Parameter Identification of Hyperchaotic Chen-Lee System Using Firefly Algorithm

Parameter identification for chaotic systems is one of the crucial issues in nonlinear science, which can be raised as a multi-dimensional optimization problem. In this article, parameter estimation of hyperchaotic Chen-Lee system was performed for the first time using the metaheuristic firefly algorithm. Firefly algorithm is a nature-inspired metaheuristic and optimization algorithm. This algorithm has been provided by Yang and inspired by natural behavior of fireflies of light emission. The efficiency of the firefly algorithm has been compared with the particle swarm optimization (PSO) algorithm. The results of the simulation conducted on the model indicated high accuracy and speed of the firefly algorithm in parameter estimation of hyperchaotic systems.


Introduction
As a characteristic, nonlinear chaotic systems have an unstable dynamic behavior, which is extremely sensitive to initial conditions and includes an infinite number of unstable periodic motions.Over the past years, the control and synchronization of chaotic system have been highly considered by researchers in different fields [1,2].The majority of these methods have been conducted only by assuming that the chaotic system parameters have been previously identified [3].
Production and hosting by ISPACS GmbH.http://www.ispacs.com/journals/jsca/2018/jsca-00096/International Scientific Publications and Consulting Services However, given the system complexity, determining the chaotic system parameters is very difficult in the real world.Therefore, parameter estimation for the chaotic system has become a highly important issue, which requires a large number of studies allocated to the field of chaotic system parameters [4,5].Among these studies are the minimization of the synchronization error using the numerical series [6,7], single feedback-based synchronization and an adaptive control method [8,9].Application of the methods based on evolutionary algorithms for solving the problem of parameter estimation has a long history, and this parameter estimation has been conducted on Lorenz chaotic system.Recently, different metaheuristic algorithms have been provided for parameter estimation in this system [10].In this regard, Dai et al. used genetic algorithm and Yang applied Particle Swarm Optimization (PSO) in for the parameter estimation of Lorenz chaotic system (estimation of one dimension of the parameter was considered) [11,12].The accuracy of particle swarm algorithm for the chaotic system parameter estimation was higher, compared to the genetic algorithm.In another study, a similar particle swarm optimization approach has been used on the chaotic system, in which all the chaotic system parameters were estimated with acceptable accuracy [13].Moreover, Lorenz system parameter was estimated using a new quantum particle swarm optimization [14].This method has higher accuracy levels, compared to the genetic and standard particle swarm algorithms.According to the literature, particle swarm optimization algorithm was first introduced in 1995 through the simulation of social behavior of birds.This algorithm can be applied as a powerful explorer and an optimization tool, which is able to search for nonlinear, non-differential and nonconvex problems [12,14].In a previous study, an algorithm based on the honey bee (artificial bee colony) was provided, inspired by the honey bee feeding behavior, accuracy of which is higher, compared to the genetic algorithm [15].In [16], a new method using the modified particle swarm algorithm has been presented to identify the variable parameters of the chaotic system.Among the advantages of this method are the easy implementation of the algorithm, high-speed convergence and parameter identification in online and offline modes and greater accuracy of the algorithm, compared to PSO and GA algorithms.In [17], using differential evolution algorithm, parameter estimation of Lorenz chaotic system has been conducted by considering the time delay.One of the benefits of this algorithm is its robustness against the white noise.This algorithm is a convenient and powerful tool for the identification of chaotic systems.In [18], parameter estimation for chaotic systems has been performed using the hybrid differential algorithm and honey bee colony.This algorithm, which is simple and could be easily implemented, has been implemented on Lorenz and Chen chaotic system.Moreover, accuracy and speed of this algorithm are higher, compared to other evolutionary algorithms (e.g., differential, honey bee colony, genetic and PSO algorithms).In [19], a hybrid collective intelligence algorithm of particle swarm optimization and ant colony has been used for parameter estimation of Lorenz chaotic system.This algorithm is a powerful tool for the chaotic system parameter estimation with high efficiency and accuracy and very low error, which has been first developed for the parameter estimation of Lorenz systems.Accuracy of this algorithm is higher, compared to the genetic algorithm and PSO, and it can be used for other applications.In [20], improved Cuckoo search optimization algorithm has been used for chaotic system parameter estimation.This algorithm performs the parameter estimation of chaotic systems in noise and noiseless conditions with high precision and reliability and is superior to the genetic algorithm and particle swarm algorithm in terms of accuracy and effectiveness of parameter estimation.Almost all optimization algorithms presented here have some drawbacks, which affect the quality of global optimization solutions.However, among all of these methods, particle swarm algorithm is more suitable for chaotic system parameter estimation due to the simplicity in the execution of the algorithm, rapid convergence and high accuracy.Nevertheless, since the number of unknown parameters is greater for hyperchaotic system parameter estimation, there is a need for an optimization method with higher accuracy and more rapid convergence, compared to PSO algorithm.http://www.ispacs.com/journals/jsca/2018/jsca-00096/International Scientific Publications and Consulting Services The hyperchaotic system is generally defined as a chaotic system with at least two positive Lyapunov exponents and their minimum dimensions include four dimensions.These systems have complex dynamics and because of this structure, they are widely used in cryptography, secure communications and electrical circuits.Numerous methods have been conducted to control and synchronize these systems, however, but no action has been taken regarding parameter estimation by collective intelligence algorithms [21].This article aimed to provide an appropriate method to estimate the parameters of hyperchaotic systems with higher accuracy, compared to similar methods, and to reach the minimum value of objective function with less number of iterations.For this purpose, the firefly algorithm has been applied, which has greater accuracy and convergence rate, compared to other evolutionary algorithms, such as PSO and genetic algorithm.This algorithm is apparently a more appropriate tool to estimate the parameters of hyperchaotic systems.Therefore, the firefly algorithm was used in this paper for the first time to identify the parameters of hyperchaotic Liu systems.These chaotic systems are recognized as a multi-dimensional optimization problem.Firefly algorithm is a collective intelligence algorithm and has been designed based on optical behavior patterns of fireflies.In this article, the estimation of parameters of Chen-Lee hyperchaotic system was performed for the first time.This system can be widely used in telecommunication systems, and accurate estimation of its parameters is significantly important.However, high number of parameters in this system increases the complexity of the process.Therefore, a high performance and accurate algorithm must be applied to simultaneously estimate the parameters of Chen-Lee hyperchaotic system with minimum errors.Therefore, firefly algorithm, which is extremely accurate and can compete with other evolutionary algorithms, is used in this research for the first time to solve this problem.Therefore, estimation of the unknown parameters of the hyperchaotic system was carried out with the help of the proposed method, which compared to the particle swarm algorithm has more computational accuracy and convergence rate.Results of the study revealed the effectiveness of this method.
The article is composed of five sections.After the introduction, hyperchaotic systems and mathematical model of Chen-Lee system are introduced in the second section.The third section describes the firefly algorithm, and the simulation results of the proposed method are stated in the fourth section.Finally, conclusion is provided in the last section.

Statement of the Problem
To introduce the hyperchaotic Chen-Lee system, the structure of nonlinear chaotic systems is initially introduced.Afterwards, we state the problem of the hyperchaotic system parameter estimation in the form of a multi-dimensional optimization problem.

Identification of Nonlinear Chaotic Systems
The identification process of nonlinear systems includes the approximation of the dynamic behaviors of the system.An n-dimensional nonlinear system can be generally explained by the following equation: X ̇= F(X, X 0 , I 0 ) (2.1) In the above equation, X is the state vector of the system; X 0 refers to the initial conditions of the state vector; I 0 is the unknown parameter vector of the system and F is a nonlinear function.The estimated model of the above nonlinear system is displayed, as follows: In the above equation, X ̇ is the estimation of the state vector of the system and  ̂0 is an estimation of the unknown parameters of the system.Collective intelligence algorithms search for the best unknown parameters of the system based on an objective function.Mean Square Error (MSE) is used as the objective function to estimate the nonlinear systems.Therefore, the objective function is defined, as follows: (2.3) In the above equation, N represents the number of samples.X K and X ̂K indicate the main state vector and estimated system in K time, respectively.MSE should be minimized to identify and estimate the parameters of Liu system.In addition, the firefly optimization algorithm is used to minimize MSE in this article.In the following figure, the block diagram of method used to estimate the unknown parameters of a nonlinear chaotic system is provided.First, an initial condition for the state vector is given to the real system and estimated model.Error is made from the output of both the real system and estimated model and MSE is given to the firefly algorithm as an objective function.

Mathematical Model of Chen-Lee System
Chen and Lee introduced a new chaotic system with the title of "Chen-Lee system" in 2004.This system is composed of three nonlinear differential equations, as follows: ẋ= −yz + ax ẏ= xz + by (2.4) ) xy + cx In the following, there are two important conditions to obtain the hyperchaotic system.The first condition is that there should be at least four minimum dimensions in a hyperchaotic, which leads to the requirement of four independent first-order differential equations.The second condition is that at least two of these equations should be unstable and one should be a nonlinear function.By introducing a nonlinear feedback controller to the third equation of relationship (2.4), the dynamic system was obtained by the following equation: ẋ= −yz + I 1 x ẏ= xz + I 2 ż= I 3 xy+I 4 z+I 5 w (2.5) ẇ= dx+ I 7 yz +I 8 w In this equation, parameter d is a constant value and determines the dynamic behavior of the system.Therefore, the controller w turns the chaotic Chen-Lee system into a four-dimensional system with four Lyapunov exponents.This system is introduced as a hyperchaotic system [22].In the above equation, x, y, z and w are the state variables, and I i (i=1,…, 8) and d are the positive parameters of the hyperchaotic Chen-Lee system.Therefore, this hyperchaotic system has eight positive parameters.http://www.ispacs.com/journals/jsca/2018/jsca-00096/International Scientific Publications and Consulting Services

Firefly Algorithm
Firefly algorithm is a nature-inspired metaheuristic algorithm.It is an optimization algorithm, inspired by the natural behavior of fireflies of light emission, and has been provided by Yang.Although the firefly algorithm is very similar to other collective intelligence algorithms and could be easily implemented, it is very effective and more applicable for solving many optimization problems, compared to the other conventional algorithms.Firefly algorithm follows three laws, as follows: First law: Fireflies are unisexual; therefore, gender plays no role in this regard.
Second law: Attractiveness of each firefly is proportional to its brightness.For two fireflies, the firefly with less light is attracted by the firefly with more light.The amount of attractiveness is proportional to brightness and has an inverse relationship with distance.By increasing the distance between two fireflies, the attractiveness rate decreases.If no firefly is brighter than the others, that particular firefly randomly selects the movement towards another firefly.

Third law:
The brightness rate of each firefly is specified by determining the objective function value, which should be optimized.Attractiveness of a firefly is obtained from the following equation: (3.6)In this equation, β 0 is the initial attractiveness in r = 0, γ is light absorption coefficient and r is the distance between two fireflies.The distance between two fireflies i and j in the situation x i and x j is obtained from the following equation:   =∥   −   ∥ 2 (3.7)However, r distance can be calculated using other distance criteria.The movement of firefly I, which is attracted by firefly j, is calculated using the following equation:   =   +  0  − 2 (  −   ) +   (3.8)The first term is the current situation of the firefly and the second term is related to the attraction law of firefly i.With regard to the evaluation of light intensity, the firefly with less light moves towards the firefly with more brightness.The third term shows the random motion of firefly in cases where there is no superiority between two fireflies in terms of light absorption.Pseudo-code for the firefly algorithms is, as follows: Step 1: Define the objective function f(x), x= (x 1 , x 2 ,…, x d ), which is the MSE in this problem.
Step 2: Initialize the population of fireflies x i and (i=1,2,….,n),randomly selected from the range of defined parameters of Chen-Lee system.
Step 3: Define the light absorption coefficient and initial attractiveness (adjusting the algorithm parameters).
Step 4: For each firefly, such as i (for each parameter, such as i, which has eight dimensions) For each firefly, such as j (for each parameter, such as j, which has eight dimensions) If I i is less than I j (the rate of light intensity of I i in x i is determined by f(x i )) {Comparing the mean square error corresponding to the selected parameters} Firefly i moves towards firefly j using the attraction law of equation (3.8)  Step 5: Evaluation of new fireflies.{Calculation of mean square error corresponding to new parameters} Step 6: Determining the best answer found.{The parameter that minimizes the mean square error is selected as the answer to the problem} Step 7: Repetition of step four in case of not meeting the termination conditions.

Simulation
Simulation has been carried out for estimating the unknown parameters of the hyperchaotic Chen-Lee system.Dynamics of the system's equations are, as follows: In the above equation, x 1 , x 2 , x 3 and x 4 are the system states and I 1 -I 8 are the unknown parameters of the hyperchaotic system.In the simulation process, initial values of the states are considered, as follows: The actual values of the parameters are, as follows:  = [ 1 ,  2 ,  3 ,  4 ,  5 ,  6 ,  7 ,  8 ] = [5, 10, 1/3, 3.8, 0.2,1.2,0.5, 0.05] In this problem, the cost function of equation ( 2.3), which should be minimized, is considered as follows: (4.10) In the above equation, f L represents the cost function of Chen-Lee system, N is the number of samples and x(k) and x ̂ (k) are the actual and estimated values in K time.The range of search variables for the parameters of system (2.4) is considered, as follows: Estimation and identification of hyperchaotic Chen-Lee system parameters were conducted using particle swarm optimization algorithm and firefly algorithm.Each simulation is conducted 20 times with the initial population of 50 and iteration number of 1000.The simulation results can be observed in the following figures: http://www.ispacs.com/journals/jsca/2018/jsca-00096/International Scientific Publications and Consulting Services Figure 2: Estimation of the unknown parameter I 1 using firefly and particle swarm algorithms (I 1 =5) Considering Figure 2, it is observed that parameter I 1 reached its final value in the iteration of about 250 in the PSO method.However, this parameter reached its actual value in the iteration of about 85 in the firefly algorithm (FA).These results could indicate that the estimated value for this parameter in FA method converges faster to the actual amount.
Figure 3: Estimation of the unknown parameter I 2 using firefly and particle swarm algorithms (I 2 =10) Considering Figure 3, we came to the conclusion that parameter I 2 converges to its final value in the iteration of about 490 in the PSO method.Nevertheless, this parameter reached its actual value in the iteration of about 85 in the firefly algorithm (FA).These results demonstrated the efficiency of firefly algorithm, compared to PSO, considering its reduced convergence time.As observed in figures 6-9, the estimated parameters I 5 , I 6 , I 7 and I 8 reached their actual value in the appropriate number of iterations.Furthermore, compared to PSO, its convergence rate is higher since it reaches the actual value of the parameter in less iteration.Moreover, the cost function value in parameter estimation for the firefly algorithm and particle swarm optimization algorithm can be seen in Figure 10.With regard to Figure 10, it is observed that in estimating the parameters of Chen-Lee system, the objective function value was minimized in the iteration of about 550 in the PSO method.Nevertheless, in FA method, the objective function value was minimized in the iteration of about 30.Therefore, the convergence rate of FA method was greater than PSO method because it reached the actual value of the parameters in less iteration.
The statistical results related to firefly and particle swarm algorithms are provided in Table 1.With respect to the figures 1-8, it can be concluded that both particle swarm and firefly algorithms were successful in estimating the parameters of Chen-Lee system.However, there was a higher level of efficiency in the firefly algorithm, compared to the particle swarm algorithm.Figure 9 revealed the convergence process of the objective function for both firefly and particle swarm algorithms.With regard to this figure, it can be concluded that the cost function value reduced faster than in http://www.ispacs.com/journals/jsca/2018/jsca-00096/International Scientific Publications and Consulting Services the firefly algorithm, compared to the particle swarm algorithm.Therefore, firefly algorithm can converge faster towards overall optimization, compared to the particle swarm algorithm.Table 1 indicated the numerical results obtained for parameter estimation of Chen-Lee system by firefly and particle swarm algorithms.As shown in the table, the worst results obtained from the firefly algorithm were better than the best results obtained by the particle swarm algorithm.Also, mean of these results and the best results obtained from the firefly algorithm were more significant, compared to the particle swarm algorithm.These results demonstrated the high accuracy of the firefly algorithm in estimation of the parameters of the hyperchaotic system.Table 2 displays the objective function values for the best results obtained by the firefly algorithm and particle swarm optimization.

Conclusion
Metaheuristic algorithms could be used to identify the parameters of nonlinear hyperchaotic systems.In this paper, two firefly and particle swarm optimization algorithms were used to identify the hyperchaotic Chen-Lee system.According to the results of the current research, the firefly algorithm was more successful in estimating the parameters of the chaotic system and had high efficiency and accuracy and very low error.In addition, this algorithm has better accuracy and speed, compared to the particle swarm algorithm.

Figure 1 :
Figure 1: Parameter estimation for the chaotic system

Figure 4 :Figure 4
Figure 4: Estimation of the unknown parameter I 3 using firefly and particle swarm algorithms (I 3 =0.33) Figure 4 showed that I 3 estimated parameter obtained by FA was very close to the true value in all experiments.It also indicated that trajectories of the estimated parameter asymptotically converge to their actual value.Again, it is clear that FA converged much faster than PSO.

Figure 5 :
Figure 5: Estimation of the unknown parameter I 4 using firefly and particle swarm algorithms (I 4 =3.8)

Figure 9 :
Figure 9: Estimation of the unknown parameter I 8 using firefly and particle swarm algorithms (I 8 =0.05)

Table 1 :
Statistical results obtained for four-dimensional Chen-Lee system with the two applied algorithms

Table 2 :
Cost function obtained from particle swarm and firefly algorithms