosð2p  wðt  1ÞÞ þ er zðt  1Þ;

ð11Þ

where t is the sample, mod is the modulus after division. The Zaslavskii map shows a strange attractor with the largest Lyapunov exponent for v = 400, r = 3, and a = 12.6695 [50]. In this case, the values of z(t) 2 [1.0512, 1.0512]. The choice of the chaotic Zaslavskii map in MTribes design is justified theoretically by its unpredictability, i.e., by its spread-spectrum characteristic and large Lyapunov exponent (a quantitative measure of chaos) [51,52]. 4. Results of numerical simulation In this section, we will illustrate the synchronization PID controller design for the above two Lozi systems given by Eqs. (6) and (8) with different initial value conditions x1(0) = x(0) = 0.1, and y1(0) = y(0) = 0.6. We solved the optimization problem with N = 25 and Ts = 1 s. In this section, to verify and demonstrate the effectiveness of the proposed method, we discuss the simulation results of synchronization between the master and slave states for different optimization approaches including classical Tribes and PSO [31,32]. Table 1 Convergence results for synchronization of discrete chaotic systems in 100 runs. Optimization method

Minimum F

Mean F

Maximum F

Standard deviation of F

Tribes MTribes PSO

2.1934 2.1776 2.2761

2.8578 2.6290 4.1784

4.0994 3.5696 10.8014

0.3526 0.2651 1.4913

638

L.d.S. Coelho, D.L.d.A. Bernert / Chaos, Solitons and Fractals 42 (2009) 634–640

Table 2 Best results of PID controller gains and performance data using optimization method in 100 runs. Parameter

Tribes

MTribes

PSO

Kp Ki Kd Mean of error signal Variance of control signal F

0.0774 0.0208 0.3385 0.0660 0.0039 2.1934

0.5027 0.1263 0.0096 0.0672 0.0032 2.1776

0.3427 0.0795 0.3273 0.0689 0.0072 2.2761

Each optimization method was implemented in Matlab (MathWorks). All the programs were run on a 3.2 GHz Pentium IV processor with 2 GB of random access memory. In each case study, 100 independent runs were made for each of the optimization methods involving 100 different initial trial solutions for each optimization method. In this paper, the optimization approaches are adopted using 3000 cost function evaluations in each run. The lower and upper bounds of the search space used in optimization methods were (Kp, Ki, Kd) 2 [0, 1]. The adopted search space is very limited. In this context, the tuning of (Kp, Ki, Kd) is very sensitive for small variations in its gains. The Tribes and MTribes approaches are free from tuning of population size and is based on different sized groups of particles. On the other hand, the PSO needs tuning of control parameters. In simulation, we set the cognitive and social components in PSO are equal to 2.05, the inertia weight is decreased over time linearly (from 0.9 to 0.4), the swarm size is 15, and the adopted stopping criterion is 200 generations. Convergence results obtained by applying the Tribes, MTribes, and PSO approaches for synchronization of two Lozi chaotic systems are summarized in Table 1. It can be seen from Table 1 that with the same maximum number of evaluations of objective function, MTribes obtained better mean and minimum F values than classical Tribes and PSO algorithms.

1.2

0.3

x(k) y(k)

1

0.2

0.8 0.1

error signal, e

0.4 0.2 0

0 -0.1 -0.2

-0.2 -0.3 -0.4 -0.4

-0.6 -0.8

0

5

10

15

20

-0.5

25

0

5

10

15

k

k

(a) states responses

(b) error signal

0.2 0.15 0.1

control signal, u

state response

0.6

0.05 0 -0.05 -0.1 -0.15 -0.2 0

5

10

15

k

(c) control signal Fig. 1. Best result using MTribes.

20

25

20

25

L.d.S. Coelho, D.L.d.A. Bernert / Chaos, Solitons and Fractals 42 (2009) 634–640

639

Table 2 presents the best results of the PID controller gains and performance data obtained using Tribes and particle swarm optimization approaches. Fig. 1 shows the state responses of the master and slave systems using the resulting PID controller gains obtained by MTribes. 5. Conclusion The Tribes algorithm is an adaptive and robust parameter searching technique. In addition, the Tribes is a parameter-free particle swarm system paradigm, where the population is divided in subpopulations, each maintaining its own order and structure. In this paper, the classical Tribes, MTribes and PSO approaches to tune PID controller gains were evaluated in a synchronization application of two Lozi systems with different initial conditions. Simulation results based on synchronization of two Lozi systems demonstrated the effectiveness and efficiency of proposed MTribes. However, more experiments, especially with larger optimization problems, may, however, be needed in order to confirm such good performance of MTribes. Further study is being conducted to investigate the effect of chaos incorporation into MTribes further and apply the MTribes for solving the tuning problems of fuzzy, sliding mode and PID controllers. Acknowledgments This work was supported by the National Council of Scientific and Technologic Development of Brazil – CNPq – under Grant 309646/2006-5/PQ. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

Strogatz SH. Nonlinear dynamics and chaos. Massachusetts: Perseus Publishing; 2000. Kantz H, Schreiber T. Nonlinear time series analysis. Cambridge: Cambridge University Press; 2004. Parker TS, Chua LO. Practical numerical algorithms for chaotic system. Berlin: Springer; 1989. Peitgen HO, Jürgens H, Saupe D . Chaos and fractals: new frontiers of science. 2nd ed. New York (NY): Springer; 2004. Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64(8):821–4. Ruan J, Li LJ. An improved method in synchronization of chaotic systems. Commun Nonlinear Sci Numer Simul 1998;3(3):140–3. Zeng XP, Ruan J, Li LJ. Synchronization of chaotic systems by feedback. Commun Nonlinear Sci Numer Simul 1999;4(2):162–5. Yang T, Chua LO. Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication. IEEE Trans Circuits Syst I 1997;44(10):976–88. Li ZG, Wen CY, Soh YC, Xie WX. The stabilization and synchronization of Chua’s oscillators via impulsive control. IEEE Trans Circuits Syst I 2001;48(11):1351–5. Sun JT, Zhang YP. Impulsive control of a new chaotic system. Math Comput Simul 2004;64(6):669–77. Xue YJ, Yang SY. Synchronization of generalized Hénon map by using adaptive fuzzy controller. Chaos, Solitons & Fractals 2003;17(4):717–22. Feng G, Chen G. Adaptive control of discrete-time chaotic systems: a fuzzy control approach. Chaos, Solitons & Fractals 2005;23(2):459–67. Nastaran V, Vahid JM. Adaptive fuzzy synchronization of discrete-time chaotic systems. Chaos, Solitons & Fractals 2006;28(4):1029–36. Lee WK, Hyun CH, Lee H, Kim E, Park M. Model reference adaptive synchronization of T–S fuzzy discrete chaotic systems using output tracking control. Chaos, Solitons & Fractals 2007;34(5):1590–8. Kim JH, Park CW, Kim E, Park M. Adaptive synchronization of T–S fuzzy chaotic systems with unknown parameters. Chaos, Solitons & Fractals 2005;24(5):1353–61. Yang Y, Ma XK, Zhang H. Synchronization and parameter identification of high-dimensional discrete chaotic systems via parametric adaptive control. Chaos, Solitons & Fractals 2006;28(1):244–51. Elabbasy EM, Agiza HN, El-Dessoky MM. Adaptive synchronization of Lü system with uncertain parameters. Chaos, Solitons & Fractals 2004;21(3):657–67. Chen S, Lu J. Synchronization of an uncertain unified chaotic system via adaptive control. Chaos, Solitons & Fractals 2002;14(4):643–7. Wen G, Wang QG, Lin C, Li G, Han X. Chaos synchronization via multivariable PID control. Int J Bifurcat Chaos 2007;17(5):1753–8. Aguilar-López R, Martinez-Guerra R. Partial synchronization of different chaotic oscillators using robust PID feedback. Chaos, Solitons & Fractals 2007;33(2):572–81. Chen HC, Chang JF, Yan JJ, Liao TL. EP-based PID control design for chaotic synchronization with application in secure communication. Expert Syst Appl 2008;34(2):1169–77. Hung ML, Lin JS, Yan JJ, Liao TL. Optimal PID control design for synchronization of delayed discrete chaotic systems. Chaos, Solitons & Fractals 2008;35(4):781–5. Chang WD. PID control for chaotic synchronization using particle swarm optimization. Chaos, Solitons & Fractals 2009;39(2):910–17. Coelho LS, Bernert DLA. An improved harmony search algorithm for synchronization of discrete-time chaotic systems. Chaos, Solitons & Fractals 2009;41(5):2526–32. Engelbrecht AP. Fundamentals of computational swarm intelligence. New York: Wiley; 2006. AlRashidi MR, El-Hawary ME. A survey of particle swarm optimization applications in power systems operations. Electr Pow Compo Syst 2006;34(12):1349–57. Sierra MR, Coello CAC. Multi-objective particle swarm optimizers: a survey of the state-of-the-art. Int J Comput Intell Res 2006;2(3):287–308. Parsopoulos KE, Vrahatis MN. Recent approaches to global optimization problems through particle swarm optimization. Nat Comput 2002; 1(2–3):235–306. Kennedy JF, Eberhart RC, Shi Y. Swarm intelligence. San Francisco (CA): Morgan Kaufman; 2001. Song MP, Gu GC. Research on particle swarm optimization: a review. In: Proceedings of the third international conference on machine learning and cybernetics, Shanghai, China; 2004. p. 2236–41. Eberhart RC, Kennedy JF. A new optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science, Nagoya, Japan; 1995. p. 39–43. Kennedy JF, Eberhart RC. Particle swarm optimization. In: Proceedings of the IEEE international conference on neural networks, vol. IV, Perth, Australia; 1995. p. 1942–8. Clerc M. Tribes – a parameter free particle swarm optimizer, Math stuff for PSO; 2002. Available from: http://www.mauriceclerc.net. Clerc M. Tribes – un exemple d’optimisation par essaim particulaire sans parametres de contrôle. In: Proceedings of the Optimisation par Essaim Particulaire, OEP, France; 2003.

640

L.d.S. Coelho, D.L.d.A. Bernert / Chaos, Solitons and Fractals 42 (2009) 634–640

[35] Clerc M. Stagnation analysis in particle swarm optimization or what happens when nothing happens, Technical report CSM-460, Department of Computer Science, University of Essex; August 2006. [36] Clerc M. Particle swarm optimization. London: ISTE; 2006. [37] Onwubolu GC. Optimization of milling operations for the selection of cutting conditions using Tribes. Proc Inst Mech Eng B: J Eng Manuf 2005;B10:761–71. [38] Onwubolu GC. Performance-based optimization of multi-pass face milling operations using Tribes. Int J Mach Tool Manuf 2006;46(7–8):717–27. [39] Onwubolu GC. Tribes application to the flow shop scheduling. In: Onwubolu GC, Babu BV, editors. New optimization techniques in engineering. Germany: Springer; 2004. p. 515–35. [40] Chen K, Li T, Cao T. Tribe-PSO: a novel global optimization algorithm and its application in molecular docking. Chemometr Intell Lab Syst 2006; 82(1–2):248–59. [41] Coelho LS, Alotto P. Tribes optimization algorithm applied to Loney’s solenoid design. In: Proceedings of 13th biennial IEEE conference on electromagnetic field computation, Athens, Greece; 2008. [42] Åström KJ, Hägglund T. PID controllers: theory, design and tuning. Research Triangle Park (NC): Instrument Society of America; 1995. [43] Cominos P, Munro N. PID controllers: recent tuning methods and design to specification. IEEE Proc Control Theor Appl 2002;149(1):46–53. [44] Coelho LS, Coelho AAR. Automatic tuning of PID and gain scheduling PID controllers by a derandomized evolution strategy. Artif Intell Eng Des Anal Manuf 1999;13(5):341–9. [45] Lozi R. Un attracteur étrange du type attracteur de Hénon. J Phys Colloque C5 1978;39(8):9–10 [in French]. [46] Hénon M. A two-dimensional mapping with a strange attractor. Commun Math Phys 1976;50(1):69–77. [47] Cooren Y, Clerc M, Siarry P. Tribes – a parameter-free particle swarm optimization. In: Proceedings of seventh EU/meeting on adaptive, self-adaptive, and multi-level metaheuristics, Paris, France; 2006. [48] Mendes R. Population topologies and their influence in particle swarm performance, Ph.D. thesis, University of Minho, Portugal; 2004. [49] Zaslavskii GM. The simplest case of a strange attractor. Phys Lett A 1978;69:145–7. [50] Coelho LS. A quantum particle swarm optimizer with chaotic mutation operator. Chaos, Solitons & Fractals 2008;37(5):1409–18. [51] Grossberger P, Procaccia I. Measuring the strangeness of strange attractors. Physica D 1983;9(1–2):189–208. [52] Russell DA, Hanson JD, Ott E. Dimension of strange attractors. Phys Rev Lett 1980;45(14):1175–80.