differential evolution algorithm-based parameter estimation for chaotic systems

Chaos, Solitons and Fractals 39 (2009) 2110–2118

www.elsevier.com/locate/chaos

Differential evolution algorithm-based parameter estimationfor chaotic systems

Bo Peng b, Bo Liu a,b,*, Fu-Yi Zhang b, Ling Wang b

a Center for Chinese Agricultural Policy, Institute of Geographical Sciences and Natural Resource Research,

Chinese Academy of Sciences, Beijing 100101, Chinab Department of Automation, Tsinghua University, Beijing 100084, China

Accepted 21 June 2007

Abstract

Parameter estimation for chaotic systems is an important issue in nonlinear science and has attracted increasinginterests from various research fields, which could be essentially formulated as a multidimensional optimization prob-lem. As a novel evolutionary computation technique, differential evolution algorithm (DE) has attracted much atten-tion and wide applications, owing to its simple concept, easy implementation and quick convergence. However, to thebest of our knowledge, there is no published work on DE for estimating parameters of chaotic systems. In this paper, aDE approach is applied to estimate the parameters of Lorenz system. Numerical simulation and the comparisons dem-onstrate the effectiveness and robustness of DE. Moreover, the effect of population size on the optimization perfor-mances is investigated as well.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

As a characteristic of nonlinear systems, chaos is a bounded unstable dynamic behavior that exhibits sensitive depen-dence on initial conditions and includes infinite unstable periodic motions. Control and synchronization of chaotic sys-tems have been investigated intensely in various fields during recent years [1–6]. Many of the proposed approaches onlywork under the assumption that the parameters of chaotic systems are known in advance. Nevertheless, in real world,the parameters may be difficult to determine due to the complexity of chaotic systems. Therefore, parameter estimationfor chaotic systems has become a hot topic in the past decade [7–16].

Some studies focused on synchronization-based methods for parameter estimation. In [7,8], the parameters of agiven chaotic dynamic model were estimated by minimizing the average synchronization error using a scalar time series.In [9], a feedback-based synchronization method and an adaptive control method (suggested in [10]) were both intro-duced to estimate parameters for several chaotic systems. Simulation demonstrated that this kind of combination waseffective and reasonably robust under noisy environment. The approach proposed in [9] was also used in [11] to estimateone parameter of the transmitter for chaotic signal communication to guarantee security. Alvarez et al. in [12] estimated

0960-0779/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2007.06.084

* Corresponding author. Address: Department of Automation, Tsinghua University, Beijing 100084, China. Tel.: +86 10 62783125;fax: +86 10 62786911.

E-mail address: [email protected] (B. Liu).

mailto:[email protected]

B. Peng et al. / Chaos, Solitons and Fractals 39 (2009) 2110–2118 2111

parameter from the two-valued symbolic sequences generated by iterations of quadratic map when its initial value wasknown. Wu et al. in [13] further studied this kind of symbolic sequences and identified parameters even when the initialvalue was unknown by adding some features of the chaotic orbits into a bi-search means. In [14], several feedback con-trol gains were introduced to synchronize the model system and the original physical system. The feedback gains werealso treated as design parameters, and then all the parameters were estimated via a minimization procedure of the syn-chronization errors. In [15], an adaptive control-based synchronization method was presented for parameter identifica-tion for a modified chaotic Van der Pol-Duffing oscillator. Besides the methods based on chaotic synchronization andcontrol theories, the linear associative memory method was applied to estimate parameters for the logistic map in [16].Additionally, a genetic algorithm (GA) and particle swarm optimization (PSO) were adopted to estimate parameters forLorenz chaotic system, respectively, in [17,18].

Recently, a new evolutionary technique, differential evolution (DE), has been proposed [22] as an alternative togenetic algorithm (GA) [23] and particle swarm optimization (PSO) [19–21,24] for unconstrained continuous optimiza-tion problems. Although the original objective in the development of DE was for solving the Chebychev polynomialproblem, it has been found to be an efficient and effective solution technique for complex functional optimization prob-lems. In a DE system, a population of solutions is initialized randomly, which is evolved to find optimal solutionsthrough the mutation, crossover, and selecting operation procedures. Compared with GA and PSO, DE has someattractive characteristics. It uses simple differential operator to create new candidate solutions and one-to-one compe-tition scheme to greedily select new candidate, which work with real numbers in natural manner and avoid complicatedgeneric searching operators in GA. It has memory, so knowledge of good solutions is retained in current population,whereas in GA, previous knowledge of the problem is destroyed once the population changes and in PSO, a secondaryarchive is needed. It also has constructive cooperation between individuals, individuals in the population share infor-mation between them. Due to the simple concept, easy implementation and quick convergence, nowadays DE hasattracted much attention and wide applications in different fields mainly for various continuous optimization problems[25]. However, to the best of our knowledge, there is no research on DE for parameter estimation of chaotic systems.

In this paper, parameter estimation for chaotic systems is formulated as a multidimensional optimization problem,and a DE approach is implemented to solve the problem. To the best of our knowledge, this is the first research to applyDE to estimate parameters of chaotic systems. Numerical simulation based on Lorenz system and comparisons withresults obtained by PSO [18] and GA [17,18] demonstrate the effectiveness, efficiency and robustness of DE.

The remainder of this paper is organized as follows. The parameter estimation is formulated as a multidimensionaloptimization problem in Section 2. Section 3 presents a brief review and an implementation for DE. Numerical simu-lation and comparisons are provided in Section 4. Finally, we conclude in Section 5 with a brief summary of results.

2. Problem formulation

Considering the following n-dimensional chaotic system:
_X ¼ F ðX ;X 0; h0Þ; ð1Þ
where X = (x1, x2, . . ., xn)T 2 Rn denotes the state vector, X0 denotes the initial state and h0 = (h10, h20, . . ., hd0)T is a setof original parameters.

When estimating the parameters, suppose the structure of the system is known in advance, and thus the estimatedsystem can be described as follows:

_Y ¼ F ðY ;X 0; hÞ; ð2Þ

where Y = (y1, y2, . . ., yn)T 2 Rn denotes the state vector, and h = (h1, h2, � � �, hd)T is a set of estimated parameters.Therefore, the problem of parameter estimation can be formulated as the following optimization problem:

min J ¼ 1

M

XM

k¼1

kX k � Y kk2 by searching suitable h� ð3Þ

where M denotes the length of data used for parameter estimation, Xk and Yk (k = 1,2, . . ., M) denote state vectors ofthe original and the estimated systems at time k, respectively.

Obviously, the parameter estimation for chaotic systems is a multidimensional continuous optimization problem,where the decision vector is h and the optimization goal is to minimize J. The principle of parameter estimation forchaotic systems in sense of optimization can be illustrated with Fig. 1.

Due to the unstable dynamic behavior of chaotic systems, the parameters are not easy to obtain. In addition, thereare often multiple variables in the problem and multiple local optima in the landscape of J, so traditional optimizationmethods are easy to trap in local optima and it is difficult to achieve the global optimal parameters.

Adjusting θ Calculating J

MYY ,...,1

MXX ,...,1

-

),,( 00 θXXFX =

),,( 0 θXYFY =

Algorithm

X0

+.

.

Fig. 1. The principle of parameter estimation for chaotic systems.

2112 B. Peng et al. / Chaos, Solitons and Fractals 39 (2009) 2110–2118

3. Differential evolution algorithm

DE is a population-based evolutionary computation technique, which uses simple differential operator to create newcandidate solutions and one-to-one competition scheme to greedily select new candidate. The theoretical framework ofDE is very simple and DE is easy to be coded and implemented with computer. Besides, it is computationally inexpen-sive in terms of memory requirements and CPU times. Thus, nowadays DE has attracted much attention and wideapplications in various fields [25].

In DE, it starts with the random initialization of a population of individuals in the search space and works on thecooperative behaviors of the individuals in the population. Therefore, it finds the global best solution by utilizing thedistance and direction information according to the differentiations among population. However, the searching behav-ior of each individual in the search space is adjusted by dynamically altering the differentiation’s direction and steplength in which this differentiation performs.

The ith individual in the d-dimensional search space at generation t can be represented as Xi(t) = [xi,1, xi,2, . . ., xi,d],(i = 1, 2, . . ., NP, where NP denotes the size of the population). At each generation t, the mutation and crossover oper-ators are applied on the individuals, and a new population arises. Then, selection takes place, and the correspondingindividuals from both populations compete to comprise the next generation.

For each target individual Xi(t), according to the mutation operator, a mutant vector Vi(t + 1) =[vi,1(t + 1), . . ., vi,d(t + 1)] is generated by adding the weighted difference between a defined number of individuals ran-domly selected from the previous population to another individual, which is described by the following equation:

V iðt þ 1Þ ¼ X bestðtÞ þ F ðX r1ðtÞ � X r2ðtÞÞ ð4aÞ

where r1, r2 2 {1,2, . . ., N} are randomly chosen and mutually different and also different from the current index i.F 2 [0,2] is constant called scaling factor which controls amplification of the differential variation Xr1(t) � Xr2(t),and NP is at least 4 so that the mutation can be applied. Xbest(t), the base vector to be perturbed, is the best memberof the current population so that the best information could be shared among the population.

After the mutation phase, the crossover operator is applied to increase the diversity of the population. Thus, for eachtarget individual Xi(t), a trial vector Ui(t + 1) = [ui,1(t + 1), . . ., ui,d(t + 1)] is generated by the following equation:

ui;jðt þ 1Þ ¼vi;jðt þ 1Þ; if ðrandðjÞ 6 CRÞ or j ¼ randnðiÞ;xi;jðtÞ; otherwise

�j ¼ 1; 2; . . . ; d; ð4bÞ

where rand(j) is the jth independent random number uniformly distributed in the range of [0, 1]. randn(i) is a randomlychosen index from the set {1,2, . . ., d}. CR 2 [0,1] is constant called crossover parameter that controls the diversity ofthe population.

Following the crossover operation, the selection arises to decide whether the trial vector Ui(t + 1) would be a memberof the population of the next generation t + 1. For a minimum optimization problem, Ui(t + 1) is compared to the ini-tial target individual Xi(t) by the following one-to-one based greedy selection criterion:

X iðt þ 1Þ ¼Uiðt þ 1Þ; if F ðUiðt þ 1ÞÞ < F ðX iðtÞÞ;X iðtÞ; otherwise;

�ð4cÞ

where F is the objective function under consideration, Xi(t + 1) is the individual of the new population. The proceduredescribed above is considered as the standard version of DE, and it is denoted as DE/best/1/bin. Several variants of DEhave been proposed, depending on the selection of the base vector to be perturbed, the number and selection of thedifferentiation vectors and the type of crossover operators [25].

The key parameters in DE are NP (size of population), F (scaling factor) and CR (crossover parameter). Properconfiguration of the above parameters would achieve good tradeoff between the global exploration and the local


exploitation so as to increase the convergence velocity and robustness of the search process. Some basic principles havebeen given for selecting appropriate parameters for DE [25]. In general, the population size NP is choosing from 5 Æ d to10 Æ d (number of dimension). F and CR lies in the range of [0.4, 1.0] and [0.1, 1.0], respectively.

The procedure of standard DE is summarized as follows:

Step 1: Randomly initialize the population of individual for DE, where each individual contains d variables (i.e.,d = N).

Step 2: Evaluate the objective values of all individuals, and determine Xbest which has the best objective value.Step 3: Perform mutation operation for each individual according to Eq. (4a) in order to obtain each individual’s

mutant counterpart.Step 4: Perform crossover operation between each individual and its corresponding mutant counterpart according to

Eq. (4b) in order to obtain each individual’s trial individual.Step 5: Evaluate the objective values of the trial individuals.Step 6: Perform selection operation between each individual and its corresponding trial counterpart according to Eq.

(4c) so as to generate the new individual for the next generation.Step 7: Determine the best individual of the current new population with the best objective value. If the objective value

is better than the objective value of Xbest, then update Xbest and its objective value with the value and objectivevalue of the current best individual.

Step 8: If a stopping criterion is met, then output Xbest and its objective value; otherwise go back to Step (3).

4. Simulation and comparisons

As a typical chaotic system, Lorenz system is employed as an example in this paper. The mathematical description ofLorenz system is as follows:

_x1 ¼ aðx2 � x1Þ;_x2 ¼ bx1 � x1x3 � x2;

_x3 ¼ x1x2 � cx3;

8><>: ð5Þ

where a = 10, b = 28, c = 8/3 are the original parameters.In our simulation, the original Lorenz system firstly evolves freely from a random initial state. After a period of tran-

sient process, a state vector is selected as the initial state X0 for parameter estimation as shown in Fig. 1. Then successiveM states (M = 300) of both the original system and the estimated system are used to calculate J. In DE, the maximumgeneration number is set to 100 (set as stopping condition), population size is set to 20, 40 and 120 when the number ofunknown parameters is 1, 2 and 3, respectively. The searching ranges of a, b and c are set to [9, 11], [20, 30] and [2, 3],respectively. Moreover, a PSO in [18] and a binary-coded GA in [17] are used for comparison, where all the parametersare the same as those used in the literature [18,17] (the length of the chromosome L is set to 20, crossover rate Pr = 0.8and mutation rate Pm = 0.1). To perform fair comparison, the same computational effort is used in DE, PSO and theGA. That is, the maximum generation, population size and searching range of the parameters in the DE are the same asthose in PSO and GA.

4.1. Simulation on one-dimensional parameter estimation

Firstly, we consider one-dimensional parameter estimation. That is, only one parameter among a, b and c isunknown and needs to be estimated. Table 1 lists the statistical results obtained by DE, PSO and GA for three cases,where each algorithm is implemented 20 times independently for each case.

From Table1, it can be seen that the best results (estimated values) obtained by DE, PSO and the GA are very close tothe true values. Nevertheless, the average and worst results obtained by DE greatly outperform those obtained by GA.

4.2. Simulation on two-dimensional parameter estimation

Secondly, the two-dimensional parameter estimation is considered. That is, two of the three parameters (a, b and c)are unknown and need to be estimated. Tables 2–4 list the statistical results by DE, PSO and GA for three cases, whereeach algorithm is implemented 20 times independently for each case.

Table 1Statistical results of different methods for one-dimensional parameter estimation

a J b J

Average result

DE 10.00000029 0.00000000031 28.00000033 0.00000000PSO 10.000000 0.000000 28.000000 0.000000GA 10.000004 0.000039 28.142774 9141.76472

Best result

DE 9.999999825 0.00000000002 28.00000005 0.00000000PSO 10.000000 0.000000 28.000000 0.000000GA 10.000000 0.000000 28.000001 0.000002

Worst result

DE 9.99999552 0.00000000098 28.00000212 0.00000000PSO 10.000000 0.000000 27.999999 0.000000GA 10.000076 0.000726 28.75000 45063.5572

c J

Average result

DE 2.666666706 0.00000000003PSO 2.666667 0.000000GA 2.652081 10829.504876

Best result

DE 2.66666668 0.00000000000PSO 2.666667 0.000000GA 2.666667 0.000014

Worst result

DE 2.666666521 0.00000000097PSO 2.666667 0.000000GA 2.624999 30714.343450

Table 2Statistical results of different methods for two-dimensional parameter estimation (aand bare unknown)

a b J

Average result

DE 9.999994871 28.00000232 0.00000000037PSO 10.003777 27.998373 0.016936GA 9.863719 28.089400 37.350294

Best result

DE 10.0000072 27.99999869 0.000000000042PSO 9.999778 28.000081 0.000191GA 10.030342 27.987013 0.231930

Worst result

DE 9.999904133 28.00004134 0.000000092PSO 10.017997 27.992305 0.081147GA 9.197794 28.479586 154.011912


From Table 2 to Table 4, it can be seen that the best, average and the worst results obtained by DE all outperformthose obtained by PSO and GA, respectively. Especially, in Table 2 and 3, the worst results obtained by DE are evenbetter than the best results obtained by the GA.

4.3. Simulation on three-dimensional parameter estimation

Subsequently, we further consider three-dimensional parameter estimation where all the parameters in Lorenz sys-tem are unknown and need to be estimated. Table 5 lists the statistical results obtained by DE, PSO and the GA, whereeach algorithm is implemented 20 times independently.

Table 3Statistical results of different methods for two-dimensional parameter estimation (a and c are unknown)

a c J

Average result

DE 10.00000414 2.666666795 0.000000000560PSO 9.999580 2.666653 0.007440GA 9.751140 2.654561 455.573793

Best result

DE 10.00000014 2.666666617 0.000000000051PSO 10.000001 2.666667 0.000000GA 10.012075 2.667048 0.206471

Worst result

DE 9.999963944 2.666667459 0.00000000099PSO 9.992503 2.666448 0.059740GA 9.029427 2.624927 1386.977496

Table 4Statistical results of different methods for two-dimensional parameter estimation (b and c are unknown)

b c J

Average result

DE 27.99999584 2.666666221 0.00000000048PSO 28.004684 2.666998 0.597665GA 27.161623 2.616623 4769.4260986

Best result

DE 28.00000035 2.666666692 0.000000000019PSO 27.999417 2.666627 0.000793GA 28.014393 2.667658 0.487866

Worst result

DE 27.99996646 2.666668285 0.00000000095PSO 28.034171 2.669051 2.909609GA 24.953003 2.499762 29226.143472

Table 5Statistical results of different methods for three-dimensional parameter estimation

a b c J

Average result

DE 10.010050 27.993870 2.666551 0.00036PSO 10.018417 27.993390 2.666281 4.18278GA 10.139783 27.742735 2.648585 943.76294

Best result

DE 10.000096 27.999999 2.666664 0.0000002PSO 9.995332 28.007146 2.667013 0.048645GA 10.067167 27.922058 2.663426 4.310715

Worst resultDE 10.054064 27.971791 2.665526 0.0016939PSO 10.608212 27.704424 2.657231 39.406026GA 10.929003 26.127605 2.562049 6461.4801


As shown in Table 5, once again it is clear that the best, average and the worst results obtained by DE are better thanthose obtained by PSO and GA, respectively. And the average result obtained by DE is even better than the best resultobtained by PSO and GA. In addition, the values of estimated parameters obtained by PSO are still very close to the

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9

10

Generation

J

Fig. 2. A typical evolving process of the objective function value J.

10 20 30 40 50 60 70 80 90 1009

9.5

10

10.5

11

11.5

Generation

Est

imat

ion

valu

e of

a

10 20 30 40 50 60 70 80 90 10027.8

28

28.2

28.4

28.6

28.8

Generation

Est

imat

ion

valu

e of

b

Fig. 3. A typical searching process for parameters a and b.


true values of original parameters. So, it is concluded that DE is more effective and robust than PSO and GA to esti-mate parameters for chaotic systems.

4.4. Discussion on searching efficiency and parameter settings of PSO

Firstly, to study the searching efficiency of DE, a case in two-dimensional parameter estimation problem (a and b

need to be estimated) is used as an example. With the same controlling parameters mentioned before, a typical evolvingprocess of the objective function J is illustrated in Fig. 2 and a typical convergence process of both parameter a and b isshown in Fig. 3.

Fig. 2 shows that the value of J decreases very fast to zero, which implies that DE can converge to the global opti-mum very quickly. Moreover, it can be seen from Fig. 3 that both parameter a and b converge to the true values rapidly,which demonstrates the great efficiency of DE to achieve global optimization.

To investigate the effect of swarm size on the performances of the DE, experiments are carried out based on theabove two-dimensional parameter estimation problem with other controlling parameters in DE fixed. Figs. 4 and 5illustrate the effect of population size on the average value of J (20 independent runs are executed) and the total numberof evaluations.

As shown in Fig. 4, when population size is too small, the results are poor because the solution space will not beexplored enough. As population size increases, the results become better at a cost of more fitness evaluations (see

20 40 60 80 100 120

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Population size

Ave

rage

val

ue o

f J

Fig. 4. The average objective value of J obtained by DE with different population size.

20 30 40 50 60 70 80 90 1002000

3000

4000

5000

6000

7000

8000

9000

10000

Population size

Num

ber o

f fitn

ess

eval

uatio

ns

Fig. 5. Total number of evaluations in DE with different population size.


Fig. 5). But there is a threshold, beyond which the results will not be affected in a significant manner. As a consequence,considering both the searching quality and computational effort, it is recommended to choose population size between40 and 60. If more parameters need to be estimated, larger population size is recommended.

5. Conclusion

From the viewpoint of optimization, parameter estimation for chaotic systems was formulated as a multidimen-sional optimization problem in this paper. A novel evolutionary algorithm, DE, was applied to solve such an issue.Numerical simulation and comparisons based on Lorenz system demonstrated the effectiveness, efficiency and robust-ness of DE. To the best of our knowledge, this is the first report of applying DE to estimate parameters for chaoticsystems. The future work is to apply DE for other chaotic systems, and to develop more effective and adaptive DEbased approaches.

Acknowledgements

This work is supported by both National Natural Science Foundation of China (Grant Nos. 60204008, 60374060and 60574072) and National 973 Program (Grant No. 2002CB312200).


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differential evolution algorithm-based parameter estimation for chaotic systems

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